Struve Fundamentals of Molecular Spectroscopy

7/15/2019 Struve Fundamentals of Molecular Spectroscopy

http://slidepdf.com/reader/full/struve-fundamentals-of-molecular-spectroscopy 1/395



FUNDAMENTALSOF MO LECULAR

SPECTROSCOPY

WALTER S. STRUVE

Department of Chemistry

Iowa State University

Ames, Iowa

WILEY

A WILEY-INTERSCIENCE PUBLICATION

JOHN WILEY & SONS

New York / Chichester / Brisbane / Toronto / Singapore



Copyright © 1989 by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of an y part of this workbeyond that perm itted by Section 10 7 or 108 of the1976 United States Copyright Act without the permissionof the copyright owner is unlawful. Requests fo r

permission or further information should be addressed tothe Permissions Department, John Wiley & Sons, Inc.

Library ofCongress Cataloging in Publication Data:

Struve, Walter S.Fundamentals of molecular spectroscopy.

"A Wiley—Interscience publication."Bibliography: p.

1 . Molecular spectroscopy.. Title.

QC454.M6S8798939'.68-5459ISBN 0 -471-85424-7

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1



To Helen and my family



PREFACE

This book grew out of lecture notes for a graduate-level molecular spectrosco pycourse that I developed at Iowa State University between 1974 and 1987. It isintended to fill a pressing need for a concise introduction to the spectroscopy ofatoms and molecules. I have tried to stress logical continuity through out, with a

view to developing readers' confidence in their physical intuition and problem-solving techniques. A suitable quantum mechanical background is furnished byth e first seven and a half chapters of P. W . Atkins' Molecular Quantum

Mechanics, 2d ed. (Oxford University Press, London, 1983): The Schriidingerequation for simple systems, angular momentum, the hydrogen atom, stationary

state perturbation theory, and the variational theo rem are all presumed in this

book. Group theory is used extensively from Chapter 3 on; it is not develop edhere, because many excellent texts are available on this subject. A one-semesterundergraduate course in electromagnetism is helpful but not strictly necessary:The concepts of vector and scalar potentials are introduced in Chapter 1 . Other

requisite material, such as time-dependent perturbation theory and secondquantization, is de veloped in the text.

Eight or nine of the eleven chapters in this book can be co mfortablyaccommodated within a one-semester course. The underlying time-depe ndentperturbation theory for molecule—radiation interactions is emphas ized early,revealing the hierarchies of multipole and multiphoton transitions that canoccur . Several of the chapters are introduced using illustrative s pectra from the

literature. This technique, extensively used by Herzberg in his classic series ofmonographs, avoids excessive abstraction before spectroscopic applications are

reached. Diatomic rotations and vibrations are introduced ex plicitly in the

context of the Born-Oppenheimer principle. Electronic band spectra are

examined with careful attention to electronic structure, angular mom entum

VI I



ViiiREFACE

coupling, and rotational fine structure. The treatment of polyatomic rotationshinges on a physically transparent demonstration of the com mutation rules formolecule-fixed and space-fixed angular momenta. From these, all of the energylevels an d selection rules that govern microwave sp ectroscopy are accessiblewithout recours e to detailed rotational eigenstates. The chapter on polyatomic

electronic spectra focuses on triatomic molecules and aromatic hydrocarbons—the former for their environmental and astrophys ical interest, and the latter fortheir illustrations of vibronic coupling and radiationless relaxation phenomena.Population inversion criteria, specific laser systems, and the principles ofultrahigh-resolution lasers and ultrasho rt pulse generation are outlined in a

chapter on lasers, which have emerged as a ubiquitous too l in spectroscopylaboratories. Som e of the higher order terms in the time-dependent perturbationexpansion are fleshed out for several multiphoton spectroscopies (Raman, two-photon absorption, second-harmonic generation, and CAR S) in the final tw ochapters. The reader is guided through the powerful diagrammatic perturbation

techniques in a discus sion designed to enable facile determination of transitionprobabilities for arbitrary multiphoton processes of the reader's choice.

I t is my great pleasure to acknow ledge the people who made this bookpossible. I am particularly indebted to my teach ers, Dudley H erschbach andRoy G ordon, who com municated to me the inherent beauty and cohesiveness ofmolecular quantum mechanics. The original suggestion for writing this bookcarne from Cheuk-Yiu Ng . David Hoffman made seminal contributions to thechapter on polyatomic rotations. I am g rateful to numerous anonymous refereesfor valuable suggestions for improving the manuscript, though, of course, theresponsibility for errors is s till mine. The line drawings w ere supplied by Linda

Emmerson, and S andra Bellefeuille, Klaus Ruedenberg, and Gregory Atchitygenerated the computer graphics . Finally, I must express deep appreciation tomy wife, Helen, whose moral support was essential to com pleting this work.

WALTER S. STRUVE

Ames , I owa

June 1988



CONTENTS

CHAPTER 1

CHAPTER 2

RADIATION-MATTER

INTERACTIONS 1 . 1 Classical Electrostatics of Molecules in

Electric Fields 1.2 Quantum Theory of Molecules in Static

Electric Fields 1.3 Classical Description of Molecules in

Time-Dependent Fields11.4 Time Dependent Perturbation Theory

of Radiation—Matter Interactions71.5 Selection Rules for One-Photon

Transitions2References8ATOMIC SPECTROSCOPY32.1 Hy drogenlike Spectra62.2 Spin—Orbit Coupling32.3 Structure of Many-Electron Atoms12.4 Angular Momentum Coupling in

Many-Electron Atoms82.5 Many-Electron Atoms: Selection Rules

and Spectra

22.6 The Zeeman Effect6References1ix



XONTENTS

CHAPTER 3 ROTATION AND VIBRATION IN

DIATOM ICS

3.1 The Born-Oppenheimer Principle

3.2 Diatomic Rotational Energy Levels andSpectroscopy

3.3 Vibrational Spectroscopy in Diatomics

3.4 Vibration—Rotation Spectra in

Diatomics

3.5 Centrifugal Distortion

3.6 The Anharmonic Oscillator

References

73

77

83

87

94

98

10 0

10 2

CHAPTER 4 ELECTRONIC STRUCTURE AND

SPECTRA IN DIATOMICS05

4.1 Symmetry and Electronic Structure in

Diatomics09

4.2 Correlation of Molecular States with

Separated-Atom States13

4 .3 L C AO — MO Wave Functions in

Diatomics21

4.4 Electronic Spectra in Diatomics364.5 Angular Momentum Coupling Cases41

4.6 Rotational Fine Structure in Electronic

Band Spectra46

4.7 Potential Energy Curves from

Electronic Band Spectra55

References61

CHAPTER 5 POLYATOMIC ROTATIONS65

5.1 Classical Hamiltonian and Symmetry

Classification of Rigid Rotors66

5.2 Rigid Rotor Angular Momenta70

5.3 Rigid Rotor States and Energy Levels73

5.4 Selection Rules for Pure Rotational

Transitions76

5.5 Microwave Spectroscopy of PolyatomicMolecules78

References80



CONTENTSiPOLYATOMIC VIBRATIONS836.1 Classical Treatment of Vibrations

in Polyatomics84

6.2 Normal Coordinates91

6.3 Internal Coordinates and the -

FG-Matrix Method94

6 .4 Symmetry Classification of Normal

Modes97

6.5 Selection Rules in Vibrational

Transitions09

6.6 Rotational Fine Structure of

Vibrational Bands136.7 Breakdown of the Normal Mode

Approximation16

References20

ELECTRONIC SPECTROSCOPY OFPOLYATOMIC M O L E C U L E S

25

7 .1 Triatomic Molecules267 .2 Aromatic Hydrocarbons

347 .3 Quantitative Theories of VibronicCoupling45

7 .4 R adiat ionless Relaxation in Isolated

Polyatomics

49References60

CHAPTER 6

CHAPTER 7

CHAPTER 8 SPECTRAL LIN ESHAPES AND

OSCILLATOR STRENGTHS678.1 Electric Dipole Correlation Functions678.2 Lifetime Broadening71

8.3 Doppler Broadening and Voigt Profiles73

8.4 Einstein Coefficients75

8.5 Oscillator Strengths77

References80

CHAPTER 9 LASERS839.1 Population Inversions and Lasing

Criteria84



XiONTENTS

9.2 The He/Ne and Dye Lasers879.3 Axial Mode Structure and Single Mode

Selection97

9.4 M ode-Locking and Ultrashort Laser

Pulses

01References

03

CHAPTER 10 TWO-PHOTON PROCESSES07

10.1 Theory of Two-Photon Processes09

10.2 Two-Photon Absorption13

10.3 Raman Spectroscopy21

References29

CHAPTER 11 NONLINEAR OPTICS

31

11.1 Diagrammatic Perturbation Theory34

11.2 Seco nd-Harmonic Generation38

11.3 Coherent Anti-Stokes Raman

Scattering41

References48

APPENDIX A FUNDAMENTAL CONSTANTS49APPENDIX B ENERGY CONVERSION

FACTORS51

APPENDIX C MULTI POLE EXPANSIONS OF

CHARGE DISTRIBUTIONS5 3

APPENDIX D BEER'S LAW57

APPENDIX E ADDITION OF TWO

ANGULAR MOMENTA59

APPENDIXF

GR OUP CHARACTER TABLES AND

DIRECT PRODUCTS63

APPENDIX G TRANSFORMATION BETWEEN

LABORATORY-FIXED AND

CENTER-OF-MASS COORDINATES

IN A DIATOMIC MOLECULE71

AUTHOR INDEX73

SUBJECT INDEX75



FUNDAMENTALS OF

MOLECULAR

SPECTROSCOPY



1RADIATION-MATTER

INTERACTIONS

In its broadest sense, spectroscopy is concerned with interactions betwee n l ight

and matter. Since light consists of electromagnetic waves, this chapter beginswith classical and quantum mechanical treatments of molecules subjected tos ta t ic ( t ime -inde pend en t) e lec t ric fields. Our discussion iden tifies the molecularproperties that control interactions with electric fields: the electr ic multipolemoments and the electr ic polarizability. Time-dependent electromagnetic wavesare then described classically using vector and scalar potentials for theassocia ted e lec t r ic and magnet ic fields E and B , and the classical Hamiltonian isobtained for a molecule in the pres ence of these potent ia ls . Quantum mechanica ltime-dependent perturbation theory is finally used to extract probabilities oftransitions between molecular states. This powerful formalism not only covers

the full array of multipole interactions that can cause spectroscopic transitions,bu t also reveals the hierarchies of multiphoton transitions that can occur. Thischapter thus establishes a framework fo r multiphoton spectroscopies (e.g.,Raman spect roscopy and coherent anti-Stokes R aman spect roscopy, which ared i s cus s ed in Chapters 10 and 1 1 ) as well as for the one-photon spectroscopiesthat are descr ibed in most of this book.

1.1 CLASSICAL ELECTROSTATICS O F MOLECULES IN

ELECTRIC FIELDS

Consider a molecule composed of N electr ic charges e n (electrons and nuclei)located at positions r „ referenced to an arbitrary origin in space . T he to tal

1



2ADIATION—MATTER INTERACTIONS

molecular charge is

q = E e „n = 1

and the elec tric dipole moment is

p = E er „

1.2)

n= 1

The latter expression reduces to a familiar expression for the dipole moment in aneutral "molecule” consisting of two point charges, e 1 = + Q and e 2 = Q (Fig.1.1). In this cas e , we have /1= + Qr +(— Q)r = Q ( r 1 — r 2 ) = Q R , which is theconventional expression for the dipole moment of a pair of opposite charges + Q

separated by the vector R . By convention, R points toward the positive charge.

For molecules characterized by e lectric charge distributions p(r) ins tead of pointcharges, the expressions for the molecular charge and dipole moment aresupersede d by

q =( r )dr1.3)

and

p =

p ( r ) d r

1.4)

where the integration volume encloses the entire charge distribution.

-Q

Figure 1.1 Dipole moment p = QR formed by charges ±0 separated by the vector

R.

*Numbers in brackets are citations of references at the end of the chapter.



CLASSICAL ELECTROSTATICS OF MOLECULES IN ELECTRIC FIELDS For a point charge e located a t posit ion r under an ex ternal electrostatic

potential Or), the energy of interaction with the potential is [1 ]*

W = e0(r)1 . 5 )

Or) can be e xpanded in a Taylor series about r = 0 (whose location is arbitrary)as

00004)(r) = 0 (0) + x — (0) + y —

ay(0) + z

O x

+ [x 8 ( 1::(0) +2: (0) + z 2 aa2z 4: (0 )

0 2 0

+ 2xy0) + 2xz a 2 4 ) (0) + 2yz —2 4 ) (0)]axayxazyaz

a 2 o n- = =V4)(0) + E x i x,•1 . 6 )zx i vx ;

where the components of r are expressed as either (x, y, z) or (x 1 , x 2 , x 3 ) . Sincethe electric field E(0) at the origin is related to 0 0 ) by

E(0) =— V4( 0)1 . 7 )

this implies that

1= 0 (0) — r • E(0) + -2 i x ; - -aS;0) + •

1E ;=r • E ( 0 ) —i x ;0) + • • •

T he interact ion ene rgy is then

a E •W=e(r) =e4(0)—er E(0) -4eE xx ;• • •

For a molecule consisting of N point charges e„ at locations r„, this become s

W =n ) 4 4 0 )r ) •E(0)

E (0 )—E (rn )i(rOi afrnxi + • • •

L nj1„r

q0(0) — p E ( 0 ) —t °Ei(°)

afroi±• • •(1.10)

( 1 . 8 )

( 1 . 9 )




(a ) (b )

(c )d )

Figure 1 .2 Examples of charge d istributions with (a) nonzero charge, (b ) nonzero

dipole moment, (c ) nonzero quadrupole moment, and (d) nonzero octupole

moment.

T he first two terms in W arise from the interaction of the molecular charge withthe scalar potential 4 ) and the inte raction of the molec ular dipole moment withthe electric field E, respectively. T he nex t terms in W are due to interactionsbetween the various electric field gradien ts D E i le(rn ) i and the correspondingcomponents [2 ]

07) = len [3(r„) (r n )i—r „ 6 i i ]1 . 11 )

of the electr ic quadrupole mome nt tens or . (Note that the term — r „ 6 i ; in V i l )drops out when Eq. 1 . 1 1 for the latter is used in Eq. 1 .10 , because V • E = 0 fo ran external field in free space.) He nce the e xpansion (1.10) illustrates how thevarious electric multipoles interact with an external field. Examples of chargedistributions exhibiting nonze ro dipole, quadrupole, and octupole moments areshown in Fig. 1 . 2 .

1.2 QUANTUM THEORY OF MOLECULES IN STATIC

ELECTRIC FIELDS

W e will be conce rned almos t exclusively with interactions between light andisolated molecules. The total Hamiltonian of the syste m is then given by

= flo + firad + w

1 .12)

1 1 0 is the Hamiltonian for the isolated molecule, "tad i s the Hamiltonian of theradiation field, and W is the interaction term describing the coupling of themolecular states to the radiation field. W e will den ote the isolated unperturbed

molecular states with IT„>; they obey the time-independent and time-dependent



QUANTUM THEORY OF MOLECULES IN STATIC ELECTRIC FIELDS Schriklinger equations

nolTn> = ET ) 1 1 1'.>

ih — = florli n >

O t

(1 .13)

(1 .14)

The unperturbed molecular states Ni t , > depend on both position (r) and t ime (t);s ince they are e igens ta tes of ilo , they can be factorized into spatial and t ime-dependent portions,

1 1 1 1 „(r, t)> = e - i . E " t / h i l 1 , , ( 0 >1 . 1 5 )

If , instead of (1.12), the total Hamiltonian for the molecule in the pres ence ofl ight we re II = R0 • grad, the e igens ta tes of the system would become simpleproducts of molecular and radiation field s tates 1 1 1 1 „(r, t)>lz r a d >, where theradiation field s ta tes IXrad> would depend on photon occupation numbers,energies and polarizations. Since no coupling of light with molecular states isimplied in such a Hamiltonian, no transitions can occur between moleculars ta tes due to absorption or emis s ion of light in this description. The s imples tHamiltonian that can account fo r spectroscopic transitions i s therefore the onein Eq . 1 . 1 2 .

Formally, the inclusion of the interaction term W requires that the Schrödin-

ge r equati ons (1 . 13 ) and (1 .14) be solved again af t er replacing 1 1 0 • gradwith thetotal Hamiltonian 1 . 1 for the molecule in the pre sence of light. A simplificationarises here because the interaction term W in Eq. 1.12 normally introduces onlya small perturbation to the isolated molecular Hamiltonian Ro . For example ,the elec tr ic field of bright sunlight is on the order of 5 V /cm. By comparison, anelec t ron spaced by a Bohr radius a o from the nucleus in a H a tom expe r ience s anelectric field of e 2 /4nc 0 ali, which is on the order of 5 x 10 9 V/ cm. Hence, W cangene rally be t rea ted as a perturbation to Radad • This approach proves to beuseful even for describing molecules subject to intense laser beams ; in such c a se s ,higher order perturbations assume unusual importance in comparison to the

situation of molecules exposed to classical light sources.For molecules subjected to static (time-independent) electromagnetic fields,the perturbed energies and e igens ta tes may be evaluated from stationary-stateperturbation theory [3]. The full Hamiltonian may be written in t e rms of aperturbation parameter (which may be s e t to unity at the end of thecalculation, after serving its usual purpose of keeping track of orders in theperturbation expansions for the energ i es and eigens tates ) as

(1 .16)

(firad has been dropped here because we are interested primarily in ho w theapplied field affects the molecular energy levels.) The time-independent eigen-




states j t / i n > , which we us e here instead ofecause we are dea ling with thestatic problem, and the eigenvalues E n are expanded as

W i n > = 1 t k T ) >> ± A 2 i t f r ( 2 ) >1 .17)

En

= E T)

+ A E V ) + )L 2 E2

+ • •

1 .18)Substituting Eqs. 1.16-1.18 into filifr n > = E n i t / i n > and using E q s . 1 . 1 3 and 1 . 1 4yields successive approximations to the perturbed energy

= E T ) + AE L 1 ) + )2 E 2 + •1 .19)

where

E T ) = < O T) 1 1 1 010T ) >

is the unperturbed ene rgy in state I I P T ) > and

o f r ( ol wi v o , ›

04° 1 wIlle> < O r ! w I t A ° ) >E( 2 ) = EnP ) — E r )

(1 .20)

(1 .21)

(1 .22)

are the leading terms in the energy corrections toAs an example of using stationary-state perturbation theory to compute the

perturbed en ergies of a molecule in a static electromagnetic field, consider an

uncharged molecule in a uniform (position-independ ent) static e lectric field E. Inthis case, the only nonvanishing term in the expansion (1 .10) is W = p • E(0).Substitution of this expression for the perturbation W in Eqs . 1 .21 and 1.22yields

E; i1 ) = — E • <On ° 1 1 1 1 t k V ) >

E E <1/4 ° ) I 0 3 ) > < O P ) 1 1 1 1 1 / 4 ° ) > • EnP ) — E r )

a) =

(1 .23)

(1 .24)

The first- and second-order corrections to the energy are linear and quadratic inthe electric field E , respec tively. It is interes ting to compare these re sults with theclassical en ergy of an uncharged molecule with permanent dipole moment P0 ina uniform, static E field: according to Eq. 1 . 10 , this would be

E = E0 — p o - E1 .25)

Here E0 , the classical energy of the molecule in the absence of the field, can beidentified with unperturbed energy E T ) in Eq. 1 .20 . Comparison of the termslinear in E in Eq s . 1 . 23 and 1.25 shows that the permanent dipole moment p o is



QUANTUM THEORY OF MOLECULES IN STATIC ELECTRIC FIELDS Table 1.1ermanent dipole moments of some

moleculesa

Molecule Po

H C 1 1 .03HBr 0 .7881 1 2 0 1.81C O 1 .2CH 3 C1 1 .9NO 2 0 . 39 9

°In units of debyes: 1 D=3.33564 x 10 -30C•m.

the expectation value of the instantaneous dipole moment operator p =1

Po = <IkT11 1 1 1 /4 ° ) >1 .26)

(Values of permanent dipole moments are given fo r several small molecules inTable 1.1.) H o w e v e r , the class ical energy 1.25 has no counterparts to the s e c o n d -and higher order terms in the perturbation expansion (1 . 1 9) fo r E.

This situation arises because the multipole expansion for W p e r se (Eq. 1 .10)has no provision for molecular polarization by the electric field. When an atom(or nonpolar molecule of sufficiently high symmetry) is subjected to an electr icfield E , the latter separates the centers of gravity of the spec ie s ' positive an dnegative charges, creating an induced dipole mo me n t p"which is parallel to E

(Fig. 1.3). For suff iciently small E , A n d is proportional to E, so that

Pind = OCE1 .27)

T he proportionality constant a i s def ined as the atomic (or molecular) polariza-bility. The total dipole moment of the polarized molecule in the electric field is

E= E0

Figure 1.3 Polarizable atom in the absence of an external electric field (left) and in

the presence of a uniform electric field E (right).




then

P = Po + Pind = PO aE1 .28)

and the classical energy becomes [4 ]

CE

E = Eo

• dE '

Jo

= E 0 — p o • E — locE E1 .29)

ins tead of Eq. 1.25. In molecules lacking special symmetry (e.g., CO2 , CH 3 OH)the induced moment /fin d does not generally point parallel to the external field E,because the electron cloud in such molecules is more eas ily distorted in certaindirections than in others (Fig. 1.4). In such cases, the polarizability is a t ensor

rather than a scalar quantity. The induced dipole moment is then given by

Pind =Œ E (1 .30)

with

(Xxxxy: X x z

O E = O C yy (1 .31)

O C ZXyPind,x

Œ ZZ

_ _

ccxxEx + ocxyEy + citxzEz_

Pind =ind,y

[

= ayxEx ± ayyEy ± ayzEz (1 .32)

Pind,z _ _ zc x E x + o c z y E y + o c z z E z _

Thus, in general, p i n d is no t parallel to E (i.e., II11r-ind,y, r ind,x 0 E y / E „ , etc.) unles s thee l e me n t s a u of the polarizability t ensor sa t i s fy special re la t ionships— as inmolecules of T, or O h symmetry.

Another property of the polarizability tensor is that it can always bediagonalized by a suitable choice of axes (x'y'z'):

C C =cXxzx'x' a4 0, , zYazz X z ' z '_

(1 .33)

This is analogous to the choice of principal axes (x'y'z') in calculating the threeprincipal moments of inertia of a rigid rotor (Chapter 5), and i t shows that only



QUANTUM THEORY OF MOLECULES IN STATIC ELECTRIC FIELDS nd

\

Figure 1.4 Polarization of an anisotropic molecule by an electric field. Since the

electronic charge distribution is more polarizable along the long axis than along the

short axis, the induced dipole moment pind is not parallel to E.

three of the components of a polarizability t ensor are independent . In moleculeswith sufficiently high symm etry, the principal polarizability axes coincide withsymmetry axe s of the molecule. In C O 2 , one o f these principal axes is the C o o

xis, while the other two may be any choice of orthogonal C2 axesperpendicular to the molecular axis. In a less sym me tric molecule like C H 3 O H ,the directions o f the principal polarizability axes must be evaluated numerically.

When the polarizability is a tenso r ra ther than a scalar, Eq. 1.29 for theclassical energy become s

E = E0—

p o • E —• E1.34)

Comparison of the terms quadratic in E in Eqs. 1 .19, 1 .22 , and 1.34 then reveals



< I P T ) 1 1 1 1 0 ( ° ) ><OrIPle>a = 2 E

.-10)(0)1* n (1 .35)

10ADIATION-MATTER INTERACTIONS

that the quantum mechanical ex pression for the molecular polarizability i s

If the spatial part 1 1 / 4 ° ) > of the unper turbed mo lecular wave funct ion i saccurately known in state n , the permanent dipole moment p o can be e valuatedfor that state using Eq. 1 .26 . H owever , Eq. 1.35 shows that an accurateknowledge of all the molecular e igens ta tes I t p r > with 1 n are normallyrequired in addition to 1 1 / 4 ° ) > to calculate the molecular polarizability in state n.In particular, the x y component of the polarizability tens or (which will benonvanishing only if the coordinate axes (xyz) are not chosen to coincide withthe principal molecular axes) will be

<44° ) I E e i x i l i P i ° ) > < 0 1 ° ) IEeiYile ) >ax y = 2 Li* n" — E T )

(1 .36)

where the index i is summe d over all of the molecular charges. Ins pection of Eq.1.35 or 1 .36 shows that large values of the polarizability tensor components arefavored by large <(4°)Ir1le> and by small ene rgy de nominators (Er — E n .For this reason, the alkali atoms , with their voluminous valence orbitals andclosely spaced energy levels, exhibit the larges t atomic polarizabilities in theperiodic table (Table 1.2), and such atoms have figured promine ntly in thedevelopment of non linea r optics. T hese express ions for molecular polariza-

bilities become useful in Chapter 1 0 , where Raman t ransi t ion probabilities arediscussed.

Table 1.2 Polarizabilities of several atoms'

Atom H. 6 6 6 7 9 3H e. 2 0 4 9 5 6

Li4 .31 . 10

0. 8 020 . 5 5 7

Ne.3946Na3 .6Ar.6 4

43.4C s9 .6

"In units of A 3 . Data taken from T. M. Miller and B. Bederson,

Adv. At. Mol. Phys. 1 3 : 1 (1977) .



CLASSICAL DESCRIPTION OF MOLECULES IN TIME-DEPENDENT FIELDS1The stationary-state perturbation theory used in this section is applicable

only to nondegenerate s ta tes 1 1 / 4° ) > ; degenerate perturbation theory mustothe rwi se be use d . The polarizability expressions developed here are only goodfo r time-independent (s tat ic) external fields. The polarizability turns out todepend on the f requency co of the applied field, s ince the elec t ronic motioncannot respond instantaneously to changes in E. Finally, since light containstime-dependent electromagnetic rather than static electric fields, the results ofthis section are not directly applicable to radiation-molecule interactions.

1.3 CLASSICAL DESCRIPTION OF MOLECULES IN

TIME-DEPENDENT FIELDS

In the classical electromagnetic theory of light, light in vacuum cons i s t s of

transverse electromagnetic waves that obey Maxwell's equations [1,2]

V E = 0

V • B = 0

V xE= —aBat

OEV B = pc ,80

(1 .37a)

(1 .37b)

(1 .37c)

(1 .37d)

where E o = 8.854 x 10- 12 c2/j m andu0

.= 1.257 x 106H/m are the electr icpermeability and magnetic susceptibility of free space .* Exam ple s of electr ic andmagnetic fields sa t i s fy ing Eqs. 1.37 are given by

E(r, t ) = E o e i( k • r -w t)1 .3 8 a )

B(r, t) = B o e l( k • r- ( D O1 .38b)

It is easy to show from Maxwell's equations that E 0 • k = B o • k =O (i.e., both

fields point normal to the direction k of propagation as required in a transverseelectromagnetic wave), that E0 • B 0 = 0, and that Eqs. 1.38 describe linearlypolarized light with its electric polarization parallel to E 0 as shown in Fig. 1.5.

Since V • B = 0 (a s magnetic monopoles do not ex is t) , B can be expressed int e rms of a vector potential A(r, t ) as [1, 2]

B=VxA1 .39)

*This book uses the International System of Units (abbreviated SI), an extension of the mks system.The units for length, mass, time, and current are meters , kilograms, seconds, and amperes,respectively. The equations in this and the following sections differ from the corresponding

equations writ ten in cgs units in that the factors of the speed of light c (Eoilo ) 1/2 = 2.998 x 10 8m / s ,hich appear in many of the cgs equations, are absent in the SI versions.




A

Figure 1 .5 Orientations of the vectors E 0 , B o ,

and k for the light wave described by E qs. 1.38.

This automatically satisfies V B = 0 in consequence of the vector iden tity

V (V x A) 01 .40 )

(The latter identity is apparent because V x A formally yields an other vector

that is normal to both V and A; hence a scalar product betwee n V and (V x A)invariably vanishe s.) From th e third of M axwe ll' s equations , we haveV x E = — a 1 3 / 0 t , which implies that

V xE= —V x

and so

V x(E+A). 01 .41 )

In the absence of magnetic fields, B and the vector potential A vanish, and theelectric f ield E is related to the scalar potential 0 by E = —V 4). Hence Eq. 1.41and the third Maxwel l equation are consistent with setting

because then

O AE= — V 0 —

e t

V xE= —V x (V0) — —

(V x A)

et— aBlat

(1 .42)

(1.43)

in view of the vector identity V x (V0) 0 for arbitrary s calar fields 0. Thisenables us to express the measurable E and B f ields in terms of a scalar potential

t ) and a vector potential A(r, t ) using Eqs . 1 .39 a n d 1 . 4 2 .T he vector and scalar potentials are not directly meas urable them se lves , since

their definition has an arbitrariness analogous to th e setting of standard states inthermod ynamic functions [2 , 5] . Suppose 0 and A are altered using some scalar



CLASSICAL DESCRIPTION OF MOLECULES IN TIME-DEPENDENT FIELDS3function X(r , t ) via

A' = A + VX

ax= at

(1.44a)

(1.44b)

Under this transformation, the magnetic field B=V x A becomes

B'=V x(A+VX)=V X A+V x(VX)

=B1 .45)

since V x (V X ) O. Similarly E = — V 4, — OA/at becomes

,E' = — V (4) — A) — —

at(A + V X )

OA= — V 4 ) +——Vg

a t

=E (1.46)

Thus, the physical E and B fields are both unaffected by this so-called gauge

t ransformat ion , and this gives us latitude to select algebraically convenientexpressions for 4 ) and A without affecting the measurable electromagnetic fields.It may be shown [5 ] that it is possible to choose the Coulomb gauge in

which V • A = O . This gauge is often used to describe electromagnetic waves infree space, where the scalar potential 4 , = O . The fields are then given by

O AE = —

a t1 .47a)

B=Vx A

1.47b)

It now remains to formulate the classical Hamiltonian for a charged particlesubjected to potentials 4, and A (or, equivalently, their associated fields E and B ).

Nonrela t ivis t ic classical mechanics is based on Newton's law of motion

F = mr1.48)

for a particle of mass m subjected to an external force F. The kinetic energy of

the particle is

T = J- m v 21.49)




and (for conservative forces) the force is derivable from a potential V by

F= —V V

Since for each Cartesian compone nt i of the particle velocity y

(1.50)

and

aT—= mv iav i

(1.51)

d (OT\ d„

dt avi)niv i ) = m r=ava x , (1.52)

it follows that

d (aT) aV+=0

d t av ix i

If one forms the Lagrangian function [6 ]

L T — V

Eq. 1.53 assumes the form of the Lagrangian equations

dLdt av i ) ax i

(1.53)

(1.54)

(1.55)

provided the external force F is cons ervative (i.e., the potential function V i sindependent of the particle velocity y ) . It may be shown that if one chooses anyconvenient set of 3 N generalized coordinates q • fo r an N-particle system, so thatfor eac h particle n the Cartesian components of its position are expressible in theform

xnn(ql, (12 1 • • • (13N)

Ynn(ql, q25 • • • q3N)

Zn = Zn(ql, q2, • • • q3N)

(1.56)

the transformation from C artes ian to gene ralized coordinates yields the morepowerful generalized Lagrangian equations

(1.57)



CLASSICAL DESCRIPTION OF MOLECULES IN TIME-DEPENDENT FIELDS5In an N-particle system with r constraints, (3N — r) of the generalized coor-dinates wil l be indepen den t , and there will be (3N — r) independent equations of

the form (1.57). The Lagrangian is treated as a function of the conjugate

variables q • and 4 i .Different ia t ing the Lagrangian function with respect to v i in Cartesian

coordinates gives (for conservative forces)

01, OT=v i =pi

ov iv i(1.58)

which is the ith component of linear momentum. In generalized coordinates, thesame procedure

(1.59)

yields the component of generalized momentum conjugate to the general izedcoordinate q • . Different ia t ion of L with respect to x i in Cartesian coordinatesyields

av— = F. =p•ax ,x, " (1.60)

which expresses the conservation law that if the Lagrangian i s independ en t o f x i ,the linear momentum component pi is a constant of the motion. The analogousequation

aL .4 ; —Pi

can also be shown to hold for general ized momenta.The classical Hamiltonian is now defined as [6 ]

3N -r

H = E p i 4 i — L

(1.61)

H(p i , q i )1.62)

and is handled as a funct ion of the conjugate variables q i , pi . It is readily shownthat the classical Hamiltonian gives the total ene rgy for a single particleexperiencing conservative forces in Cartesian coordinates:

3 = E— E Imsq +V=T+V (1.63)



1 6ADIATION-MATTER INTERACTIONS

A similar result can be obtained for the Hamiltonian (1.62) in generalizedcoordinates.

The Lorentz force F expe r i enced by a charge e of mass m in the pr esen ce ofelectr ic and magnetic fields E and B i s given in SI units by [1, 2, 5]

dF = e[E +V x B] = (niv) (1 .64)

This nonco nse rvative (v — dependent) force is not derivable from a potentialene rgy V in the manner of Eq. 1 .50 . If a Lagrangian funct ion can never theless befound that obeys Eq. 1.57, the formulation of the Hamiltonian using Eq. 1.62will still be valid. Re cas t ing Eq. 1 . 64 in t e rms of the vector and scalar potentials(Eqs. 1 .39 and 1.42), we obtain

aAF = e [ — V(/)

v x (V x

at

This expression for the force can be simplified with the identi t ies

dA aA+(v • V )A

d tt (1 .65)

(1 .66)

x (V x A) V ( v • A) — (v • V)A1 . 67 )

to give

F = e[—V(4) v A) — —A] d

dt = (mv)(1 .68)

W e are n o w in a position to show that the Lagrangian for th is sys tem happens tobe

L =-1-mv 2 + e(v A) — e1 . 69 )

Substitution of (1 .69) into the Lagrangian equations (1.57), using the particleCartesian coordinates for the q i , yields

—

( i n v . + e A . ) — e a ( v • A — 4 ) ) = 0d tx i

(1 .70)

since the external vector potential A = A(r, t ) depends only on position andtime. This res ult is in fac t jus t Eq. 1 .68 in component form, which confirms thatthe

sys t em Lagrang ian i s cor rec t ly g iven by Eq. 1 . 69 . Accord ing to Eq. 1 . 62 , the



TIME-DEPENDENT PERTURBATION THEORY O F RADIATION-MATTER INTERACTIONS7system Hamiltonian becomes

3

H =E p i 4 i — L

= 1

3

i 1

C E ) . Ei - MV- (v A) + e d 9=5Ci

= MV2 e(v •A) — imv 2 — e ( v • A ) + e i c k

1= Imv 2 + e4) = —

2m(mv) 2 + eck (1 .71)

The Hamiltonian is conventionally written in t e rms of the position coordinatesq • (x i if Cartesian) and their conjugate momenta pi . From Eq. 1.58, the latter are

Lmi =k + eA i

ex i(1 .72)

or p = mv + eA . Substitution of (p — eA) fo r mv in Eq. 1.71 then produces theclassical Hamiltonian for a charged particle in an electromagnetic field as afunct ion of the conjugate Cartesian variables r and p,

H= --p_ e A ) 2 + eq2m

(1 .73)

(the r — d e p e n d e n c e in H stems from the r — dependence in the scalar andvector potentials). Physically, the conjugate momentum p does not equal mvbecause the particle's linear momentum in the electromagnetic field i s inf luence dby the vector potential A. This Hamiltonian is straightforwardly modified if, inaddition to the ex t e rna l fields we have just treated, the particle experiences aconservative potential V(r) (e .g. , that aris ing from electrostat ic interactions withother charges in a molecule). The correct Hamiltonian in this case is given by[7,8]

H =2 7 m

1 .74)

1.4 TIME-DEPENDENT PERTURBATION THEORY OF

RADIATION—MATTER INTERACTIONS

The quantum mechanical Hamiltonian operator corresponding to the class icalH amil tonian (1 .74 ) i s

1 hV V2m i

— eA) + e( t) + (1 .75)




which may be e xpande d into

-=-- —2m

[ - h 2 V 2 -7( V • A )2he

( A • V ) + e 2 A 2 1

+e4 + V1 .7 6 )

T he parenthese s surrounding the quantity V A in Eq. 1.76 indicate that Voperates only on the vector potential A immed iately following i t . The choice ofCoulomb gauge ( V • A = 0) fo r electromagnetic waves propagating in free space(0 = 0) reduces the H amiltonian to

h22 A 2 he=[-—

2mV 2 + V(r )1 + 1—A • V)1 11 0 + W (1.77)

2m im

where the terms have now been grouped in the form of Eq. 1 . 1 6 . fio

representsthe zero-order H amiltonian for the particle unpe rturbed by the external fields,and the terms arising from the radiat ion-matter interaction have been isolatedin the pe rturbation W For particles bound in atoms or molecules e xperiencingordinary electromagnetic waves, the internal electric fields due to V ( r ) are ordersof ma gnitude larger than the external fields, with the consequence that eA « p .T he quadratic term e 2 A 2 in W then become s negligible ne xt to e h A - V , with theres ult that the pe rturbation is we ll approximated by the linear term,

ie hW = —

m

A- V (1.78)

As a concrete example, the vector potential for a linearly polarized mon o-chromatic electromagnetic plane wave with wave vector k may be written

A(r, t) = R e ( A o e i( k • r -w t)

= A o cos(k • r - w t)1.79)

where k and the circular frequency co are related to the wavelength A andfrequency v by

lki = 27r11 . 80a)

= 27tv1 .80b)

Since 0 = 0, the electric field i s

E(r, t) = - —A

= --(DA ° sin(k - wt)e t

= -clklA 0 sin(k r - w t)1 .8 1 )



T1ME-DEPENDENT PERTURBATION THEORY OF RADIATION-MATTER INTERACTIONS9so that th e electric field points antiparallel to A. The magnetic field associatedwith the light wave is

B(r, t) =V x A= —k x A o sin(k r — cot)1 .82)

Another way o f writing Eq. 1.82 is

B(r, t ) =

tjiI

a a aex ay az

A z Ay Az

(1 .83)

Comparison of Eqs . 1 .81 and 1 .82 shows that E and B are mutually orthogonal,and the latter e quation requires that B is orthogonal to th e wave's propagationvector k. Hence th e vector potential in Eq. 1.79 describes a line arly polarizedtransverse electromagnetic wave. T he wave propagates at th e speed o f light c ,

because E and B are both functions of (k • r — c o t ) r — 01 ) =- 11(1r — ct ) according to Eq s . 1 . 80 .Since th e vector potential A in Eq. 1.79 depen ds e xplicitly on time, the

perturbation W = ( i eh/m)A- V is time-dependent as well. A perturbation theorybased on the time-dependent Schriklinger equation (1 .14) must therefore be usedto des cribe th e radiation— matter coupling. The Hamiltonian is assumed to haveth e form of Eq. 1.77, ex cept that the perturbation W(t ) is now explicitlyacknowledged to depen d on time. The molecule has zero-order e igens ta tesI t i / („Nr , t ) > obeying Eqs . 1 . 1 3 through 1 .15 . It is ass ume d initially (at t = --cc)

that th e molecule is in state 1 k > le>. We then turn on the perturbation W(t),which can cause the molecule to undergo a transition to some other s ta te

1 t , /4 ) > because of its interaction with the radiation field. W e wish tocalculate th e probability that the molecule ends up in state im> by some latertime t .

In general, the state of th e interacting molecule—radiation system 1 1 1 1 (r , t ) > willnot coincide wi th o n e of th e zero-order s ta tes 1 '1/Jr,

0 > i X r a d > ,because th e

Schrödinger equation is modified by the presence of the coupling term W(t). (Inwhat follows, we will drop ixr a d > from our discuss ion, since including it couldonly tell us how many photons of each type (ene rgy, polarization, etc.) will beabsorbed or emitted in a given transition, and we have other ways of obtainingthis information. By focusing on the molecular states I t l i n fr , , we gain the farmore interesting information about what happen s in the molecule.) If we have acomplete orthonormal set of zeroth-order (i.e., isolated-molecule) e igens ta tesr1 1 „(r, t ) > of fi o , the mixed state itli(r, t ) > can always be e xpresse d as

itP(r,E cn(t)IT,,(r , t)> c(t) exp( — iE;,° ) t /h) I n >1 .84)




by a suitable choice of coefficients c„(t). T he c(t) are assume d to be normalizedand to obey the initial conditions

c k ( — co) = 1 }co ---- b n k

c n *k ( — c o )=n(— )0

(1 .85)

l c , ( t ) 1 2 = 11 .8 6 )

Equation 1.85 states that 1 1 1 1 (r,o)> = ex p(— ia) t /h)1k>; i . e . , th e molecule isinitially in state 1 k > with ene rgy Ek Er. The expansion (1.84) can besubstituted into th e time-dependent Schrödinger equation to give

/h[fi + W(t)] E c n (t)e — i E " t / h i n > = ih —a E n ( t ) e —iktin>

a t n(1.87)

Using th e fact that R o ln> = kin>and multiplying on the left by the bra <ml, wehave

E„t /h in > acn e _ o h l n ><ml E c n ( t)W(t)e -i

' n at

ih Ex< m i n >a t

= ihe-iE m t lh a C ma t (1.88)

The latter follows since <On> = 6„ ,„in an orthonormal set, so that only th e termwith n - - = m survives in the -summation. He nce , th e time-dependent coefficientsobey th e coupled equations

dcm=

d th E C „ ( t ) e t / h < M I W(t)In>n

(1.89)

This expression i s exact. W e can now introduce the spirit of the perturbationtheory by assum ing that the transition probability from the initial state lk> tosome other state 1 m > is sma ll. This would imply that c(t) 6k n at all times t, notjust at t — co. He nce, as a zeroth approximation, we take [9]

c(t) = 11 .90)

c („°) k (t ) =01 .91)

a n d , using w n „ , to denote (E n — E m )lh and substituting Eq. 1.90 into th e right side



e-iw k . t 2< n l W(t2)I k > d t 2

TIME-DEPENDENT PERTURBATION THEORY OF RADIATION—MATTER INTERACTIONS1of Eq. 1 .89 , we can ge t an ex press ion for the next order of approximation to c, n (t):

dew 1m— E b n k e i œ n in t <ml W(t ) In>

dth „1

—ih

e -iwk m t <mIW(t)lk> (1 .92)

From this, considering the initial condition that C m (t) c ( n ) ) (t ) as t co (andhence that c(t) -+ 0 for m k in this limit), we can integrate Eq. 1 .92 to obtain

1 i t( 1 )c m (t) = —ihk - t '<miW( t i ) l k>d t i (1 .93)

For a second approximation to c, n (t) , we can place a(t) into the right side of E q.1.89 to obtain

dc!!dt

)=I c(i)o)e-im—`<miW(t)In>

ih „

1= -ja4 m t <m1W(t) In>-i"-"<ni147 0 1 ) 1 k > d t i1 . 9 4 )

c o

Integrating this with th e proper initial condition leads to

1tc(t) = 2 E- i--"<miw(toin>citi( h) j'tix (t2)1k>dt2-

Iterat ion o f this process will show that [9]

C m (t ) = c(t) + C(t) + C(t) + • • •

jk m ± — f el a h m t 1 <m 1 W(t i )lk>dt,ih _ o

+ ( i h ) 2 En

t

etiM W ( t 1 ) I n > d t 11

—

1

▪

( • h ) 3 nn'

e i w n ' t2 < n 1 W i n ' > d t2-c o

f 2

x-i"- 3<ti l W I k > d t 3

(1 .95)

( 1 . 9 6 )




In the absence of an y perturbation W(t ) , c m (t ) is given by c („ ? ) = b k „ „ and notransitions can occur from the initial state 1 k > . T he next term c2 1 ( t ) correspondsto one-photon pro ce s s e s (absorption an d emission of single photons), and coversmost of classical spectroscopy. The two-photon processes (two-photon ab-

sorption and Raman spectroscopy) are contained in the second-order termc („P(t) , the three-photon processes (e.g., second-harmonic generation and three-photon absorption) correspond to c2 ) (t), and s o on. We will concentrate on theconsequences of the first-order (one-photon) term e(t) in the next few chapters.Higher order terms like c(t) and 4, ) (t ) require intense electromagnetic fields(i.e., lasers) to gain importance, and indeed the practicality of Raman spec-troscopy bloomed dramatically with the advent of lasers.

Under the normalization and initial conditions (1.85) and (1.86), th e .proba-bility that the molecule has reached state 1 m > at time t is equal to km(t)l2. In firstorder, c m ( t ) is given by

1c2 - ) (t) = —

ih j-t1<m1W(ti)lk>dt, (1.97)

and so we must have <m[W(t)ik> 0 0 for an allowed k mone-photontransition. The transition is otherwise said to be f orb i dden . To calculatemolecular transition probabilities more concretely and to derive generalselection rules fo r allowed t ransi t ions, we need only to substitute specificexpressions fo r W(t) .

1.5 SELECTION RULES FOR ONE-PHOTON TRANSITIONS

Heuristic selection rules for on e-photon transitions may be obtained by usingE q. 1 .9 or 1 . 10 for the pe rturbation Win the e xpression for c;»(t), E q. 1.97. Thisprocedure yields the matrix element

OniwIk> = <mleOlk> — <Op Elk>

OE ;— <ml — e E x i x ,ik> + • • •

ii

1E ;= 0 — E E em ix i xiik> + • • •

2(1.98)

which controls the probability of transitions from state k to state m. The f i rs tterm vanishes (<mlk> = 0 ) due to orthogonality between eigens tates of no

igenvalues. The second term results from interaction ofthe instantaneous molecular dipole mo men t with the external electric field E ,and leads to electric dipole (El) transitions from state k to state m. T he third termarises from in te ract ion of the instantaneous molecular quadrupole mo men t

tensor with the electric field gradients aEi/ax i ; it is responsible fo r electricquadrupole (E2) transitions from state k to state m . Our qualification that it is theinstantaneous (rather than permanent) moments that are critical here is



SELECTION RULES FOR ONE-PHOTON TRANSITIONS3/Ix

zY

Figure1.6 Orientations of the E and B fields associated with the linearly polarized

light wave described by Eqs. 1.99. The E field, directed along the x axis, interacts

with the x component of the molecule's instantaneous electric dipole moment; the B

field, directed along the y axis, interacts with the y component of the molecule'sinstantaneous magnetic dipole moment.

important, since, fo r example, El transitions can occur in atoms (e.g., in Na and

Hg lamps) even though no atom has any nonvanishing permanent dipolemoment p o . The foregoing discussion can be summ arized in the followingselection rules:

For allowe d El transitions, Onlpik> 0 0For allowed E 2 transitions, <mix i xi lk> 0 0 for some (i , f)

W hile this discussion based on electrostatics ignores the t ime depende nce inW(t) and omits the effects of magnetic fields ass ociated with the light wave, i tdoes anticipate some of our f inal results in this section regarding e lectric dipole

and electric quad rupole contributions to the matrix e lements <nil W(t)lk>. Ityields no insight into magnetic multipole transitions or into the nature of thetime-ordered inte grals in the Dyson ser ies expansion of Eq. 1 .96 .

Next we calculate the matrix e leme nts us ing the correct time-dependentperturbation W = ( iehlm )(A • V), Eq. 1.78. W e assume for clarity that the vectorpotential is that for a line arly polarized plane wave (Eq. 1.79) with A o = A o randk =114 This vector potential points along the x axis and propagates a long thez axis (Fig. 1.6); results for the more general case are given at the end of thisdiscuss ion . Following Eqs. 1.81 and 1.82 , the electric and magnetic fieldscorres ponding to this vector poten tial are

E(r, t) = -41A rsin(kz - cot)1 .99a)

B(r, = -fiklA o sin(kz - cot)1.99b)

so that the E and B fields point along the negative x and y axes, respectively. The

matrix e lement Ortl W(Olk> becomes

ih e<m lW( t )i k> =niAoe i(k•r-cat), v i k >

the A o e -'•m le i k Vik>

iheA o e' • <ml (1 + ik-r +

(ik • r) 2=—

2• • •) V1 1.0 ( 1 . 1 00 )




The quantity k • r is equivalent to j k l I l l co s 0 = 2nIricos 0/A, where 0 is the angleformed between the vectors k and r. The matrix elements < m l W(t) lk> limit I ri tothe molecular dimensions over which the wave functions 1 k > and < m l areappreciable, i .e. , In 10A in typical cases. The shortest wavelengths A us ed inmolecular spectroscopy are on the order of 1 0 3 A for vacuum-ultraviolet light,and are of course much longer for visible, IR , and microwave spectroscopy.H e n c e k-r is typically much less than 1 , and the se r ies expansion of exp(ik r )

converges rapidly. In the special geometry we have a s s u m e d for our vectorpotential,

ih e< m 1 W ( 0 1 k > =o e -k" <mli — lk>

e x

ihe +o e - "<mkikz) — 1k>ax

t h e ikz) 2 a+ —m

Aoe' t <rn1 1k> + • •2 ax

(1 .101)

The f irs t term in < m l W ( t ) li c > requires the matrix element <mla/axik>. This can beobtained by evaluating the commutator

E flo ,1 r

= [P2 /27/1 TAX), x] yi LP2

> x]

1 x_r n _ 2—

2m

LPx, x] —

2 m

[ p r , x]x , x]kx

ih— — P x Hx — x i -1 0

rn(1.10

where we have u s e d the commutator identity [ A B , C] = A[B, C] + [ A , C ]B[3]. Then

and s o

a iiI1, ,__ _ m 37 i. . .h 2 px — — 4 2 (H x — x i? )

a x h x

a<m lk> = — 1 1 1 < mitiox — xfi o lk >a x2

= — —h 2

< M I E" '

X XEklk>

= — p - (E n , — E k)<mlx1k>

(1 .103)

MWmk

< m 1 x 1 . 1 s > ,h (1.04)



SELECTION RULES FOR ONE-PHOTON TRANSITIONS5The second t e rm in < 1 W(t)1k> requires

0 oniz—x 1 k >1k> +1<miz —x + x -s1 k >

hah <m la zx - — lk> + 1 < mlz — + x — 1k>

axa axz

i 2h OnlLy ik> + (m l — + x — 1 k >axz (1.105)

since the y component of the orbital (not spin or total) angular momentum isL y = zi)x - xfr z . The las t matr ix e leme nt on the right side above can be obtainedusing the commutator

[fio, xz]1=132

2mxz]

T h e n

+xz]- 2m

- 2 1 3]z + -2m

2 /5zxu

z, z]

2m

- ——

ih(pz + xpz)

h 2 7 a + x az )— m ax

(1.106)

0<rniz

a x >m lfioxz - x zfi o l k >

= — (Ek - En)<mlxzlk>h 2

""km

h Onlxzlk> (1.107)

Collecting these results for the first and second t e rms in < m 1 W ( t ) 1 k > , wesummarize that

< i n1 W ( O l k> = + i ec o ,„ A o e'<mlxik>

ik eeco k mA o e't(m1Cy lk> +o e'<mixzik>

2m ih k 2eA o e -i f<rnIz 2k> ±•2mx (1.108)




For a single particle, the electric dipole operator is p = er; the first term o nthe right side of Eq. 1 . 1 0 8 is therefore

iwk.Aoe i '<mip ik>

and i t represe nts the electric dipole (El) contribution to the total transitionprobability. Only the x component of p appears here, because in our exam plethe E field of the light wave has o nly an x component (Fig. 1.6), and the electricdipole interaction behaves as p • E. The second term in Eq. 1 . 108 can be recast in

terms of the magnetic dipole m o m e n t operator in SI units

p m = eL/2rn1 . 1 0 9 )

(the orbital angular mome nt L of a moving charged particle physically gives riseto a proportional magne tic dipole moment p m in the same direction as L for

e > 0 ), and so i t become s

—ikA0e -i"<m1(u . ) y i k >

This corresponds to the magne tic dipole ( M 1 ) contribution. The energy of amagnet ic dipole mo me nt p. in a uniform magnetic field B is • B , and themagnetic field of our current problem is d irected along the y axis (Fig. 1 .6)—which is why only the y component of p m appears in this term. The third term in

Eq. 1 . 1 0 8 embodies the elec tric quadrupole (E2) xz compone nt, which is the onlycontributing electric quadrupole tensor component since E in our exam ple hasonly an x component that spatially depends only on z (all of the other aE i lax i

are zero). The succee ding terms not shown in Eq. 1 . 108 describe higher order(electric octupole, magne tic quadrupole, e tc.) transitions; their importancedecre ase s rapidly with increasing order because of the increa singly high powersin (kz). Since the E 2 and M 1 transition amplitudes contain a factor of kz that isabsent in the El term of <m l W(t)110, they are inherent ly much weake rtransitions than elec tric dipole transitions. The vast m ajority of one-photonspectroscopic transitions that are exploited in practice are El transitions.

W e now give without proof the matrix e lement of W(t) for the more generalcase of an incident plane wave A(r, t ) = A o exp(ik - r — cot) in which k and A o

point in arbitrary d irections (A 0 must be normal to k to give a physically rea llight wave in vacuum, however):

(m1W (t)11 0 =th ew k m <mj(I t o • r)lk>t 1

— — < m i L lik> +imcok .

< m l ( k • OA ° ' 1 0 1 k > + • • .12h e -"C (1 .110)



SELECTION RULES FOR ONE-PHOTON TRANSITIONS7in which L 1 denotes th e component of orbital angular mome ntum about an axisnormal to both A o and k. These terms in order correspond to the El, Ml, andE2 k m transition amplitudes.

From Eq. 1 .96 , th e k mone-photon transit ion probability becom es

e -iwk r n t1 <m1W(t1)lk>dt11 2

2

40 ) k m +0 ) t id t i

meaning that the transition probability is pro portiona l to the absolute valuesquared of the weighted sum of matrix eleme nts in Eq. 1 . 1 1 0 . T o s e e th e

significance of the time integral, we may take the limit as t --+ + oo (thecontinuous-wave limit) to get

4 n2 10 2

Ic 2 ) ( 0 1 2 =26 (wk. + co)] 2

since th e integral representation of the Dirac delta function is

1(x) = Ye't dt1 . 1 1 3 )

Hence, th e k m transition in this limit cannot occur unless= c o b ? , = + o ) „ ,k = (E. — E,3/h. T he frequency in the external radiation field

mus t exactly match the energy level difference between the initial and finalstates, in accordance with th e Ritz combination principle. Thus, th e timeintegral leads to an "energy-conserving" delta function. This energy-matchingcondi t ion should not be take n too s er iously at this point, because in fact th eenergies E. and Ek themselves are not sharply de fined in gene ral owing tolifetime broadening [10] (sometimes referred to as the "time-energy uncertainty

principle"). Rather, th e time integral in Eq. 1 . 1 1 1 expresses the co-dependence ofthe transition probability in the idealized case when th e energies of the two levelsare infinitely sharp. It is interes ting to note that if the upper integration limit t i s

se t to som e f in ite num ber ra ther than + oo , th e function 6(co k ,„ + co) will bereplaced by s ome funct ion g(co) with a fini te , rather than zero, w idth. Thiscor re sponds to the fact that a light wave with les s than infinite len gth has s omeuncertainty in its frequency co, so that its center (o r most probable) frequencycan be detuned from (E. — E k )/h and still have some finite probability ofeffecting th e molecular transition.

T he se lection rules we have derived in this sec t ion form th e basis for all of the

one-photon spectroscopies treated in Chapters 2 through 7 of this book. Theymay be succinctly summ arized as follows for general one-photon transitions




from state k to state m:

Electric dipole (El):

Magnetic dipole (M1) :

Electric quadrupole (E2):

<mlillk> 0 0

<nalk> 0 0

Onlxixilk> 0 0

fo r s o m e (i, j)

Obtaining the se lec t ion rules for two-photon an d higher order multiphotonprocesses requires analysis of the expansion coefficients 4, ) (t), c(t), . in E q.1 . 9 6 . This is don e e xplicit ly for two-photon proces s es in Chapter 1 0 , where two-photon absorption and Raman spectroscopy are di scu ssed . This formal i smbecomes increasingly unwieldy when applied to three- and four-photon pro-cesse s , and diagrammatic techniques then become useful fo r organizing the

calculation of the pertinen t transit ion probabilities (Chapter 11) .

REFERENCES

1 . M . H . Nayfeh and M. K. Brusse l , Electricity an d Magnetism, Wil e y , New York, 1985.

2 . J. D. Jackson, Classical Electrodynamics, Wil e y , New York, 1962.

3 . E. Merzbacher , Q ua n tum Mechanics, Wiley, New York, 1 961 .

4 . K. S . Pitzer , Q ua n tum Chemistry, Prentice-Hall, Englewood Cliffs, NJ, 1953 .

5 . W. K. H . Panofsky and M. Phillips, Classical Electricity an d Magnetism, 2d ed. ,Addison-Wesley, Reading, MA, 1962 .

6 . J. B. Marion, Classical Dynamics of Particles an d Systems, Ac a dem ic , New York,1965.

7 . R. H . Dicke and J. P . Wittke , Introduction to Q ua n tum Mechanics, Addison-Wesley,Reading, MA, 1 9 6 0 .

8 . W . H . Flygare, Molecular Structure and Dynamics, Prentice-Hall, Englewood Cliffs,NJ, 1978.

9 . A. S. Davydov, Q ua n tum Mechanics, NE0 Press, Peaks Island, ME, 1 966 .

1 0 . P. W. Atkins, Molecular Q u an t u m Mechanics, 2d ed. , Oxford Univ . Press, London,

1983 .

PROBLEMS

1 . For the electric and magnetic fields given in Eqs . 1 .38 , show that Maxwell'sequations in vacuum (E q s . 1 . 3 7) require that E0 B = E 0 • k = B o • k = 0.

2 . A vector potential is given by A(r, t) = A o ( f + fc ')cos(k r — c o t ) , in which thewave vector k = 41. Compute E(r , t ) and B(r, t) in the Co ulomb gauge, and

show that these fields obey Maxwell's equations in vacuum.



PROBLEMS 29

3 . T he evaluation of ground-state atomic or molecular polarizabilities usingEq. 1.35 requires accurate k nowledge of all of the molecular eigenstates inprinciple. This proves to be unneces sary in the hydrogen atom ( A . Dalgarno andJ . T. Lewis, Proc. R. Soc. London, Ser. A 233: 70 (1955); E. Merzbacher , QuantumMechanics , Wiley, New York, 1961) , where the second-order perturbation sum(1.35) can be e valuated ex actly. In this problem, we evaluate az z for the is groundstate 1 0 >

< 0 1 z 1 / > < 1 1 z 1 0 >az z 2e E

4i E 1 — E 0

in which I i> den otes an exci ted s tate in hydrogen and the summation i sevaluated over all such states .

(a ) V erify by substitution that the function

pao (rF

h 2 2+ ao ) z

satisfies the comm utation relation

z10> = (FIL — ILF)10>

Here t t and ao are the hydrogen atom reduced mass an d B ohr radius, andIL is the hydrogen atom H amiltonian.

(b ) Show that this res ult lead s to

pt a0 e 2r= h2 (01 — + a o ) z 2 1 0 >2

so that no information about excited s tates Il> with 1 0 i s required tocompute the polarizability in the hydrogen atom.

(c ) Compute a3 . Compare this value with the polarizabilities o f H e(0.205 A') and Li (24.3 A 2 ) and discuss the differences.

4 . An electromagnetic wave with vector poten tial

A(r, t) A o (f+ cos(kx — cot)

i s incident on a i s hydrogen atom.

(a ) Calculate the E and B fields, assuming 0 ( r) = 0 .

(b ) W rite down all of the nonvanishing terms in the matrix eleme nts <1s1WI2s>,< 1 , 5 1 W 1 2 p x >, <1s1W12p y > , and < 1 s 1 W 1 3 d „ z > fo r this vector pote ntial up to




first order in ( k • r ) in Eq. 1 . 1 0 0 . [I n terms of the hydrogen atom stationarystates I t l f , , i , n (r)>, Ils> is 1 4 / 1 00 ( r)> , 12 s> is 1 0 20 0 ( r)> , 12 p x > is the linearcombination 2 -1 / 2 ( — 1 0 2 1 1 > + 1021, - 1 ›), etc. Use elementary symmetryarguments to determine which of the matrix elem en ts will vanish.]

(c) For this particular vector poten tial, which of the transit ions is - - - * 2 s ,is -+ 2p x , is -+ 2p y , an d is 3d El-allowed by symmetry? E 2 -allowed? Ml-allowed?

5. Combine M axwell's equations in vacuum with Eqs. 1.39 and 1 .42 togenerate the homogeneo us wave equat ion for the vector poten t ial in theCoulomb gauge,

32

(v2 — po e o(r , t) = 0

Show that the most general solution to this wave equation is

A(r, t) =f(k - r — wt)

where f is any function of the argumen t (k • r — cot) having first and secondde rivatives with respect to r and t. W hat physical significance does this functionhave in general?

After expanding the exponential in the matrix element <mlexp(ikz)(0/0x)Ik>in Eq. 1 . 1 0 1 , we de mons trated that the first-order term <ml(ikz)(0/0x)1k> breaksdown into a sum of con tributions proportional to (miLy lk> and <mlxz1k>. Theseaccount for the M 1 and E2 transit ion probabilities, respectively. Reduce thesecon d-order term <ml(ikz) 2 (0/0x)1k> into a similar pair of physically interpre-table matrix eleme nts , using the identity

Z 2 a1[2Z " a ôx-ac — x -6 xz

az

Determine which types o f multipole transit ions are embodied in this second-order term. W hat kinds of electric and magnetic field gradien ts are generallyrequired to e ffect these types of transitions? By what factor do these transitionprobabilities diffe r from those of M 1 and E 2 transitions?

7. T he t ime-dependent perturbation theory developed in Section 1 .4 is usefulfo r small perturbations, and is widely applied in spectroscopy. T he contrastingsi tuation in which the pe rturbation is n ot sm all compared to the energyseparat ions be tween unper turbed levels i s of ten more difficult to treat. Asimplification occurs when the Hamiltonian changes sudden ly at t = 0 from Ri



PROBLEMS1to Hf , where fi i and Hf are t ime-independent Hamiltonians satisfying

H I s ;

i> =Es I s ; i>H ik; f > = Ef ik; f >

It can be shown that in the s ud d en approximation (which is applicable when thetime T during which the Hamiltonian changes satisfies T(E k — Es ) « h), a systeminitially in state s of ii i will evo lve into state k of fif after t = 0 with probability

Ps—qc = Ks; ilk; f> 1 2

A is tritium atom undergoes 18 keV )6 decay to form He. With what

probabili t ies is H e + formed in the is, 2s , and 3d states?



ATOMIC SPECTROSC OPY

Atomic spectra accompany electronic transitions in neutral atoms and in atomicions. One-photon transitions involving outer-shell (valence) electrons in neutralatoms yield spect ra l lines in the vacuum ultraviolet to the far infrared regions ofth e electromagnetic s pectrum (Fig. 2.1), corresponding to wavelengths betweenseveral hundred angstroms and several meters. Transitions involving the moret ightly bound inne r-shell electron s give rise to spectra in the X -ray region atwavelengths below — 1 0 0 A ; we will not be conce rned with X -ray spectra in thischapter.

Atomic em ission spectra a re commo nly obtained by gene rating atoms in theirelectronic e xcited states in a vapor and analyzing th e resulting emission with aspectrometer. Electric discharges produce excited atoms by allowing ground-state atom s to collide with electrons or ions that have been accelerated in anelectric field. Such collis ions convert part of the ion's translational kinetic energyinto electronic excitation in the atom. Low-press ure mercury (Hg) calibrationlamps operate by this mechanism. Atomic excited s tates may also be producedby exci ta t ion with lasers (Chapter 9 ) , which are intense, highly mon ochromaticlight sources . T his monochromaticity permits s elective laser excitation of singleatomic states, a feature that is not poss ible in ordinary electric discharges . A lesscommon m ethod o f gene rating excited atoms is by chemical reactions, and theresulting emission is ca lled chemiluminescence. An important exam ple is th ebimolecular reaction between sodium dimers and chlorine atoms,N a 2 + C l — > Na* + N a C 1 , which creates electronically excited sodium atomsNa* in a large number of different excited states. T he photodissociation processCH 3 I + hv -4 C H 3 + I* init iated by ultraviolet

light is an ef f ic ient method ofproducing iodine atoms I* in their lowest e xcited e lectronic state, which cannotbe reached by El one-photon transitions from ground-state I .

33



34TOMIC SPECTROSCOPY

Visible

Radio

icrowaveRV-raysamma rays

3000

30 km300mmcm.3mmm04.3 4I 10

3060 90 1 20 1 50 20

KHzHzHzHz

Figure 2.1 The electromagnetic spectrum. IR and UV are acronyms for infrared

and ultraviolet, respectively; the abbreviations kHz, MHz, GHz, and THz stand for

kilohertz, megahertz, gigahertz, and terahertz.

Light em itted a t di f ferent w avelengths i s s pa t ia lly disperse d in gratingspectrometers , and emiss ion spect ra may be recorded o n photographic film.Alternatively, the grating instrume nt may be ope rated as a scanning mono-chromator that transmits a single wavelength (more precisely, a narrowbandwidth of wavelengths) at a time. Em ission spectra may then be recorde dusing a se ns i t ive photomultiplier tube to detect the emiss ion transmitted by th emonochromator while the latter i s scann ed through a range of wavelengths.

Atomic absorptionspectra

may be obtained by pas sing light from a sourcethat emits a continuous spectrum (e.g., a tungsten filament lamp, whose outputspectrum approximates that of a blackbody em itting at the filament temper-ature) through a cell containing the atomic vapor. The transmitted continuum i s

then d ispersed in a grating spectrometer, and may be reco rded e i ther photo-graphically or electronically using a vidicon (television camera tube) or linearphotodiode array. C haracterist ic absorption wavelengths are associated withoptical de ns ity minima in developed photographic negatives, or with trans-mitted light intens ity minima detected on a vidicon or photodiode array grid.Emiss ion spectroscopy is preferable to absorption spectroscopy fo r detection o f

atoms in trace amounts , s ince em itted photons are readily monitored photo-ele ctrically with useful signal-to-noise ratios at atom concentrations at whichabsorption line s would be barely detec table in samples of reasonable size.

Re presentat ive em ission spectra are shown sche matically in Fig. 2.2 forhydrogen, potassium, and mercury on a comm on wavelength scale from thenear infrared to th e ultraviolet. Un der the coars e wave length res olution of thisfigure, the em itted l ight intensi t ies are concen t ra ted a t single , well-definedemission lines. In H , th e displayed e mission consists of four conve rgen t series oflines, th e so-called Ritz-Paschen and Pfund series in the near infrared, th eLyman series in the vacuum ultraviolet, and the Balmer series in the visible.

Johann Balmer, a schoolteacher in Basel in the la te n ine tee nth century,



ATOMIC SPECTROSCOPY 35

Ritz-Paschen Balmer

H

DiffusePrincipal 1 1 1 1 1

Fundamental

Hg

25000 50000 75000 100000

Lyman

v(cm- 1 )

Figure 2.2 Schematic emission spectra of H, K, and Hg atoms. These are plotted

versus the line frequencies ij 1b in units of cm where the 2 are the emission

wavelengths in vacuum. Only the strongest emission lines are included, and relativeline intensities are not shown. Line headers for H and K denote series of lines

resulting from transitions terminating at a common lower level. Line headers are

omitted for the Pfund series in H (which appears at extreme left) and for the sharp

series in K, which closely overlaps the diffuse series.

discovered that the wavelengths in angstroms of l ines in the latter se r ie s close lyob ey the remarkably simple formula

/1. = 3645.6

(m 2 —2)

with m = 3, 4, 5, .... In the limit of large m, this expression converges to a serieslimit at A = 3 6 4 5 . 6 A . (This limit is not directly observable in spectra like that in

Fig. 2.2, because the line intensities become weak for large m.) The wavelengthsof l ines in the Ritz-Paschen series are similarly well approximated byA = 8 2 0 2 . 6 m 2 /(m 2 — 3 2 ) with m = 4, 5, 6, .... Hydrogenlike ions with atomicnumber Z 1 (He, Li',etc.) exhibit analogous series in which the emissionwavelengths are scaled by the factor 1 /Z 2 relative to those in H . Thecompactness of the analytic expressions (cf. Eq. 2.1) for spectral line positions in

hydrogenlike atoms is, of course, a consequence of their simple electronicstructure.

m 2

(2.1)



36 ATOMIC SPECTROSCOPY

Though potassium (like hydrogen) has only one valence electron, i tsspectrum appears more complicated. It can be analyzed into s everal overlappingconvergent series (historically n amed the principal, sharp, diffuse, and funda-mental series) as shown by the l ine headers in Fig. 2.2. Potassium exhibits a

larger n umber of visible spectral lines than hydrogen, and their wavelengths are

not accurate ly given by an alytic formulas as simple as Eq. 2.1. These differencesare caused by interactions between the valence electron and the tightly boun dcore e lectrons in the alkali atom— interactions that are absent in hydrogen. Nodiscrete absorption line s occur at en ergies higher than about 3 5 , 0 0 0 cm -1 , the

ionization potential of potassium.The mercury spectrum is even less regular. The electron configuration in Hg

consists of two valence elec t rons outs ide of a closed-shell core ...( 5 s ) 2 (5p) 6 of) 1 4 (5 d , 1 0 .) The Hi spectrum features that are not anticipated in H or

K arise from electron spin multiplicity (i.e., the formation of triplet as well as

singlet excited states in atoms with even numbers of valence electrons) and fromspin-orbit coupling, which assumes importance in heavy a toms l ike Hg

(Z = 80 ). The me rcury spectrum in Fig. 2.2 has been widely used as a spectralcalibration standard.

In this chapter, we re view electron ic structure in hydrogenlike atoms anddevelop the pertinent selection rules for spectroscopic transitions. The theory of

spin-orbit coupling is introduced , and the electronic structure and spectroscopyof many-electron atoms is greated. These discussions enable us to ex plain de tailsof the spectra in Fig. 2.2. Finally, we de al with atomic perturbations in staticex ternal magnet ic f ields , which lead to the normal and anomalous Zeeman

effects. The latter furnishes a useful tool for the assignment of atomic spectrallines.

2.1 HYDROGENLIKE SPECTRA

The unperturbed H amiltonian for an electron in a hydrogenlike atom withnuclear charge +Ze i s

h 2e 2Ro =22, uire o r

(2.2)

The atomic reduced mass p is related to the nuclear mass m N and electron massm e by p = m e m N I(m e + m N ), V 2 operates on the electronic coordinates , and r i s theelectron-nuclear separation. The eigenfunct ions 0, O D and eigenvaluesE n of this H amiltonian ex hibit the properties

/loll n i m ( r , 1 9 ,Enitknar, 1 9 , O D2.3)

n = 1, 2, ...

1 = 0 , 1 ,(n - 1)

m=0, +1,..., +1



HYDROGENLIKE S PECTRA7I t k n i m ( r , 0 , 01>n1(r)Y1.( 0 ,

plZ 2 (e 2 /47re 0 ) 2

The e igenfunct ions factor into a radial part R 1 (r) and the well-known sphericalharmonics Y I „ , ( 0 , 0); n, 1 , and m are the principal, azimuthal, and magneticquantum numbers, respectively. T he energy e igenvalues E „ depend only on theprincipal quantum number n. Since the total number of independe nt sphericalharmonics Y i n ,(0, 4 ) ) fo r 1 (n — 1 ) is eq ual to n 2 for a given n , each energyeigenvalue E n is n 2 -fold degene rate. While n controls the hydrogenlike orbitalenergy as well as size, the quantum numbers 1 and m govern the orbitalanisotropy (shape) and angular mom en tum. Since the o rbital angular mo -

men tum operators L 2 and L z commute with the Hamiltonian 1 1 0 , theeigenfunct ions 'Cu m > of IL are also e igenfunct ions of L 2 and L z ,

L 2 } m (0 , 1)h 2 Y1,0 , (k)2.6)

C z Y i , n (0 , 4 )) = m hY b n (0, 4))2.7)

(The radial part R 1 (r) of 1 t f r n i n i > cancels out in Eqs. 2 .6 and 2.7, because L 2 and L zoperate only on O and 0 .) This implies that the orbital angular momentumquantities £2 an d L z are cons tants of the mo tion in stationary state I t k n i .> with

values 1(1 + 1 )h 2 and mh, respectively. A common notat ion for one-electronorbitals combine s the principal quantum number n with the letter s, p, d , orf ororbitals with 1 = 0, 1, 2, and 3, respectively. (This notation is a ves tige o f thenomenclature sharp, principal, diffuse, and fundamental for the emission seriesobserved in alkali atoms, as shown fo r K in Fig. 2.2.) An orbital with n =2, 1 = 0is called a 2s orbital, one with n = 4, 1 = 3 a 4f orbital, an d so on. Numericalsubscripts are occasiona lly adde d to indicate the pert ine nt m value: the 2p 0

orbital exhibits n =2, 1 = 1 , and m = O . Chemists frequently work with real(rather than complex) orbitals which transform as Cartesian vector (or tensor)components. A normalized 2 p x orbital is the linear combination( — 1 2 p 1 > + 12 p _ 1 >)/li ( —1211> + 121, — 1>)/.1i, because the spherical har-monics Y1 1 and 1 7 1 , _ I are given in Cartesian coordinates by

=2n 2 h 2

(2.4)

(2.5)

Y1 , 1=

— 1 =

3 x + iy

8 7 r 3 x — iy

8 7 r (2.8)

with r = (x2 ± y 2 4 _ z 2 \ 1 / 2 .) Con tours of some of the lower-energy hydrogenlikeorbitals ar e shown in Fig. 2.3.

In accordance with the selection rules developed in Chapter 1 , one must have



3s 3p z

2pzs

is

38



H Y D R O G E N L I K E S P E C T R A93d e

d x,

Figure 2.3 Contour plots of several low-lying H atom orbitals. Curves are surfaces

on which the wavefunction exhibits constant values; solid and dashed curves

correspond to positive and negative values, respectively. The outermost contour in all

cases defines a surface containing —90% of the electron probability density. The

incremental change in wavefunction value between adjacent contours is 0.04, 0.008,

0.015, 0.003, 0.005, and 0.003 bohr -312 respectively for the 1s, 2s, 2p, 3p, and

3dorbitals. Boxes exhibit side lengths of 20 bohrs (1s, 2s, 2p) and 40 bohrs (3s, 3p,

3d), so that orbital sizes can be compared. Straight dashed lines in 3p, and 3d,2 plots

show locations of nodes.

fo r allowed one-photon transitions from state l i k „ i m > to state l t , G „ T „,)

< 0.1.1ol l1'„T. , > o 0,l transitions

< 0 . 1 . 1 1 , 10.T. , >,1 transitions2.9)

o 0 ,2 transit ions

To cons ider the specific selection rules on electric dipole (El) transit ion s inone-electron atoms, we evaluate the matrix eleme nts of the pert ine nt electricdipole moment operator p=E e i r i =ZerN —er', where rN and r ' are thepositions o f the n ucleus and electron referenced to an arbitrary origin in space.Then

OknImIPIOnTm') = KOnlmIZerN er ' ItfrnTne>

= (- Z —e<iknindrItfr.T.•>

= O — e<0 .1 .1 r ICTne>2.10)

since the hydrogenlike states are orthogonal and depend only on the electron's



4 0 ATOMIC SPECTROSCOPY

coordinates r = r' — r N relative to the position of the nucleus. Using

r

r

sin 0 cos 0 I

r=in sin c P

r cos 0

the transit ion dipole m o m e n t integral is

< t f r nintlfil0 n'l' m'›

( 2 . 1 1 )

2 .= —e f r2 dr rR„ ,(r)R i ,(r) f sin 0 d0 f d 4 ) Y r , n ( 0 , 4 ) )

o

_sin 0 cos 0

s in 0 s in 0

r. , (0 , 0 )

2 . 1 2 )

cos 0

It can be shown that the angular part of this integral vanishes [1 ] unlessAl 1' — 1 = + 1 (not zero), and unless Am E - 7 m ' — m = 0 or + 1 . Evaluation of

the radial part is difficult; this factor is nonzero regardless of An n ' — n,provided Al = + 1 . Hence we can summarize the El selection rules for o n e -photon transitions in hydrogenlike atoms,

Al=+1

2 . 1 3 a )

Am = 0 , ±1

2 . 1 3 b )

Since the hydrogenlike energy levels are given by Eq. 2 . 5 , the transitionenergy accompanying emission of a photon of freque ncy c o will be

— hco = E „ , — E_itz 2 (e 2 1 4 7 t E 0 ) 2 [ 1]2h ) 2

= — h c lA2 . 1 4 )

in going from state 1 0 „ i n i > to state 1 0 „ , , , , „ ) . The corresponding photon wavel-ength is then

8 h 3 cn 2 (0 2A = tiz2(e 2/ E 0 )2 n 2n r) 2 ( 2 . 1 5 )

which has a form identical to Eq. 2 .1 if n' = 2. The visible H atom lines in theBalmer series thus result from transitions from n = 3, 4, 5 , . .. down to n' = 2.



HYDROGENLIKE SPECTRA1Other H atom series arise from transitions terminating in different values of n '(Fig. 2.4). Since the hydrogenlike energy levels En are independent of land m, the

se lection rules (2.12) do not preclude the appearance of a spectral line for anyenergy separation (E n , — E„), and lines appear for all combinations (n', n). Thesefacts quantitatively acco unt for the wavelengths of all of the H atom spectral

lines in the low-resolution spectrum of Fig. 2.2.Another cons equence of the selection rules (2.13) is that a hydrogenlike atom

cm

oo 54

3

2

roro c J N-

.co (0 vrtrr 6 .0 dr

7—• C O 0to CS _ rn

co oco ( . s ô

Ritz-

• Brackettseries

Balmer

series

Paschen

series

—50,000

— 1 00,000

Lyman

series

Figure 2.4 Hydrogen atom energy levels and transitions. The Lyman, Balmer, Ritz-Paschen, an d B rackett series occur in the vacuum ultraviolet, visible, near-infrared,

and infrared regions of the electromagnetic spectrum, respectively.




in any excited e lectronic s tate exce pt the 2s state can emit a photon sponta-ne ously by electric dipole radiation, and thereby relax to some lower ene rgystate. For any s uch exci ted s ta te IC erne > , there exists a lower state 1 1 / / „ / .> fo rwhich A l = ±1 an d A m = 0 , ±1. T he 2s state is the exception, because the only

s ta te with lowe r en ergy than the 2s s tate is the is state, and the 2s isfluoresce nce transition (A/ =0) is El-forbidden. T he 2s state is therefore ca lledmetastable sin ce it will e xhibit an unusually long lifetime, lacking an electricdipole-allowed radiative transition to any lower state. Metas table states are notsimply states which have no El-allowed transition to the ground state : to bemetastable, a state must have n o El-allowed transitions to any states of lowerenergy. Such atomic states play an important role in the efficiency of H e / N elasers (Chapter 9 ) .

Since the spherical harmonics Y i „,(0, 0) also describe angular mome nta for thesingle valence electron in alkali atoms (Li, Na , K, Rb, Cs ), the selection rules

(2.13) apply equally we ll to valence electron transitions in such atoms (but not totransitions involving core e lectrons, where an gular mom entum coupling canbecome important). In this case, (nlm) and ( nT m) denote the initial and final setsof quantum numbe rs in the valence orbital. Valence—core interactions in alkaliatoms split the n 2 -fold degeneracy of en ergy levels belonging to a given principalquantum number n , so that the ene rgies now depend on 1 as well as n. This i sillustrated in the en ergy level diagram fo r potassium in Fig. 2.5. (Each of thelevels is still (2/ + 1 )-fold degene rate, since the m sublevels fo r given n and 1 arei soenerget ic in the absence of external magnetic fields.) The larger numbe r ofdistinct levels resulting from this degene racy-breaking in K does complicate the

emission spectrum. However, the selection rules (2.13) still limit the observed Eltransit ions to a small subset of the total numbe r that could conce ivably occur.T he allowed transitions (A l = + 1) are res tr icted to o nes that conne ct levels inadjacent vertical groups of levels in Fig. 2.5, where the levels are organized incolumns acco rding to their 1 values. T he principal series in K arises from np 4stransit ions with n 4 ; the sharp series re sults from n s — > 4 p transitions withn 5; the diffuse series o ccurs in n d 4p transitions with n 4; and thefundamental ser ies is produced by n f — > 4d transitions with n 4. T he origin ofeach of the lines in the low-resolution potassium spectrum (Fig. 2.2) can now bequalitatively unde rstood with refe ren ce to Fig. 2.5.

Each of the series in the H and K spectra converges in principle to a serieslimit, as the discrete atomic ene rgy levels must converge when the onset of theionization continuum is approached (Figs. 2.4 and 2.5). T he lines are very weakin the neighborhood of the se ries l imits , because their intens i ties are pro-portion al to the absolute value s quare of the transi t ion dipole momen t . Thelatter contains the factor

1 . 3 d rR n i (r)R r(r)f

2

which falls off rapidly with (n — re) fo r given ( 1 , 1 ) [1 ]. Fo r example, this quantity



e m -1

5.000-

10,000 -

15,000 -

20,000 -

25,000

30,000 -

35,000-

3

2

1Ii

I ///

I I

I/

o

Volts

4.32

II

.4 0II1/

/ /,/4

SPIN—ORBIT COUPLING3Figure 2 .5 Energy levels an d observed transitions in K. This type of diagram iscommonly referred to as a Grotrian diagram. All of these low-lying energy levels arise

from electron configurations of the type (1 s

)2(2s)2(2p)6 ( 3s )2(3p) 6 (no 1. .The 2 9 1 1 2 level labeled "1", th e 2 P 1 1 2 and 2 P 3 1 2 levels labeled "2", th e 2 D 3 1 2 and 2 D s /2

levels labeled "3", and the 2 F 5 1 2 and 2 E7 /2 levels labeled "4" correspond to theelectron configurations • • • (4s) 1 , • • (4p) 1 , • 4d) 1 , an d • • • (40 1 respectively.

Reprod uced, by permission, from G. Herzberg, Atomic Spectra and Atomic Structure,

Dover Publications, Inc., New York, 1944.

equals 0 .464 , 0 . 075 , and 0 .026 A 2 , respect ively , for the is 2p, is 3p, andis 4p transitions in hydrogen, and so good sensitivity is required to observel ines nea r the se r ies limit.

2.2 SPIN—ORBIT COUPLING

If the low-resolution potassium spectrum in Fig. 2.2 is reexamined using ascanning monochromator that can distinguish between wavelengths that are 5 Aapart, each of the lines becomes split into closely spaced multiplets or groups ofl ines. Lines in the principal and sharp series become doublets, and l ines in thediffuse series appear as t r iplets . This fine structure ar i ses f rom the interactionbetween the orbital angular momentum L and the spin angular momentum s ofthe valenc e electron. Analogous splittings occur in the hydrogen spectrum, but

much higher resolution is required to observe them in this atom, and otherrelativistic effects have comparable importance in hydrogen.



fl.= — Ps • B

(_ s es( ZeL2 m e Aitom e c 2

Zg s e 2(L • s)8n e o m !c 2 r 3


W e can de r ive the interaction between the electron's intrinsic spin angularmomentum s and its orbital angular m ome ntum L classically for an electronmoving in a circular orbit aroun d the nucleus. In this picture, the electron isinstantaneo usly moving with a velocity V e perpend icular to a line that connectsit with the nucleus. In the electron's rest frame, the nucleus appears to be moving

in the oppos ite direction, V N = — Y e , relative to the electron. So the electronexperiences a magnetic field from the apparent moving charge on the nucleus[2],

1 B = (VN x E) = — 4 (V e x =+ (E X p)m e c

(2 . 16)

where m e is the electron mass and E is the electric field at the electron due to thenucleus. Since

ZereE =47r8 0 r 3

=4nEor2

the magnetic field is

ZeeB(r x p) — 4n c o m e c 2 r 37tg o m e c 2 r 3

(2.17)

(2 .18)

where L i s the electron's orbital angular mome ntum. The e lectron's intrinsic spin

s carries an associated magnetic dipole mome nt [3]

— g s esPs =

2m e

(2.19)

where g s is the electronic g factor (g, = 2 according to Dirac, 2 . 0 0 2 3 according toSchwinger [4]). T he ne gative sign in Eq. 2 . 19 i s due to the ne gative charge on theelectron. T he energy of interaction o f the electron spin magnetic moment withthe magnetic field B due to the moving nuclear charge is [2]

(2.20)

Since the potential energy of attraction between the electron and nucleus is

V(r) = Ze 2 /4ne 0 r



SPIN—ORBIT COUPLING5the spin-orbital ene rgy is

g sav(L • s )_ s o = 2m 2e c 2 r ôr (2.21)

This actually overestimates the spin-orbital energy by a factor of 2 , because wehave neglected the fact that an electron in a circular or elliptical orbit does nottravel at a uniform velocity V e , bu t experiences acceleration. T he effect ofcorrecting for this is to cance l [5] (or nearly cancel, according to Schwinge r) theg factor g s , and we write

fiso=v (L s )

2,u 2 c 2 r O r

(2.22)

where m e has been replaced by the atomic reduced mass t t . W e note that sinceV(r) = - Ze 2 /4ne 0 r, the magn itude of the spin-orbital H amiltonian increase swith atomic n umber Z.

In hydrogenlike atoms, the total electronic H amiltonian now become s

fi = fi0 + f

h 2e 2 = - —/ 1 V 24 nE o r H"

(2.23)

In the limit where rl s o can be t rea ted as a stationary perturbation, the energycorrected to first order become s

pZ 2 (e 2 /47c6 0 ) 2E n i .

2n2h2w imInsoltiinim! (2.24)

The latter matrix elemen t requires an expression for L - s according to Eq. 2.22.It also re quires knowledge of the total angular mo me ntum states that can arise

in an atom with orbital and spin angular mom enta L and s (Appendix E). T hespherical harmonics in the atomic s tates t f r „ „ n are eigenfunctions of L 2 and L z(Eqs . 2 .6 , 2 .7). The electron spin states ism s > obey

: i 2 Is m 5 > = s(s + 1)h 2 Is m 5 >2.25a)

k I s ms > = msh Isms>2.25b)

with s = and m s = +1 for a sin gle electron [3]. T wo alternative commutingsets of angular mom entum operators are then 1 3 , L z ,s, and 13, 2 P

where the total angular m omen tum J is defined asJ = L + s.

Eigenfunctions ofthe first commuting se t form the uncoupled represen ta t ion Ihn i sm s > Ilmi>lsms>.Since f2 does not commute with L z or & (Appendix E), these uncoupled s tates




are not eigenfunctions of P. Eigenfunct ions of the second commuting s e t formthe coupled represen ta t ion I lsjm>. The total angular momentum squared J 2 must

be a constant of the motion in an isolated atom, which is then appropriately

described using the coupled representation. In this representation, the s ta tes

lisim>are eigenfunctions of :i2 with eigenvalue j(j + 1 )h 2 , where the possible

values of j are

j=i+s,/+s-1,...,11-s12.26)

Since

L • s = -1(J 2 - L 2 - s2 )2.27)

the energy 2.24 corrected for spin-orbit coupling becomes

-itZ2(e2/47c80)2

±0 1 7

/-1-r a. t2 -1 s 2 ) 1 1 /1 nt.>

1 OVItiini.>

(2.28)

(2.29)

(2.30)

. . n2h2;2c2 \wnimi -(J2 -r-ar 2

- pZ 2 (e 2 14 ne 0 ) 2j(j + 1) - 1(1 + 1) - s(s + 1)]h 2= 2n2h2 Letting

h 2

4g2c2

13Vdr„ 1 (r )2

<Iiiniml ; w

- s(s + 1)]/2

Ant = 2p2 c 2

the corrected energies are

- pZ 2 (e 2 1 4 7 r 8 0 ) 2+ An i [j(j

0r —ar

+ 1) - 1(1 + 1)2n2h2

The possible values of the quantum number j = 1 + s, il - si reduce toj = 1 + for 1 0 0 in hydrogenlike a toms . As an example , the possible j values

for 2p states in hydrogenlike atoms are j = 1 + 4, 1 - =4,1. The correspond-ing energies of the j sublevels would then be *

_It Z2(e 21 4 7 r E 0 )2

E312h 2 A 21 al + 1 ) - 2 - -D/2 for j =

*It can be shown that for hydrogen like atoms, the spin— orbit coupling constant A n i is given by

Z 4 e 2 h 2 =2 p 2 c 2 ag n 3 1 ( 1 +1) (1 + 1)

T he radial wave functions R 1 (r) do not have closed-form express ions in many-electron atoms(Section 2.3), and so A n i is not given by simple formulas in such atoms. Note the sensitivity of A n t tothe atomic number Z; this gives rise to

largespin— orbit coupling

in heavy atoms.



SPIN-ORBIT COUPLING 47

pz2( e 2 47 r e 0 )2E 1 1 2

h2 A 2 l [4a ± 1) — 2 — i]/2o r j = 42.31)

and the energy splitting between these spin—orbi tal states is

E312 E112 = 3A2 1/22.32)

This i s an ex ample of the Landé interval rule (1933) for the energy separation ofspin—orbi tal states with success ive j values:

E. — E 1 = jAn i2.33)

To keep track of the different angular momenta in atomic states, t erm sym bolsare used to speci fy the values of 1 , m , and j:

2 s+ 1 L2.34)

W h e n 1 = 0 , L i s den o t ed wi th an S ; when l = 1 , L is denoted with a P, and so o n .Similar t e rm sym bols are used to notate the angular momenta in many-electrona toms. For the hydrogenlike 2p sublevels with ] =4 and 4 , the term symbols are

2 2 P 3 1 2 and 2 2 P 1 1 2 respectively, where the f i rs t digit indicating n = 2 i s useful forspecifying the principal quantum number of the valence electron in hydrogen-

like and alkali atom States.Since each of the hydrogenlike levels with 1 0 0 is no w spli t into doublets withj = l + 4, it becomes necessary to augment the El se lec t ion rules on An , Al, andAm with El se lec t ion rules on A j . It turns out that these are A j = 0 , + 1 . W e willdemonstrate this selection rule in the case of n 2 S 1 1 2 n1 3 ; transitions, where ]can be 4 or 4 . The coupled (total) angular momentum states 1 / s j m > Um> in

a t oms can be e xpres s e d as a superposition of uncoupled statesl/m i sm s > weightedby Clebsch-Gordan coef f ic i ent s [1, 3] ,

L i m > =E ilmisms></mismskim>2.35)mim

s

For the n ' 2 Pi states, it is understood that 1 = 1 , s = 4 ; the possible m i values are0, + 1 and the possible m s values are + 4 . The four components of the 2 P 3 1 2 state(m = — 4 through +4 ) are then

fini ni s(2.36a)

i > = 1 1 , I>

1 4 , > + or2I1, -4>2.36b)

1 4 ,(4)" 2 1 — 1 , 1> + (4) 1 1 2 1 0 , — 1>2.36c)

1 4 ,1 — 1 , —2.36d)




The first of Eqs. 2.36 arises because there i s on ly one combination of m i and m s ina 2 P 3 i 2 state that can give a total m of (m 1 = 1 , m s =I) . The second can beshown using the fact that the raising/lowering operator J+ has the effect

.\/./G + 1 ) — m(m ± 1 ) 1 . / ( m ± 1 ) >

when applied to state Lim>, and that

J+ = L+ +S

(2.37)

(2.38)

with

= .0(1 + 1 ) — m l ( m i + 1 )1/(m i ± 1))2.39a)

S ± Ism s > = N /s(s + 1 ) — m s ( m s ± 1 ) 1 s (m s ± 1 ) >

2.39b)

The U m > s ta te I I , — 4 > can thus be obtained by applying the J_ operator toll, 4 >and using Eqs. 2.37 and 2.39.

The two components of the 2 P 1 1 2 state are given by

m1 n i sI%= VI 2 10 , 4> -6 - W 2 1 1 , - 4 >

1 1\2 \)

1 / 2 1I_ 1 , 4>1/210, -4>

t3

(2.40a)

(2.40b)

Thes e fol low because the L i m > state ll, i> must be normal ized and orthogonal tothe U m > s tate II , 4 > in E q. 2.36, and because I I , — 4 > can be obtained from 1 4 ,4 >by application of the J_ operator and use of Eqs. 2.37 and 2.39. Finally, then 2 S 1 1 2 state (1 = 0 ) has the two components (m = m s = +1)

1 0 , i> and 1 0 , --4 >2.41)

The elec tric dipole transition moments for the various fine structure transitionsbetween the n 2 S 1 1 2 and n' P 1 1 2 3 1 2 21/2,3/2 manifolds (i.e. , groups) of levels can now beevaluated:

(41= + 1 , Am = +1)

< 2 S 1 / 2 , 1 / 2 I P I 2P 3 /2 ,3 /2>° , 1 1 0 1 1 ,I>

2.42a)

(4i= + 1 , Am = 0 )

< 2S 1 /2,1 /21P1 2 P3/2,1/2>i) 1 1 2 11 /1 1 ° , 12> ± (4) 1 1 2

X <0, 1101 1 ,

-I>2.42b)



SPIN-ORBIT COUPLING9(Aj = + 1 , Am = — 1 )

< 2 S1/2,1/2101 2 P3/2, - 1/2> = ( )1"2<O 1 1 0 1 - l,4>o1 1 1 20, lipio, --21-->2.42c)

(Aj = + 1 , Am = — 2 )

< 2 S1/2,1/41 2 P3/2, - 3/2>09 1- 101 - 1 9 -1>2.42d)

In evaluating these , we note that

<nlims1P1m;InD = <miltilmDbm s ,m ;

since p does not depend on the spin coordinates. Further, this matrix elementvanishes unless Am / = —m , = 0 or + 1 . H e n c e , the second t e rms on the rightsides of Eqs . 2 .42b and 2.42e both vanish, but none of the total transitionmomen t s in E q s . 2 . 38a -2 . 38c vanish. T he only El-forbidden transition is th eone whose (zero) transition moment is given by Eq. 2.42d.

For the n 2 S 1 1 2 n1 3 1 1 2 transitions, we have

(Aj 0, Am = 0 )

< 2 S 1 / 2 , 1 / 2 1 / 1 1 2 P1/2, 112> = ar/ 2 < 0 , 1 - 1 / 1 1 0 , 1 >

—(i) 1 / 2 < 0 , (2.43a)

(Aj = 0 , Am = — 1 )

< 2 s 1 , 2 , 1 / 2 1 1 1 1 2 1 ) 1 , 2 , - 1 , 2 > = (2)1/2<0 iI —1 , 1 >

—( ) 1 / 2 < 0 , 4 1 P 1 0 ,2.43b)

T he f irs t term s on both right sides are nonzero, and so these are both allowedt ransi tions . A se t of equations analogous to (2.42) and (2.43) can be obtained fo rt ransi t ions from I S2 - 1/2, - 1/2>> and will not be included here. Equations 2.42 and2.43 typify the El selection rules Al = + 1; Aj = 0, ±1; and Am = 0, + 1 fo relectronic transitions in hydrogenlike a toms .

These angular momentum selection rules figure prominently in the finestructure of alkali atom spectra. The filled-shell core electrons have zero netorbital and spin angular momentum, so the term symbols 2 2 S 1 1 2 , 3 2 S 1 1 2 , 4 2 S 1 1 2 ,5 2 S 1 1 2 , and 6 2 S 1 1 2 of ground states Li, Na, K, Rb, and Cs respect ively arecomposed from th e angular momenta of the single valence elec t ron. T heprincipal series o f alkali atomic lines ar ises f rom n 2 S 1 1 2 n1 3 1 1 2 , 3 1 2 transitions;



3

sharp


since Aj = 0, + 1, there will always be two fine structure componen ts (e.g.,3 2 S 1 1 2 32. s

r11 2 an d 3 2 S 1 1 2 -+ 3 2 P 3 1 2 in Na) in this series . (Be ar in mind that the

various m sublevels of any Urn) state are degen erate , so that only transitionsfrom a given level to final states with di f ferent j values will give rise to more thanone spectral l ine—unless an applied magnetic field splits the m sublevels.) T he

di f fuse series n2 P 1/2,3/2 re 2 D3 2 , 5 1 2 always yields triplets ( e . g . , 3 2 P 1 1 2 3D 3 1 2 ,3 2 P 3 1 2 3D3 1 2 , and 3 2 P 3 1 2 3D 5 1 2 ; bu t not 3 2 P 1 1 2 3D5 1 2 , for whichAj = +2.) Doublets occur in the sharp series n 2 P 1 1 2 ,3 1 2 re 2 S 1 1 2 . These E l-allowed fine structure transit ions are all summarized in Fig. 2.6 .

In the alkali atoms , the spin-orbit coupling is a small perturbation to thezero-order e lectronic H amiltonian. In N a, the ene rgy se paration between the3 2 P 1 1 2 and 3 2 P 3 1 2 spin-orbit sublevels of the lowest excited 3 2 P state is only- 17 cm 1 , versus 1 7 , 0 0 0 cm -1 for the difference between the 3 2 P and groundstate (3 2 S) levels. At the other extreme , the 5 2 P ground state of the I atom is s plitby spin-orbit coupling into 5 2 P312 and 5 2 P 1 1 2 sublevels which are about8 0 0 0 cm -1 apart! In this limit, the spin-orbit subleve ls behave m uch likedifferent electronic states— which they are, because the spin-orbit coupling isno longer a small perturbation to the electronic structure.

When R s 0 is n ot a small perturbation, it become s important to know whichdynam ical o bservables are still cons erved in the atom. In the absence of spin-orbit coupling, the electronic H amiltonian 1 1 0 and the angular mome nta obeythe commutation relationships

[ 1 1 0 , 1 : 2 ] = Ofl o , P] = 0n o , gz ] = o

[ fio, s2] = 0

n o , L z ] = 0R 0 , fz ] = 02.44)

2 P1/2,3/2312, 5/25/2, 7/2

Figure 2.6 Schematic Grotrian diagram showing fine structure transitions in Na.

The spin-orbit splittings are greatly exaggerated: The 32p112-32P312 splitting is only

17 cm -1 , as compared to 16,961 cm -1 for the 3 2 S 1 1 2 3P 1 1 2 transition.

2

S 1/2



STRUCTURE OF MANY-ELECTRON ATOMS1When Il s o is turned on , th e total Hamiltonian 1 - 1 becomes 1-? 0 + fi s t, gaining aterm proportional to L • g . In this case , it can be shown (Problem 2.4) that

[fi ,£2]

:=i = o

fi,

[fi, Pi = 0II, t z] 0I?, fz ] = 02.45)

so that m i and m s are not good quantum numbers (and L , and S z are notconserved) in the pres ence of spin-orbit coupling.

We now briefly consider th e magnetic dipole (M1) selection rules fo rtransitions in hydrogenlike and alkali atoms. The relevant matrix element is

(not as has sometimes been implied, because th ederivation of the M1 selection rules in Chapter 1 makes it clear that only th eorbital part of the angular momentum enters in this matrix element). Using [3]

=E)2.46a)

- L = —2 i

(L + - L_)2.46b)

L z = L z2 .46c)

and using Eq. 2.39 immediately shows that since all matrix elements of L arediagonal in 1 (i.e., proportional to S), th e Ml selection rule on Al i s Al=-- 0 . To

obtain the M1 selection rule on Aj, one must again expand the coupled 1 1 m >states in t e rms of the uncoupled states (e.g., Eqs. 2.36 and 2.40) and then getexpress ions analogous to Eqs. 2.42 and 2 .43 , with L replacing p . An example ofa n M 1 (but not El) allowed atomic transition is th e 2P112

_÷ 2 - m 312r spin-orbitalt ransi t ion bet ween the lowest two levels in the I atom, which forms the basis ofthe 1 .2 kan CH 3 I dissociation laser. That such a laser works at all is somewhatstartling, because M1 transitions are inherently weak, and the overwhelmingmajority of laser transitions (e.g., in the He/Ne laser) operate on strong Eltransitions.

2.3 STRUCTURE OF MANY-ELECTRON ATOMS

W e n o w e x t e n d o u r discussion of hydrogenlike atoms to complex atoms with atotal of p elect rons . The nonrelativistic Hamiltonian operator for such atoms inthe absence o f ex t e rna l fields is

h 2e2 22me

V+Ei=inc o rine 0 rii

(2.47)

where ri is the distance of elec t ron i from the nucleus and ri; is the separationbetween elect rons i and j. This Hamiltonian consists of a s u m o f p one-electron




hydrogenlike Hamiltonians

h2e2=

2 m ,4 ne 0 r i(2.48)

combined with a s um of electron—electron repulsion terms of the forme 2 /4ns0 r i i . Were it not fo r these pairwise repulsion terms, the many-electronHamiltonian R0 would reduce to

1 1 0 =E nii =

whose eigenfunctions are simply products of p hydrogenlike s t a te s 1 0 m

1 0 0 ,, • • • ,10.0 1 m 1 ( 1 )0 n 2 1,(2 ) • • • 0.,,ip.„(P»

(2.49)

(2.50)

Such expressions incorporating hydrogenlike states do not in fact provideuseful approximations to electronic wave functions in many-electron atoms: theelectron repulsions have a large effect on the total energy , and the wavefunction (2.50) is not properly antisymmetrized (see below). The conce p t o fwriting many-electron wave functions as products of generalized one-electronorbitals nonetheless provides a viable starting point for developing accurateapproximations to the true nonrelativistic wave functions. As a prototypeexam ple , we cons ide r the neutral He atom, for which the Hamiltonian operatoris

no -n i + n2 + e 2 /4nE 0 r 1 22.51)

Since the Schrödinger equation using this two-electron Hamiltonian cannot beexact ly solved, we use as a trial wave funct ion fo r ground-state He the product ofone -elect ron orbi tals 14 ) 1 (1)> and 102(2 )> ,

wo =0 ( 1 , 2 )I(1, 2)>

is bounded from below by the true ground-state energy E 0 ,

E0 < W0

As a first approximation to 1 0 1 , 2)>, we may start with

1 4 ) 1 ( 1 )02( 2 )> = 1 0 1 0 0 ( 1 ) 0 1 0 0 ( 2»

1 0 0 , 2» = 101 ( 1 ) 4 ) 2 ( 2 )>2.52)

According to the variational theorem [3], the trial ene rgy

<0 (1 , 2 ) 1 f i l +IL + e 2 1 4 neo r 120 ( 1 , 2 )> (2.53)

(2.54)

(2.55)



STRUCTURE OF MANY-ELECTRON ATOMS3where 1 0 1 0 0 (i)> = N exp(— Zr i lao ) is the normalized hydrogenlike is orbitalwith Z = 2 fo r electron i . This arbitrarily places both electrons in identicalorbitals which are undistorted by electron repulsion. Substitution of this zeroth-

order wave function into Eq. 2.53 for the trial energy in He yields

W o = <01( 1 ) 1 1 1 1 1 0 1 ( 1 )> + <0 2( 2 ) 1 H 2 1 4 ) 2 ( 2 ) >

< 0 1 ()02(2 )1 e 2 /4 ncori2101( 1 ) 4 3 4 2 ( 2 ) >

= 2<0100( 1 )1R1P100( 1 )>

+0 0 ( 1 )0100(2)1 e 2 /4n6 0r1210100( 1 )0100(2)>

= 2E1 + (e 2 1 47 rgo)<0 no(1 ) 0 1 0 0 ( 2 ) 1 1 / r 1210 10 0 ( 1 ) 0 1 0 0 ( 2 )>2.56)

where

tiZ 2 (e 2 / 4 . 7 r e o ) 2E = —4(13.6058 eV)

2h 2(2.57)

i s the exact nonrelativistic ground-state energy of the hydrogenlike ion H e +

(Z = 2 ) . The matrix e l emen t of 1 /r 1 2 can be evaluated [6 ] to yield5ttZ(e 2 /4ne 0 ) 2 46 n 2 = 3 4 .0 1 4 5 eV . Then Wo becomes — 74.832 e V , as compared tothe experimental ene rgy 79.014 e V required to remo ve both electrons from a Heatom. While Wc , so computed is obviously a large fraction of the true electronic

ene rgy, it s error of 4.18 e V is of the s a m e o r d e r as exci ted-state en ergyseparations in He (cf. Fig. 2.11); a more sophist icated treatmen t is clearlynece ssary to obtain results of spectroscopic accuracy.

An improved wave function can be o btaine d by replacing the fixed atomicnumber Z = 2 in 1 0 1 0 0 ( 0 > with a single variational paramete r The trial energyW o is then calculated in a manne r analogous to Eq. 2.56, and is minimized withre spect to by setting W0 /0C = 0 . This procedure yields = 2 7 /1 6 = 1 . 6 8 8 in

H e; this is phy%ically smaller than Z = 2 , because each electron scree ns part ofth e nuclear charge from the other electron. The corresponding trial energyW c , = — 77.490 eV is a closer approximation to true en ergy, but its error is stilllarge. It is then logical to con sider trial wave functions with more flexibility thanNexp( — Cr i lao ) , which has only one variational parameter. An example of such awave function is the Slater-type orbital (STO), which has the gene ral form

= Nr7-i. ( 0 i,2.58)

STOs exhibit no radial node s (unlike hydrogenlike orbitals fo r 2s and higherene rgy states), but both n and can s imultane ously be varied to simulate the

behavior of the outer (largest-r) lobes of orbitals in many -elect ron a tom s.

Optimizationof n and

4 to minimizeW

o , again using identicalS T O s for

bothelectrons in ground-state He (with 1 = m = 0 ), yields n = 0.955, = 1.612, andWc , = 77.667 eV. This is still close r to the true e nergy, but the error has bee nreduced by a factor of only 0.88 over that in the previous approximation. The



-75

ENERGY, V

- 77

- 79


use of arbitrarily flexible functions in variational calculations that restrict bothelectron s to occupying iden tical orbitals in He yields no trial energies lower than— 77.8714 eV (Fig. 2.7). The remaining error in the energy is(79.014 — 77.871) eV = 1 . 1 4 3 eV, and is called the correlation error. It arises

physically from the electrons' tende ncy to avoid each other in order to minimizetheir average Co ulomb repuls ion en ergy— a tende ncy that i s ignored whenidentical hydrogenlike or Slater-type o rbitals are used for both electrons.

To reduce the trial energy Wo below —77.8714 eV, the electrons must beplaced in functionally distinct orbitals, or trial functions more gene ral thansingle products of the form (2.52) must be introduced. In pursuing the first of

these alternatives, the electrons could be placed in hydrogenlike orbitals

I t A lôdrip =1 . 1

1 0 1 24(ri)> = N 2e - r 2 1 1) 2(02.59)

with the two independe ntly variable parameters C I and C 2• Such a calculation

requires explicit construction of two-electron wave functions that are ant isym-metric [6 ] with respect to exchange of the electrons (which are fermions). Sincethe Pauli princ iple d em ands tha t no two electrons with the same spatialquantum numbers (n, 1 , m ) can have the same spin, the electrons in ground-stateHe (1s) 2 must have opposite spin, m s = +1 (a) and m s = (i ) . An acceptableant isymmetr ized trial function for He using the individualized orbitals (2.59) isthen

0 ( 1 , 2 ) = 1 [ 0 1 ( 1 ) 4 9 2 (2) + 0 1 (2)0 2 (1)] [a(1) ) 3(2) — a(2)/3(1)]2.60)

(Antisymmetrization of trial functions placing the two electrons into identicalspatial orbitals 4 ) 1 ( i ) was unneces sary, s ince the use of the correctly ant isym-

He

74.832 (hydrogenlike 40's, Z=2)

, -77.490 (hydrogenlike AO's, optimized)

77.667 (STO's, n and C optimized)

—777.8714 (SCF limit)

correlation error =1.143 eV

—--- 79.014(experimental)

Figure 2.7 Energies obtained from variational calculations on the He atom.



STRUCTURE OF MANY-ELECTRON ATOMS5metrized function 0 1 (1)0 1 (2)[a(1) 1 3 ( 2 ) — a ( 2 ) / 3 ( 1 ) ] in place of 0 1 (1)0 1 ( 2 ) in E q.2 . 5 3 does not influence the value of the trial energy.) The simultaneousoptimization of the orbital exponents ( 1 and C 2 for 0 1 and 02 in E q. 2.60 yields

( 1 = 1 . 1 8 9 , ( 2 = 2.173, and Wo = 78.252 eV. It is clear that this simplecalculation removes a substantial part of the correlation error, but the res idualer ror o f 0.762 e V still does not begin to approach spectroscopic accuracy. Betteraccuracy can be achieved by us ing trial functions that are linear combinations ofmany antisymmetrized functions like (2.60), in which a set of linearly indepen-

dent basis functions of spherical symmetry (e.g., 2s , 3 s , 4s , . .. hydrogenl ikeorbitals in addition to 1 s ) i s used for ( / ) , and 4 ) 2 • This procedure corrects the so-called radial correlation error. Incorporation of a sufficient number of suitablenonspherical orbitals (i.e., with higher order spherical harmonics Yi n , in theangular part) as basis functions in such calculations removes the angular

corre la t ion error , and spectroscopic precision has been achieved in this mannerfo r many atoms.

Elect ronic structure calculations in atoms with more than two electronsrequire properly antisymmetrized trial funct ions . For a closed-shell atom with p

electrons having paired spins, such a function can be written in the form of anormalized Slater determinant:

0 1(1)0 (1 1 (2 )*2 ))1(P) 6 (P )

0 1 ( 1 ) ) 6 ( 1 )1 ( 2 ) f l ( 2 )(P)fl(P)

1

2(1)00) 02( 2 0(2 )

2(P)Œ(P)

N/P! Op12( 1 *( 1 )pi2(P)Œ(P)

4p/2( 1)/3(1)p12(P)13 (P )

(2.61)

This determinant vanishes if any two rows are identical, which occurs if any twoelectrons are ass igned the s a m e spatial and spin s tates . The determinant changessign if any two columns are interchanged, corresponding to exchange of tw oelec t rons . Hen ce , both the Pauli principle and antisymmetrization are built into

the Slater de terminant . Note that the determinant (2.61) ass igns the s ame spatialfunction c k i to both electrons in each orbital. Since the many-electron Hamil-tonian has the form

noP

= E Hi E 2 147rE 0 rii

it can be shown that that trial energy becomes [6 ]

(2.62)

wo =<00, 2, • • • , 0 1 /010( 1 , 2 , • • • ,

<0 ( 1 , 2 ,) 1 0 ( 1 , 2 , • • ,

=2H i i +—)]2.63)i ==1




with

Hu= <4itHikki> (2.64)

<WU> = <0i(1)(ki( 2 ) 1 e 2 /4 n sor i210 i ( 1 ) ( 1 ) ; ( 2» (2.65)

> = < Oi(1)0;(2)1 e2/47reori 210;(')02» (2.66)

The mat r ix e l ement s in Eqs. 2 .6 4 through 2.66 are re ferred to as the core integral ,the Coulomb integral , and the exchange integral, respectively. Minimization of

the trial energy by varying the (P i under the constraint that the basis functionsremain orthonormal leads to the Hartree-Fock equat ion [6 ]

p/2

{H a + E [2<4 ; (2)1e 2 /42te o r 1 2 1C(2)> — <0;(2 ) } e 2 /47rEoriziOi( 2 )>]}(ki(1)i= 1

{H ii +2 .1 ; ( 1 )— K ; (1)]} OM) =( 1 )g ii2.67)=11where the c are e l ement s of a (p/2) x (p/2) matrix having units of energy. This

equation is frequently abbreviated as

= E2.68)

where F, called the Hartree-Fock operator, depends on the basis functions (P i

he Hartree-Fock equation can be solved numerically by

setting the equal to e ii S i i , computing the F operator from an assumed set ofbasis function (P i , and us ing Eq. 2 .68 to compute a n e w set of functions (P i . Thesenew functions are us ed in turn to compute a n e w F operator. This cycle isrepeated until the ( P i u s e d for calculating the Hartree-Fock operator converge tothe final solutions ( P i to within desired precision. When this self-consisten t f ield(SCF) limit has been reached, the orbital energies e i may be evaluated from Eq.2.63 ,

gi=Hu+ E ( 2 < m > -

and the optimized trial ene rgy (Eq. 2 .63 ) becomes

p/2

Wo =E (H i i + e )i =1

(2 . 69)

(2 .70)

The spherical harmonics are ordinarily used for the angular part of the orbitals,and the Hartree-Fock equations are solved to obtain the radial wave funct ionsnumerically. The Hartree-Fock wave functions are the bes t radial wave

functions that can be obtained in the form of the Slater determinant 2.61. For



STRUCTURE OF MANY-ELECTRON ATOMS7He, the Hartree-Fock wave funct ion i s equivalen t to the infinite-parametervariational wave function (2 .52) with both electron s in identical spatial orbitals;th e SCF energy of He is —77.871 eV (Fig. 2.7). Slater determinants l ike (2 .61)

and the derivation of the Hartree-Fock equations given here are specific toclosed-shell atoms. For open -shell atoms (e.g., K, F) and for electronicallyexcited atoms, different procedures must be followed .

The difference between the SCF energy and the true nonrelativistic ene rgy isth e correlation error. For first-row atoms , the correlation error is less than 2 % ofth e true ene rgy. This implies that the physical picture offered by the Hartree-Fock t reatmen t—in which each electron e xperiences a centrosymmetric fielddue to the averaged interactions <1/ru > with other e lectrons— accounts for themajor portion of the electronic ene rgy. H owever, the absolute correlation e rroris so large ( 1 . 14 eV for He) that the differences between SCF ene rgies computed

in atoms do not agree well with spectroscopically m easured ene rgy separations.A common ly used me thod of recouping part of the correlation error isconfigurational interaction (CI). In this technique, the ground-state wavefunction is expanded as a linear combination of dete rminants , rather than asingle determinant as in closed-shell Hartree-Fock theory. One of these is theHartree-Fock determinant wave function (2 .61) , and the remainder are determi-nants for excited e lectron configurations. For He, a CI wave funct ion may beexpanded as

1 1 t ( 1, 2) = C 0 A(1s 2 ) + C I A(1s2s) + C 2 A(2s 2 ) + C 3 A(1s3 s) + • • •2 .71)

where A(1s 2 ) is the determina nt des cribing two electrons in iden tical is orbitals,

Table 2.1.s

2s 2p

3pd

4s 4pd 4f

175s 5pd 5f




etc. The e xpansion coefficients C o , C 1 , etc., are optimized in a variationalcalculation. Inclusion of many e xcited configurations in CI determinan t ex pan-sions like (2.71) reduces the radial correlation error to an arbitrary level, as theexci tat ions permit the electrons to becom e more se parated and decrease theaverage electron repulsion e nergy.

This se ction barely scratches the surface of many-electron atomic structurecalculations. T hey have mushroomed in com plex ity to multiconfigurationalSC F calculations in which linear combinations of atomic orbitals (L CA0 s ) areused fo r each of the spatial orbitals ( k i . W ith atomic orbital basis s ets of sufficientsize, close agreeme nt is obtained with experiment.

T he Aufbau or building-up principle by which elec trons are place d insucce ss ive atomic orbitals in many-electron atoms has bee n ex perimen tallyestablished to follow the pattern shown in T able 2.1.

2.4 ANGULAR MOMENTUM COUPLING IN MANY-ELECTRONATOMS

Since filled s hells do not con tribute to the net orbital or spin angular mo men tumin atoms , one ne eds to cons ider only the electrons in unfilled orbitals whencalculating the possible angular mo men ta for a given electron configuration.The simplest case is an atom with two e lectrons in unfilled orbitals. W e us e 1 1

and 1 2 to denote the orbital angular mom enta of the individual electrons, an dlikewise us e s 1 an d s 2 for their spin angular mome nta. In the limit of weak spin—orbital coupling, the total atomic angular mome ntum is compose d by f i rs t

coupling the above vectors to obtain resultants for the total orbital and totalspin angular momenta,

L= 11+ 1 2

S = si s 2.72)

According to the rules for composition of angular mome nta, the possiblequantum numbers L and S for the resultant orbital and spin angular mom entaare

L =1 1 + 1 2, • • • , 1 1 1

S = s 1 + s , s i — s 2 1

T he total angular mome ntum J is then the vector sum

J = + S

and exhibits the possible quantum numbers

(2.73)

(2.74)

(2.75)



ANGULAR MOMENTUM COUPLING IN MANY-ELECTRON ATOMS9This coupling scheme is known as Russel l -Saunders coupling. As an example, wet reat the equ i valen t p 2 configuration in which the two valence electrons have thesame principal quantum number n (e.g., the ground state of a carbon atom thatha s the configuration (1s) 2 (2s) 2 (2p) 2 ). In this case / 1 = 1 2 = 1, so that

L =/ 2 , l i t — / 2 1 = 2, 1, 0

S = s 1 + s2 , i s, — s 2 1 = 1 , 02 . 7 6 )

In an equivalent p 2 atom we thus expect to find S , P , and D s tates (i .e . , L = 0 , 1 ,2 ) ; some will be singlet states (S = 0 ) and some will be triplets (S = 1). Schematicvector diagrams illustrating these resultant angular momenta are shown in Fig.

2.8 . At first sight, one might predict that all possible combinations of L and S

could appear ('S, 3 S, '13 , 3 P, 1 1 3, 3 D)—but some of these combinations willviolate the Pauli principle, and the possible states must be considered more

directly. One may count the ways in which two electrons can legally bedistributed among the equivalent (degenerate) p states; for example, the allowedconfiguration

=— 1 in t =0 m 1 = +1

contributes one state with ML

= m +1 12 = e l ,

A/4 = r n s 1

n s 2 =O .

The totaln u mbe r s of states counted for all combinations of ML and Ms may be tabulated

II1 1 1 1 2

L= 2= I= 0

S= I=O

Figure 2 . 8 V e c to r addition of orbital and spin angular momenta for two valenceelectrons with /1 = 1 , / 2 = 1 in the Russell-Saunders coupling scheme.




in the Slater diagram

AIL

+2+ 1

0

— 1

— 2

1

11

1

1

2

3

2

1

0

1

11

+ 1 Ms

which can be viewed as the sum of the diagrams

1

111 11 +111+1

111 1

1

('S)3 P)D)

T he last diagram, for example, represents a ID state, because it restricts M s tozero (i.e., it is a singlet state) but allows M L to range f rom +2 to — 2. Thus , th epossible states in an equivalent p 2 configuration are I S, 3 P , and D. In each of

these, the possible J values range between L + S and I L — S I ; i .e . , th e allowedterm s ymbols are 1 S 0 , i D2 , 3 P2 , 3 1 ) 1 , and 3 P 0 . T he J subscript is superflous

in singlet term symbols (since only o n e J value is possible for a given L in singletstates , namely J = L) and it is often om itted. The total number of states in anequivalent p 2 configuration is

isPDE ( 2 L + 1 ) (2 S + 1 ) = 1 ( 1 ) + 3(3) ± 5(1) 15L,S

which must equal th e sum of the integers in the Slater diagram. Another way of

summing th e states i s to count (2J + 1) fo r each 2 S + I L ./ term symbol ,

iS 0 12 3 P2 3 13 1 3 P 0

E (2J + 1) = 1 + 5 + 5 +3 + 1 = 15

For nonequ ivalen t p 2 configurations in which th e two electrons are in differentshells (e.g., th e (2p) 1 (3p) 1 excited state of carbon), states e xist fo r all combinationsof the allowed L and S values generated in Eqs . 2 .76 . Slater diagram tabulationsof the allowed (M L , Ms ) combinations are then superfluous; th e nonequivalent

p 2 configuration gives rise to 1 s, 3s, 1p, 3 -1-%,

r i D, and 3 D term sym bols. Slater



ANGULAR MOMENTUM COUPLING IN MANY-ELECTRON ATOMS1diagram tabulations are readily ex tende d to open-shell configurations withthree or more electron s (although they can becom e ted ious). A simplificationoccurs fo r nearly filled (n, 1 ) shells: The possible term sym bols are iden tical to

those for configurations consisting of the absent electrons. For example,equivalent p 5 and p' configurations both exhibit only P- 1/2 and 2 P 3 1 2 termsymbols, and equivalent d 2 and d 8 configurations both yield th e term sym bols1 S 0 , 3 1 3 0,1,2 , 11329 3F 2,3,4 , an d i G4.

T he ene rgy level scheme for a p2 atom can now be described for the case ofsmall spin— orbit coupling (Fig. 2.9), where R usse ll-Saunders coupling applies. Inthe absence of electron—electron interactions, all of the configurations countedin the Slater diagram would have the same energy, because the one-electron p

orbitals with m 1 =0, + 1 a re degenera te . W hen the electron— electron inter-actions are turned on, the s tates with term symbols 2 S + 1 - Lj have differentenergies fo r different L, S . By Hund's rule (the states with th e highest multiplicityfrom a given configuration—in this case p2 — will be lowest in energy), the3 P states empirically lie below the others. W hen th e spin—orbit coupling isturned on, 'the energy levels assume a J-dependence of the form E(L, S)

+ A[J(J + 1) — L(L + 1) — S(S + 1)]/2; in this case A is not ana lytic, becausethe radial wave functions in a many-electron atom are of course no longerhydrogenic.

Rus se ll-Saunde rs coupling applies on ly when th e spin—orbital coupling issmall enough to be treate d as a perturbation to th e electronic Hamiltonian. Inthe limit of large 1 1 an alternative scheme called jj coupling applies. In thiscase , the total angular mo men ta of each electron a re adde d to form th e resultanttotal angular mo men tum J. For a two-electron configuration, we have

+ s i = j i

+ S2 = j22.77)

=2Figure 2.9 Qualitative energy levels for a

/3 2 atom under Russell-Saunders coupling.Electron correlation splits the energy levels

of states with different (L, S); spin—orbitcoupling further splits levels with d ifferentJ.

electron

correlation

spin-orbit

coupling



6 2 ATOMIC SPECTROSCOPY

HSO

J.

I S (0) D(2)

3P

2

O

Oigure 2.10 Correlation between levels underRussell-ussell-Saunders coupling (left) and jj

ouplingSaunderscouplingj-couplingright).

In a p 2 configuration, each electron i can have j = / i + s i , . . . , l — s i l, or j

Then the possible J values are j 1 + 1 2 9 • • •121; this lead s to J =0, 1 , 2 , 3 .However, the last of these values violates the Pauli principle. (Allowing J = 3

would require ji = 1 2 = 4; this me ans that / 1 =1 2 = 1 and s 1 = s 2 = 1, whichwould require that one of the possible states has m 1 1 = ml , = 1 andm s , = nis2 = 1. This would place two electrons with the same spin into the same

orbital). Hence , the only allowed Jvalues in an equivalent p 2 configuration are0 , 1 , and 2. This is reasonable, since J2 is conserved even for large li „ accordingto the commutation relationships in Eqs . 2 .45 . Hence , the J values that are

possible for a given electron configuration in the limit of Russell-Saunderscoupling will be the same as the ones accessible in the limit off/ coupling, and aconceptual correlation diagram can be drawn between J states in the Russell-Saunders and f/ limits (Fig. 2.10) . In the latter limit, the splittings between termene rgies depend primarily on J, rather than on L and S. The noncrossing rule

states that no two ene rgy curves re present ing sta tes with the same symme try(which is indicate d by J, M in the full rotation group of spherical atoms) can

intersect, and this rule is obse rved in Fig. 2 .10 . Most atoms are Russell-Saundersor intermediate coupling cases; few atoms are if-coupled.

2.5 MANY-ELECTRON ATOMS: SELECTION RULES AND

SPECTRA

An interesting exam ple of a many-electron spectrum is that of He, in which theshown low-energy transitions involve orbital jumps of one of the two electrons.For this case our one-electron atomic se lection rules (Al = + 1, Aj = 0, + 1) hold

for the electron involved in the transition. The He electronic spectrum resembles

o



's lp'D 3D 3 F

20,000 -

40,000 -

60,000

80,000 -

100,000 -

120,000 -

140,000 -

160,000 -

180,000 -

—00 ,00V-

MANY-ELECTRON ATOMS: SELECTION RULES AND S P E C T R A3a superposition of two independent alkali spectra [7]: It exhibits two indepen-dent principal ser ies , two independen t sharp ser ies , and so on (Fig. 2.11). Thisoccurs because in He, where th e spin—orbit coupling is very small (Z = 2), th e

electronic states have nearly pure singlet or triplet character. S ince th e electricdipole operator p doe s not contain any spin coordinates, the spin selection ruleA S = 0 is s t r ic t in He . He nce there are two families of E l transitions, one amongthe singlet levels (S = 0 ) and the other amon g the triplet levels (S = 1 ) . T heseparations betwee n lines within fine structure multiplets in this light atom aretoo small to depict in Fig. 2 . 1 1 , and the J subscripts in the level term symbols areomitted in this figure.

T he spin— orbit coupling is much larger in Hg (Z = 80), whose st rongesttransitions are shown in Fig. 2.12. This complicates th e emission spectrum intwo ways. The fine structure levels arising from e ach triplet multiplet ( e . g . , the3 Po, 3 P 1 , a n d 3 P 2 levels arising from the (6 s) 1 (6p) 1 configuration) are now wellseparated in ene rgy, increasing th e number of spect ra l lines observed unde r low

Volts24.47

22

20

1 8

1 6

1 4

1 2

10

8

6

4

2

O

Figure 2.11 Grotrian diagram of energy levels and observed transitions in He. All

shown levels arise from electron configurations of the type (1s) 1 (n0 1 . The 1 S level

labe led "1" is the (1s) 2 ground state. The 1 S, 3 S levels labeled "2" arise from

(1s) 1 (2s) 1 configurations, the l P, 3 P levels labeled "2" arise from (1s) 1 (2p) 1configurations, and so on. Reproduced by permission from G. Herzberg, Atomic

Spectra and Atomic Structure, Dover Publications, Inc., New York, 1944.




Volta

10.38

10

9

Singlets riplets

l - ' . l. F ; 1D, 1F„ 1 3 S, 3P 3P 3, 3D, 3 D2 3D, 3 1 ' c r o '

i_- : - - -. = - -- = -= ,9d fil0 s,9 9p----9p-d8d-6f_8 d7-6-8d

7 d VI" —89.....---89-.--.-...89--- 7 dddf

._0s10,000-'

•:) \ l,6d9dI

I

8

7

6.67

6

5.43

54.86

4.64

4

3

2

1

0 4.

Figure 2.12 Grotrian diagram for low-lying Hg levels with the configuration • •

(6s) 1 (n1) i . Excited levels are labeled with the quantum numbers of the valence

electron which is excited; for example, th e 1 P 1 level labeled "7p" arises from the

electron configuration • (6s) 1 (7p) 1 . Reproduced b y permission from G. Herzberg,

Atomic Spectra and Atomic Structure, Dover Publications, Inc., New York, 1944.

resolution. The large spin—orbit coupling in Hg also endows the electronicstates with mixed singlet/triplet character, with th e consequence that manyintersystem transitions ( A S = + 1) are observed. Thes e factors lend the Hgemission spectrum in Fig. 2.2 an appearance of irregularity which is absent in theemission spectra o f H and K.

Strong El transitions in many-electron atoms a re observed only when on eelectron changes it s orbital quantum numbers; fo r this e lectron, th e selectionrule Al = + 1 must be obeyed (cf. our discuss ion following E q. 2.12). T oappreciate this, we rec all that spatial wavefunct ions in many-electron atomsmay be expressed (Sect ion 2.3) in terms o f products ik(1, 2 , ... , p) = 0 1 (1)0 2 (2)

0 1 ,(p) of one -electron orbitals C(1 ), 4 ' 2 (2), , O p (p). Since the pertinent

electric dipole operator is p = — eZ r i , the El transition moment fromelectronic state 0(1, 2, ... , p) to state On 2, ... , p) = 4 ( 1 )0 '2 (2) ... O p (p )



MANY-ELECTRON ATOMS: SELECTION RULES AND S PECTRA5behaves as

—e <4 1(l)4' 2 (2) • • O p ( p) C (1 ) 4 Y 2 ( 2 ) • 4 );p ) >

= — e < 0 1 ( 1 ) 1 1 . 1 1 0 / 1 ( 1 ) > < 4 ) 2 ( 2 ) 1 02(2) >

—e<02(2)1 r2 1 0 ` 2 ( 2)X 4 ) 1 ( 1 ) 144( 1 )> • • < O p ( P ) 1 4 Y „ , ( P ) >

—e<Op( p) I r p l d ) ;( p)>< C( 1 ) 1 44 ( 1 ) > • • < O p- tp — 1 ) 1 44-1(p — 1 ) >

Each of these terms contains a factor <0 1 ( i ) t r i l4i ( i ) > analogous to the hydrogen-like transition moment (2.10), multiplied by orthogonality integrals< ( / ) i ( j ) 1 4 4 jp fo r all other electrons j i . Hence, if electron i jumps to a new

orbital, i.e., (ni ) NO, orthogonality requires that all other electrons remain intheir original orbitals because <C(j )1(/);(j)> = 0 unless ( / ) . ; = 4.

T h e E l selection rules may also be formulated in terms of the quantumn u mbe r s L , S , and J for the many-electron orbital, spin, and total angularmomentum. These prove to be

El

AL = 0 , ±1xcept L = 0 — x — =

AS = 02.78)

AJ 0, ±1xcept J = 4 — x — * J' =0

Not all t ransi tions con sistent with these selection rules are El-allowed, sinceAl = +1 must be simultaneously obeyed by the electron that jumps. Forexample , a transi t ion from a state with L = 3 , / 1 = 3, 1 2 = 1 to a state withL = 2 , l = 1, 1 2 = 1 is forbidden because A/ 1 = —2 and A/ 2 = 0, even thoughthe AL = 0 , + 1 selection rule is obeyed.

The corresponding M 1 and E2 selection rules are

M1

E2

except J =0' =0

(2.79)

A. / = 0, + 1, + 2 and (J + J') 2AL = 0 , ±1, + 2 except L = 0 4 — x — L =02.80)

AS = 0



66T6mIC SPECTROSCOPY

In all case s, the AS = 0 se lec t ion rule becomes re laxed when th e spin—orbitcoupling is large.

T o illustrate one final point, we return to the example of the equivalent p 2

configuration, for which the possible atomic term s ymbols are 1 S, 3 1 3 0 , 1 , 2 , and

'D . The El select ion rules in Eq. 2.78 indicate that there are no El-allowedtransitions conne ct ing any two of these leve ls with different L if the spin—orbitcoupling is small. This is an example of a gene ral rule that no El t rans i t ionsconne ct two term symbols aris ing from the same electron configuration (in thiscase, p 2 ) . E l-allowe d transi t ions can , howeve r , occur betwee n s ta tes f romdifferent electron configurations, say s i p' p 2 . This proves to be a usefulprinciple that carries over to the electronic spectroscopy of diatomic andpolyatomic molecules.

2.6 THE ZEEMAN EFFECT

W h e n a to m s are subjec ted to e xternal magne t ic fields, their spectral linesbecome spli t in to several components whose s epara tions de pend on the fieldstrength B . In atoms with zero net electronic spin (S = 0), the splittings turn outto be iden t ical fo r all spectral lines. For weak, s tatic magnet ic fields B , theinteraction energy for a spinless atom o f angular momentum J = L with thefield is

W = —p„,-B. 2m e LB2.81)

and i t may be t reated as a stationary perturbation to the atomic H amiltonian1 1 0 . The first-order correction to the electronic energy of any level characterizedby angular momentum quantum numbers (L, M L ) in the presence of a magneticfield B = B is

— 2me< LM L I L - B I L M L > =

2me

Bz<LMLILILML>

eh=

M

2m e zL(2.82)

Since ML ranges from + L to — L , a level with orbital angular momentum Lbecom es split into (2L + 1) components se para ted in en ergy by the amountehB z /2 m e . A spectral line that appears at en ergy AE in zero magnetic field thenappears at energy

ha) = AE +h B AM2m e (2.83)



oI P

M L

2e tt B z /2me

0

—

—2

TH E Z E E M A N E F FE C T7in the presence of the field, where AM L is the change in M L accompanying the

electronic transition. The El selection rules on AM L (Eq. 2.78) then im ply thattwo ne w l ines w ill appear for AM L = + 1, in addition to an unshifted line

correspon ding to AM L = 0 (Fig. 2.13). The only exception to this n om inallyoccurs when both electronic levels in the transition are S states with L = 0, butsuch a transition would be El-forbidden to begin with. Such splittings werepredicted c lassically by Larmor many ye ars before they we re observed in atomswith S = O . This phenomenon is historically known as the normal Zeeman effect.

In atoms with nonvan i shing spin as well as orbital angular mome ntum, the

situation is far more interesting. The Hamiltonian for such atoms in externalmagnetic f ields assumes the form

11=flo +f (r)L -S—p m •B2.84)

where the second and th ird te rms represe nt the spin—orbit and ZeemanHamil ton ians , respectively. (The function f(r) in the spin—orbit H amiltonian

B z = 0 Figure 2.13 Transitions observed between 'D andtates under the normal

Zeeman effect. Since the separation between adjacent sublevels is invariablyAt , 1 2m e and the selection rule restricts AML to0, +1, spectral lines appear at only

two newfrequencies when the magnetic field is turned on.




depends on ly on the electronic radial coordinates.) T he atomic magneticmoment Pm i s now given by

eLgeS

=2m, 2me

!IL+ P s

2.85)

where the electron spin g factor is approximately given by g s c s . _, 2 . 0 . Th e Z eemanHamiltonian

= — fi. • B2 .86)

fo r B =ould be taken to be

= (L. +me (2.87)

which could then be use d in principle to calculate the ne w levels using first-orderperturbation theory in weak magne t ic fields. The problem here is that in thepresence of spin—orbit coupling the atomic stationary states are noteigenfunctions of L z and S z , due to the commutation rules (2.45). He nce, Eq. 2.87cann ot be use d to calcula te the f irs t-order Zeeman ene rgies directly. In the

classical view of R usse ll-Saunders coupling, L and S prece s s about J, which inturn preces s es about the magnetic field direction 1 as shown in Fig. 2 .14 .(This corresponds to the quantum mechanical picture in which J z is station ary,bu t L z and S z are indefinite.) Since P L and ps are proportional to L and S,respectively (Eq. 2.85), PL and ps also prece s s about J, with the consequence thatonly the component s of P L and ps directed along J con tribute to the expectation(averaged) value of pm = P L + ps . T he vector along J with len gth equal to sumof these compone nts is

1 k 1 =s)•JP/J 2

2.88)

T h e Z e e m a n H amiltonian then e ffectively becom es

flz = because components of PL and lis normal to J average out to zero duringprece ss ion. Now the fact that

1/ . + Ps) • Je (L + gsS) ' • yi2me (2.89)



T H E Z E E M A N EFFECT98,

Figure 2.14 Angular momentum vectors L, S, J (left) and their associatedmagnetic moments pL , fis' p, (right). Under Russell-Saunders coupling, the orbital

and spin angular momenta L and S precess about the total angular momentum J. The

magneticmoments 0,, ps (which are

antiparallel to Land

S) undergo similarprecession. Since gs 2, we have Ip s i/ISI 2Ip L VI1I, with the consequence that the

vector sum O h + ps ) is not parallel to J. The effective g factor g, is the projection of

(pt + i 's) upon the direction of J, since precession rapidly averages the other

components of Oh + f is) tozero.

combined with th e identities

allows us to write

S 2 = (J — L) 2 = J2 + L 2 — 21 4 -. I

L 2 =(J — S) 2 = J2 +52_

2S J (2.90)

Pr =

e J2 + L 2 — S 2 + g ,(J 2 + S 2 — L 2 ) jJ2(2.91)

Replacing J 2 , L 2 , an d S 2 with h 2 J(J + 1), h 2 L(L + 1), an d h 2 S (S + 1 ), re -spect ively, yields

g jeJ (2.92)J = 2m e




with the e ffective Land& g factor

(2.93)

Using the approximate value 2 .0 for the electron spin g factor g s res ults in thecommon expression [1]

J(J + 1) + S(S + 1) — L(L + 1)(2.94)

2J(J + 1)

Finally, the Z ee man correction to the ene rgy in a state with angular mome ntumquantum numbers L, S, J, M, becomes in the weak -field limit

eg,—<LSJM,Iii n i -BILSJM„> = —

2 m ,< L S J A / JIJ • BILSJM,>

g jeh=z M,

2 m ,(2.95)

T he remarkable property of the effe ctive g factor g, is that it depends on L and Sas well a s on J (Eqs . 2 .93 and 2.94). In the limits whe re L = 0 and S = 0, g,reason ably approaches g s and

unity, respectively. Ifg s

were unity rather thanapproximately 2 .0 , g, would also be unity according to Eq. 2.93; p i n would thenbe pa rallel to J. Since the orbital and spin angular momenta have different gfactors, pn , doe s no t point along J (Fig. 2.14), and the Z eeman energies dependon the orbital and spin as well as total angular mom entum quantum numbers.They are independent of the radial quantum numbers. Since atomic spectral linesin the abse nce of external fields do not appea r with little flags e xclaiming "I camefrom J =4, L = 3, S = 1 ," this anomalous Zeema n ef fect observed in atoms withnonvanishing L, S has proved to be an invaluable experimental tool forassigning them. T he fact that g, depends on ly on L, S, and J i s known as

Preston's Law.Figure 2 . 15 illustrates the anomalous Z e e m a n effect i n Na atom transit ionsfrom the 3 2 S 1 1 2 ground state to the lowest lying ex cited fine structure levels3 2 P 1 1 2 an d 3 2 P 3 1 2 . (These are the well-known D line transit ions which occur at5 895 .93 and 5889 .96 A in zero field.) According to Eq. 2.94 , the respec tive 9,values for the 3 2 S 1 1 2 , 3 2 1 1 1 1 2 , and 3 2 P 3 1 2 terms sym bols are 2, 4 , and 4 ; a weakmagnetic field splits the fine structure levels into 2J + 1 = 2, 2, and 4componen ts , respectively. T he El transitions displayed in Fig. 2 . 15 obey theselection rules AJ = 0, ±1 and AM . , = 0, + 1. T heir wavelengths may beanalyzed using Eqs. 2.94 and 2.95 to deduce the angular mom entum quantum

numbers a priori.

( 1 + gs)J(J+ 1 ) + (1 — g s ) [ L ( L +1 ) — S ( S + 1 ) ]=J(J + 1)



2P3/2

M j=

3/2

1/2

- 1/2

- 3/2

g = 4/3

1/2

g = 2/3

PROBLEMS12P

1/2 _

2S1/2

-1/2

g = 2

Figure 2.15 Transitions observed between a 2 S 1 1 2 state and 2 P 1 1 2 and 2 P 3 1 2 states

under the anomalous Zeeman effect. Since the Lande g factor gj= 1 +

[J(J +1) + S(S + 1) L(L + 1 )]I2J(J +1) has a different value in each of these

electronic states, a unique frequency is associated with each of the shown

transitions. These frequencies can be analyzed to infer the angular momentum

quantum numbers for each electronic state.

REFERENCES

1. E. U. Condon and G. H . Shortley, The Theory of Atomic Spectra, CambridgeUniversity Press, London, 1970 .

2. M. H. Nayfeh and M. K. Brussell , Electricity and Magnetism, Wiley , New York, 1985.

3. E. Merzbacher , Quantum Mechanics, Wiley , New York, 1 9 6 1 .

4 . K. Gottfried, Quantum Mechanics, Vol . 1 , W. A. Benjamin, New York, 1 9 6 6 .

5 . W . H. Furry, Am. J. Phys . 2 3 : 517 (1955) .

6 . W . H . Flygare, Molecular Structure and Dynamics, Prentice-Hall, Englewood Cl i ff s ,NJ, 1978.

7. G . Herzberg, Atomic Spectra and Atomic Structure, Dover , New York, 1944.

PROBLEMS

1 . Consider the 2 D 3 1 2 and 2 D5 1 2 f ine st ructure levels in a one-electron atom.

( a ) Express the coupled s tates lisjm> in terms of the uncoupled sta tes 1 / m 1 sm s >

fo r these levels by wo rking out the Clebsch-Gordan coefficients using




angular momentum raising and lowering operators (there are 1 0 of thesecoupled states in all).

(b) Determine which of the 2 4 transitions between 2 D 3 1 2 and 2 D512 finestructure levels are M 1 -allowed. Can you extend your reasoning to obtaingeneral selection rules on A i and Am for M 1 transitions?

2 . In a hydrogenlike atom, the magnitude of the unperturbed Hamiltonian Ro

is on the order of Ze 2 /47re0 r, and that of the spin—orbit Hamiltonian ils o iseze2/87re0me2c2r3. Considering that r is typically equal to a o /Z (where a o is theBohr radius), calculate the numerical magnitude of fi s cilio , and comment on the

validity of using perturbation theory to describe spin—orbit coupling. (The fine

structure constant a =e 2 /47re 0 hc is equal to 1 / 137 .0372 ; the Bohr radius isao = 47rE o h 2 /m e e 2 .)

3. Which of the Hg atom levels shown in Fig. 2.12 are metastable?

4 . Use the angular momentum commutation rules [L x , L y ] = ih L z , [L ,

L z ] = i h L „ , [Li , Lx ] = ih L y , [L, L2 ] = 0, along with the analogous rules for the

components of S and J, to prove the commutation relationships (2.45) that holdin the presence of spin—orbit coupling. Take the spin—orbit Hamiltonian to be

H 0 = f ( r)L • S, with f(r) a function of the radial coordinates only.

5. Determine the atomic term symbols 2 S + 1 L, that may be obtained from theequivalent p 3 configuration us ing a Slater diagram. Ascertain explicitly whetherthere are any El-allowed transitions among sublevels of this p 3 multiplet.

According to Hund's rule, what should be the term symbol fo r ground state N?6. Which levels in K could be reached by E 2 transitions from the 4 2 S 1 1 2ground state?

7 . An El transition a t 3 1 2 5 . 6 6 A connects the 6 p 3 P 1 and 6 d 3 D 2 fine structurelevels in Hg. A weak magnetic field spli ts these levels via the anomalous Zeemaneffect . Evaluate the g j factors for both levels, indicate all of the El transitionsthat will occur using a schematic energy level diagram, and draw the resulting"stick spectrum" on a wavelength scale.



3RO TATION AND VIB RATION

IN DIATO M IC S

M a n y of the features that are prominent in atomic spectroscopy have closeanalogies in electronic s pectroscopy of diatomic molecules . Like atoms , dia-tomics ex hibit electron corre lation, s pin— orbit coupling with its asso ciated fine

structure and angular mom entum coupling schem es , and the Zeeman effect. Indiatomics as in atoms, the symmetry of the electronic H amiltonian plays amajor role in the elect ronic s ta te degen eracies and se lect ion rules. Thedistinctions encountered here in going from atom s to diatomics are rather trivialone s that accompany the reduction of spherical symm etry in atoms into theand 1),o t point group sym metries in diatomics.

These analogies fail to ex plain the coarse structure observed in diatomicelec t ronic spec tra unde r low resolut ion (Chapter 4), because the additionalnuclear coo rdinates int roduced by the second atom in a diatomic molecule ABcreate new internal modes that have no counterpart in atoms. The nuclearpositions relative to an arbitrary origin fixe d in space can be specified using theCartesian vectors R A and R B (Fig. 3.1). The y may be e quivalent ly described interms of the coordinates

R e m = (MA R A + M B R B )/(M A + MB)3.1)

and

R RB R A3.2)

where M A and MB are the nuclear masses. The vector Re m locates the nuclearcenter of mass; R is a vector whose length R gives the internuclear separation

73



74OTATION ANDVIBRATION IN DIATOMICS

Figure 3.1 The nuclear positions in a diatomic

molecule may be specified in terms of either thelaboratory coordinates RA, RB, or the center-of-mass coordinates R c m , R .

and whose direction k gives the orientation of the in ternuclear axis in space. It

can readily be shown [1 ] (see also Appendix G ) that th e kinetic energy

T =-IMA l k i2t + IM B I k 43.3)

of nuclear motion in Cartes ian coordinates becomes transformed using Eqs. 3.1and 3.2 into

T =IN12=

Im1tc2 in It1 2 IiR20 23.4)

in terms of the center-of-mass coordinates R c f f , and R . The masses appearing in

Eq. 3.4 are the total nuclear mass

M = MA ± M B3.5)

and the nuclear reduced mass

MAMBAMA + M B )3.6)

The first term in the kinet ic energy (3.4), asso cia ted with cente r of m a s stranslation, does no t affect the internal ene rgy levels in the molecule. The 1 1 1 N / 1 2

term is the vibrational kinetic en ergy arising from changes in the diatomic's bondlength R . The last term, 1 1 . L N R 2 6 2 , is th e nuclear rotational kinet ic energyconnected with molecular rotation through an angle 0 about an axis normal tothe molecular axis . The usefulness of the center-of-mass coordinates in Eq. 3.4 isthat they se parate the total nuclear kinetic energy into contributions from theoverall molecular translation and the internal modes (vibration and rotation).These in te rnal modes in diatomics are responsible fo r spectroscopic transitionsoccurring in the microwave to near-infrared regions of the electromagneticspectrum.

An example of a far-infraredabsorption spectrum of a diatomic gas is shownin Fig. 3.2. The vertical coo rdinate is l ight inte ns i ty trans mitted by an H C1



ROTATION AND VIBRATION IN DIATOMICS51 0 0 200

wavelength (0

Figure 3.2 Far-infrared absorption spectrum of HCI gas. The vertical coordinate is

light intensity transmitted by the gas sample; the horizontal coordinate is wave-

length. Reproduced by permission from D. Bloor et al., Proc. London Ser. A 260: 510

(1961).

sample after passing through a scanning monoch romato r operating at wave-lengths A betwen —9 0 and 3 0 0 p m . The intensity envelope peaking near 130 ,umis determined primarily by wavelength-dependent light transmission (techni-

cally, the grating blaze) in the monochromator . The characteristic HC1 ab-sorption lines appear as minima at = 2 4 0 , 1 6 0 , 1 2 0 , and 96 pm . The regularityin this sequence of l ine s is more striking when these wavelengths are convertedinto frequencies i , = 1 / .1 = 41 .7 , 6 2 .5 , 8 3 .3 , and 1 0 4 cm', respectively: Thesef requencies are uniformly spaced by 20.8 cm'. The ene rgy changes associatedwith these absorption lines, which arise from transitions between differentrotational levels in HC1, are smaller than those observed in the atomic linespectra in Fig. 2.2 by factors of typically 1 0 3 .

The far-infrared spectrum in Fig. 3.2 exhibits poor signal-to-noise ratios(S/N) and low resolution in comparison to other forms of molecular spec-

troscopy that will be studied later. As far-infrared spectra go, it representednearly state-of-the-art technology in 1 9 6 1 . The experimental difficulties of far-infrared spectroscopy stem f rom the low intensities available from continuumlight sources at such wavelengths (a 1 0 4 K blackbody emits only 1 part in 1 0 8 ofit s total output be tween 150 and 35 0 pm ) and f rom the low efficiency ofspecialized light detectors (e.g., Golay thermal detectors, superconductingbolometers) that can be used in this regime. The linewidths displayed in Fig. 3.2(def ined as full width at half maximum, or fwhm) are on the order of 5 cm '.They are not intrinsically a property of the H C1 gas, but arise from the neces s i t yof relaxing the monochromator resolution (using wider entrance and exit slits)

to bring the signal to a detec table level . Far higher resolution is now routinelyavailable in this wavelength regime using Fourier transform techniques.



3000900

800

700

HC 1

1 8 0.-040\

1 6

14 2

1 2R(0) NI)

8 6 WAVE NUMBER,cm— I

Figure 3.3 Near-infrared absorption spectrum of HCI gas: transmitted light

intensity versus frequency in cm'. Reproduced by permission from Pyler et al., J.Res. Nat . Bur. Stand. A 64 : 29 (1960).



THE BORN-OPPENHEIMER PRINCIPLE7HC1 gas also exhibits a ser ies of absorption lines in the near- infrared, as

shown in Fig. 3.3. The S/N in this spectrum is considerably higher thanin the far-infrared spectrum; the fwhm linewidths in Fig. 3 . 3 are approximately1 cm'. This reflects on the relative facility of generating and

de t ec t -ing light with near-infrared wavelengths; the ene rgy changes associated withthe transitions in Fig. 3.3 are larger than those in the far-infrared spectrum byfactors on the order of 10 2 . The absorption l ines in the near-infrared HC1spectrum are grouped into closely spaced doublets separated by 2 cm -1 . Itwill be shown that these doublets appear because HC1 naturally occurs in tw ocommon isotopes, H"Cl and H 3 7 C 1 , with 75.5 and 24.5% abundance, re-spectively. (Under sufficient resolution, the far-infrared absorption l ines in Fig.3.2 would similarly be split into isotopic doublets.) The near-infrared spectrumresults from simultaneous rotational and vibrational level changes in HC1 , and

is called a vibration-rotation spectrum. The frequency spacings of — 2 0 cmbe tween successive absorption l ines are similar to those occurring in the far-infrared spectrum, although they are not nearly as constant as in the purerotationspectrum of Fig. 3.2. The vibration-rotation spectrum shown in Fig. 3.3is centered near 29 00 cm -1 ; similar (but weaker) families of vibration-rotationlines can also be observed at frequencies near multiples of 29 00 cm '.

These spectra can be analyzed to yield detailed information about themolecular bond length, about the potential energy function that governs the

vibrational motion, and about the interactions between vibrational and rota-tional motions which résult in smal l corrections to the total molecular energy. A

remarkable property of pure rotational spectra is that no knowledge ofelectronic structure is required to interpret them. This situation contrastsstrikingly with that in atomic line spectra, where extensive configurationalinteraction calculations must be performed to correctly predict spectral linepositions. The implication here is that electronic and nuclear rotational motionsare in some way essentially decoupled. Vibrational and electronic motions of tenprove to be separable in a similar sense, although details of electronic structuredo influence vibrational motion via their effect on the vibrational potentialenergy function. The concept of separability between electronic and nuclearmotions fo rms the basis of the Born-Oppenheimer approximation, which we

di scus s in Section 3.1. In the remainder of this chapter, we build on thefoundation provided by the Born-Oppenheimer principle to develop theeigenfunct ions , energy levels, and spectroscopy of diatomic rotations andvibrations.

3 .1 THE BORN-OPPENHEIMER PRINCIPLE

The Born-Oppenheimer principle is a cornerstone of molecular spectroscopy, anorganizing principle that vastly simplifies the assignment of di f ferent spectral

features to different types of molecular motion. Without it , electronic and

nuclear motions would be scrambled in complicated molecular Hamiltonians ,



78OTATION AND VIBRATION IN DIATOMICS

and exte nsive num erical calculat ions would be nece ssary to extract even the

most qualitative features of vibrational and rotational st ructure— as must nowbe done to compute ene rgy levels that are influenced by correlation in many-electron atoms. The well-known approximations to diatomic vibrational androtational e ne rgy levels [E, ho(v + I) and E, hcBAJ + 1)] would have no

reality. The Born-Oppenheimer principle some times breaks down, and the casesin which it does so have attracted considerable research interest from peopleinterested in molecular dynam ics and energy transfer. Examples of suchbreakdowns are vibronic coupling e ffects in electronic spectroscopy of polynu-clear aromatic hydrocarbons , and nonadiabatic transitions in reactive mole-cular collisions.

The total Hamiltonian for a diatomic molec ule with n electrons is

22 —h2 2fl = E+ E+

i = 1 2m,

7rE c ,

r i — ril

= 1 2MN

1 n2N e 2ZAZBe2E ER s o3.7)4 ne 0 i= 1 N = 1 Ir i — R N I 4720 IR A — R B I

These contributions to il include the nuclear kinetic ene rgy, the electronickinetic energy, the electron— electron repulsions, the electron— nuclear attrac-tions, the nuclear—nuclear repulsion, and the spin— orbit coupling. A priori, theelectronic coordinates r and nuclear coordinates R appear to be inse parablymixed in H , and the electronic and nuclear coordinates are strongly coupled.The Schrödinger equation for the diatomic become s

M r,)> = EIT(r, R)>3.8)

where r and R represent the sets of electronic and nuclear coordinates ,respectively.

W e now def ine the electronic Hamiltonian fi e l as22

He i HN = 1 z-MN'N SO

3.9)

R e ' differs from II by lacking the nuclear kine t ic energy and spin—orbitoperators Î and Ito . It is possible to find eigenfunctions of IL for fixed R ( i . e . ,for nuclei motionlessly clamped in position), such that

1 1 .1(1 , R ) M a r ; ek(R)10k(r; R)>3 .10 )

The resulting e lectronic states 1 1 / / k (r; R ) > will depend parametrically on R in thatthe choice of fixed nuclear positions influences the electronic states (physically,

pulling the nuc lei apart will naturally dis tort the molecule's electron cloud). Thek(R) are the diatomic potent ial ene rgy curves — the electronic e nergies of the



T k =

Tk" k , =

U k k , = < 11l ±ollfr k'›lc(R) (5 k k '

- h 2.7Wt., • VR

211N

—h 2kk '

dk i e —

iV

'›

ntfrklViltPle>

(3.14a)

(3.14b)

(3.14c)

(3.14d)

(3.14e)

THE BORN-OPPENHEIMER PRINCIPLE9various s tates k as functions of the internuclear separation R . Numerouspotential energy curves are shown fo r several diatomic molecules in Figs. 3.4,3.10 , 4 .2 , 4 .6 , 4 .7 , 4 .8 , 4 .9 , 4 .14 , 4 .29 , 4 .30, and 4.31 in this book.

For the solution s Y(r, R)> to the full diatomic H amiltonian , we n ow t ryex pansions of the form

R ) > =k l i k k ( r ; R ) > I X I A R ) >3 .11)

where the l x k (R)> are the diatomic nuclear (vibrat ional— rotat ional) wavefunctions in electronic state k and the a k are e xpansion coeff icients . If we havecomplete sets ' ,P k > and l x k > , IT(r, R)> can always be expanded as in Eq. 3 .11 . Itis only when IT(r, R)> can be well approximated by a single term of the forml i / i k > l x k > that one can be said to have achieved separation betwee n electronic an d

nuclear motion, by factorizing the total wave funct ion in to e lect ronic andnuclear parts. Such a factorized state is called a Born-Oppenheimer state.

If we subst itute from Eq. 3.11 fo r jkli(r, R)> into the diatomic Schrödingerequation using the Hamiltonian of Eq. 3.7, we obtain [2 ]

ak

[

„ . 7 4 2 Ukk — EllX k(R»N = 1 ,LMN

= - E T Z k 'kliXk'(R)>3.12)k ' * k

According to Eq. 3.4, the nuclear kinetic energy operator may be recast in termsof center-of-mass coordinates,

2222V=_B 2

N = 1 2MNM AMB

h 2çj

V

v2

2M' 2 p . N R(3.13)

Sincethe electronic and nuclear wave functions are

independentof the

center-of-mass position R c „„ the term in T involving V e r n drops out in Eq. 3.12. T he newquantities introduced in Eq. 3.12 are defined as




T he operators VR and V i a re given by

a a k . VRR +0 + Rin 0 0 0

1( i 12004._in22" R 2—3 .15 )R — R 2 ORR/ ± R2 s in 0 00 n 00) where (0, 4)) define the orientat ion of the internuclea r axis in space, R is the

MX

M - FX-

Figure 3.4 Lowest two bound potential energy curves for an alkali halide molecule

MX. The zeroth-order ionic and covalent curves, which intersect at the crossing

radius R = Re , are shown at left. The adiabatic curves e. (R) and i 2 (R) are shown at

right; the avoided crossing at R = R e is caused by mixing of the degenerate ionic and

covalent states, which have the same symmetry. The lower adiabatic state Ji1 (r; R)>

describes ionic II/1+X - for R < Re , but correlates with uncharged separated atoms

M + X for large R. The higher adiabatic state 1 0 2 (r; R)> describes covalent MX

(which is far more weakly bound that ionic MX) for R < Re , and correlates with

M + X - at large separations.



THE BORN-OPPENHEIMER PRINCIPLE1internuclear separation, and ( r , f k') in this con text are unit vectors associatedwith the coordinates (R, 0, 4)).

Equation 3 .12 is actually an infinite s e t of coupled equations that tell us thatnuclear motion generally takes place in all electronic states simultaneously

(since motion in a given state k is coupled by the right side to motion in all otherelectronic states V ). W hen this is true, the electronic and nuclear motions cannotbe separated.

W e now brief ly consider the special case in which the right sid e of Eq. 3 . 12 isnegligible. The quantity Tja, on the left side of Eq. 3 .12 is usua lly sm all (i.e., the

electronic states 1 1 / 4 > do no t norm ally os cillate violen tly with internuclear

separation R! ) and so in this case we obtain

[—2 V iz ± k k (i.) — Liak(it)

2 [ I N

(3 .16)

Wh en the spin—orbit coupling is small, U k k (R ) = s k (R) according to Eq. 3 .14a:

U k k (R ) = <441 1 /0 1 1 f rk> = s k (R). Then E q . 3 . 16 becomes a Schrödinger equation

Figure 3.4 (Continued)




fo r nuclear motion (vibration and rotation) in the potential U k k (R ) specified bythe potential ene rgy curve e k (R ) of electronic state k. This eigenvalue equationthen leads to the dia tomic rota t ional and vibrational s tate s as they arecommonly understood in molecular spectroscopy. W hen Eq. 3.16 is approx-imately valid, the Born-Oppenheimer principle applies .

W e now consider reasons why the B orn-O ppenheimer approximat ion canbreak down , i.e. , why the right side of Eq. 3.12 can become appreciable. SinceUkk' = klitoliPk'› fo r k k', large s pin-orbital Hamiltonians Ito can be aculpri t—and Bo rn-Oppenheimer violations can inde ed re sult f rom large spin-orbit coupling. T he term Tk k r in Eq. 3 .14 is gene rally small, for the same reasonthat Tk is usually negligible. So we now focus on rkk. This term can becomelarge when the potential en ergy curves fo r two electronic s tates k and k 'approach closely, causing nohadiabatic transitions from nuclear motion in statek to nuclear motion in state k '.

As an exam ple, we discuss the approach of an a lkali atom M and a halogena tom X . T he lowest two potential en ergy curves e i (R) and 8 2 (R ) fo r the alkalihalide M X a re schem atically shown in Fig. 3.4. T he lower of the two states1 1 / / 1 (r; R)) correlates with the ground-s tate ne utral alkali atoms M + X , and ha s'V symmetry. T he upper state 10 2(r; R ) > correlates w ith the ions M + Xand also has 1 E + symmetry . T he two potential en ergy curves s 1 (R ) and 8 2 (R )exhibit an avoided crossing at R = Rc , in consequence of the noncross ing rule[3 ] for curves corresponding to states of the same symme try. Because of thisn o n c r o s s i n g rule, the adiabatic s tates M I > and 10 2 > rapidly change in characternea r R = R. In particular, 1 0 1 > describes ionic M 4 " X - fo r RR—since thedeep well in e l (R ) res ults primarily from Co ulomb attraction

between the ions atsmaller separat ions— but corresponds to covalent MX for R Z,R c , wherecovalent M X has lower ene rgy than ionic M ÷ X -. Conversely, W I 2 > switchesfrom covalent to ionic character as R, i s passe d going outward; s 2 (R) ha s ashallow well since covalent M X i s cons iderably m ore weakly bound than ionic

In the neighborhood of R c where the potential curves 8 1 (R ) and 8 2 (R)approach closely, Tk k, become s large fo r k = 1, k' = 2, because the quantity

d 1 2 = < l lVRI2> =r 1 0 2 >

3.17)

is large in this region where 1 4 / 1 > and 1 0 2 > are switching betwee n ionic andcovalent be havior.

The implications o f this for collisions between the neutral atoms M + X areas follows. T he collision starts out at large R in state It P i > (corresponding toM + X ). As the atoms approach at R » R c , the system remains in state 100,because in this region the right side of Eq. 3.12 is sm all. But as R approaches R c ,

d 1 2 become s appreciable, and then

- h2

r121 1 , N U12 • VR (3.18)



DIATOMIC ROTATIONAL ENERGY LEVELS AND SPECTROSCOPY 83

can become important if the atoms approach with sufficient relative velocity(which is pro portion al to V R ) . At this point, Eq. 3.12 informs us that the systemha s a finite probability of hopping to s tate 11//2> near R = R , . If i t does, th e atoms

will mom en tarily formcovalent MX (following curve c 2 (R)) on closer approach,

R R, . If the curve-hopping fails to occur, the system remains adiabatically instate Ili/ > , and forms ionic M + X when R become s s maller than R , . T heprobability of this curve-hopping (which amounts to a breakdown of the Born-Oppenheimer principle) depends on the relative nuclear velocity: the larger V R

is, the more probable the nonadiabatic jump. The widespread use of Eq. 3.16 inmolecular spectroscopy rests on the ass umption that such nonadiabatic trans-i tions ne ver occur, and that nuclea r motion i s re s t r ic ted to motion on onepotential ene rgy curve (or surface in polyatomic molecules) corresponding to asingle electronic state. This is valid in the limit of small spin—orbit coupling, and

when the elec t ronic potent ia l energy surface in question is not closelyapproached by other surfaces at the total (electronic plus nuclear) energies ofinterest.

Interested readers a re referred to the chapter by J. C. Tully in Dynamics ofMolecular Collisions, Part B, edited by W . H . Miller (Plenum, New York, 1976)fo r additional information on this subject .

3.2 DIATOMIC ROTATIONAL ENERGY LEVELS AND

SPECTROSCOPY

In the remainder o f this chapter , we assume that the Born-Oppenheimerapproximation is good, and that Eq. 3.16 holds. In this sec t ion we consider th erotational motion of a idealized rigid diatomic rotor, in which R is f ixed at R o .

Then U k k (R ) = U kk. o ,1 is a constant that we m ay s e t to zero for convenience.Using Eq. 3.15 for V I in the Schrödinger equation (3.16), we imme diately have

•- 2[ 1 (R2 a2LR2 0RR) h2R2iix-t )

J22 1 2 N R6

I xrot( 0 ,Erotlx,00, (3.19)

where we have used th e fact that rotor's rotational angular mom en tum operatorPis

122 = —h2[ 1 a (sin e a ) +3.20)s in e a eein 2 e 802

and the fact that terms in aiaRdrop out when R is held constant. He n ce the rigid

rotor Hamiltonian is

'tot = .12 /2/1NR6 = P/213.21)



Erot = 2/

J(J + 1)h


where I is the rotational moment of inertia, I = , t iN n. The eigenfunctions ofare the spherical harmon ics

I X , o t ( 0 , 4)) > = 1 7. ( m ( 0 , 0)

J =0, 1, 2, ...M =—J3.22)

and it s e igenvalues are

J =0, 1, 2, ...3.23)

According to Eq. 3 .22 , the rotat ional levels are (2J + 1)-fold degenerate.

Equation 3.23 gives th e rotational levels in joules (J) when h is in J • s and I is inkg • m 2 . They are more commonly expressed in cm' numbers) via

E r o t (cm -1 ) = BJ(J + 1)3.24)

where B (the rotational constant in cm ') is

B = h2 /2hcI = h /87r 2 cl3.25)

The rotat iona l en ergy spacing in cm between level J and level ( J — 1 ) is

BJ(J + 1) — B(J — 1)J = 2BJ, which increas es line arly with J (Fig. 3.5 ).W e now de r ive th e pure rotational se lection rules (i.e. , in the absence ofvibrational or electronic transitions) for the rigid rotor. If a rotor of fixed lengthR o has charges + 6 glued to its e n d s , it has a permanent dipole moment

Po =45R = 6Ro ft3.26)

along the molecular axis. The E l transition moment for the J M — > J 'M 'rotational transition is then 6 R o < Yjm (0 , yri g ,(0, O D, which vanishes unlessAJ J' — J = +1, M M' — M = 0 , ±1, and 6 0 0 . Accordingly, El trans-

itions occur only between adjacent rotational levels fo r whichAE, = E ( J ) — E ( J — 1 ) = 2 B J . Note that this El transition moment integral isformally identical to the angular part of the El transition moment forhydrogenlike atoms— which is why it yields iden tical selection rules. The factorof 6 me ans that only heteronuclear diatomics with perma n en t dipole moments(6 0)—such as HBr and CO — can exhib it a pure El rotational spectrum.

Since th e energy changes in cm for transitions from state ( J — 1) to state Jare equa l to 2BJ for a rigid rotor, the observed rotational absorption lines arepredicted to be uniformly separated by the spacing 2 B (Fig. 3.5 ). This is borneout to within experimental error in the far-infrared spectrum of H C1 , Fig. 3 .2 . I fthe nuclear masses (and hence the nuclear reduced mass p t N ) are independently



DIATOMIC ROTATIONAL ENERGY LEVELS AND SPECTROSCOPY5E j

J =0

2B8B8Figure 3.5 Rotational levels in a diatomic molecule. Allowed absorptive trans-itions (6al = +1) are shown by vertical arrows. Schematic absorption spectrum is

shown at bottom.

known, the rotational constant B inferred from such a spectrum can be use d tocalculate the apparent internuclear separation

R o = (h2 12hc 1N B ) 1 1 23.27)

Indeed, a principal application of rotational (and vibrational—rotational)spectroscopy is de termination of molecular bond lengths (and bond angles in

polyatomics, Chapter 5).T he intensities of the abso rption lines in a far-infrared spectrum like Fig. 3.2



9 0 0 K0 0 K

5 1 0

Figure.6 Relativeotationaltateopulations,ivenygjexp[ -hcBJ(J + 1 )1kTyljgjexp[-hcBJ(J +1)101ersusotationalquantum number J for B = 10.6 cm'.

Table 3 .1

-cm 1

Rotational constants of diatomic molecules,

Molecule B e Œ e D e

"Ar 2 0 . 0 5 9 7 5 0 . 0 0 3 7 5 11 .3E-7138 B a 1 6 0 0 . 3 1 2 6 1 4 0 0 . 0 0 1 3 9 2 1 2 .724E-7"C l6 0 1 . 9 3 1 2 8 0 8 7 0 . 0 1 7 5 0 4 4 1 6 . 1 2 1 4 E -61 H 2 60 .853 3 .0 62 4 .71E-21 1 - 1 3 5 C 1 1 0 .593 41 0 . 3 0 7 1 8 5 .3194E-41 2 7 1 2 0 . 0 3 7 3 7 0 . 0 0 0 1 1 3 4.2E-9127 1 3 5 c1 0 . 1 1 4 1 5 8 7 0 . 0 0 0 5 3 5 4 4 .03E-823 Na 3 5 C 1 0 . 2 1 8 0 6 3 0 0 . 0 0 1 6 2 4 8 3 .12E-7

4 . 9 0 1 2 0 . 1 3 53 3.32E-4"Na 0 . 1 5 4 7 0 7 0 . 0 0 0 8 7 3 6 5 .81E-714N2 1 .99824 0 . 01731 5 .76E-614N1 6 0 1 . 6 7 1 9 5 0 . 0 1 7 1 0 .5E-61 6 02 1 .44563 0 . 0 1 5 9 4 .839E-6'6011 1 8 .91 0 0 .7242 19.38E-4

Data taken f rom K. P . H u b e r a n d G . Herzberg , Molecular Spectra

and Molecular Structure: Constants of Diatomic Molecules, V anNostrand-Reinhold , New Y o r k , 1 9 7 9 .

86



VIBRATIONAL SPECTROSCOPY IN DIATOMICS7a re also of interest. T he optical de ns it ies of absorption lines in a sample atthermal equilibrium will be weighted by the Boltzmann factors

Th exp(—.E . I i I kT) = (2 .1 1 + 1)exp[ — h c B J I (J + 1 ) 1 k 7 ]3.28)

where J is th e rotational quantum number of the lower (absorbing) level.(Optical densities a re defined in Appendix D.) This causes the pure rotat ionalspectrum to peak i ts intens ity at som e J determined by th e sample temperatureand rotat ional moment. In Fig. 3.6, relative rotat ional s tate populations areshown fo r J = 1 to 10 in H C1 fo r two contrast ing temperatures.

Most values of heteronuclear diatomic rotat ional constants B fall betweenth e extremes of HF (20.9 cm -1 ) and IC1 (0 . 11 4 cm -1 ) . For these species , thefrequencies of the J =0 -+ 1 rotational transitions are 41 .8 and 0 .228 cm -1 ; bothare in the

far-infrare d to microwave region of the electromagnetic spectrum.Some represen tative rotational constants are listed in Table 3.1.

3.3 VIBRATIONAL SPECTROSCOPY IN DIATOMICS

With the aid of Eqs. 3 .15 and 3.20, the Schriidinger equation (3.16) fo r nuclearmotion in the Born-Oppenheimer approximation becomes

h 2 (1 [2ri‘\n\tnx

„2k k ( R )2 1 . 1 1 ,1 R 0R(R) h 2 R

2 )11Xk l n ) / EdiXklIt)J

where l x k (R)> i s th e wave function fo r nuclear motion in electronic state k. If wemake the substitution IX ,JR)> = S k (R)Y , k { (0 , 4))IR, and rec all that j Yjm ( 0 , 4 9 )

= J(J + 1)h Yjm (0, 4)), we o btain

d 2 S k (R ) 21IN [E Uk (R ) j(j)h2d R 22k (R)= 03.30)2tI N R 2

This is the Schrödinger equation for vibrational motion, with eigenstates S k (R )

represe nting vibrational wave functions in electronic state k, under the effective

vibrational potential

, D, J(J+ 1 ) h 2i b (R) — u k w./ +

2 1 1 N R 2(3.31)

W e rec all that U k k (R) = Ktkklfleilth> +<OM oltk 1); it is the electronic potentialene rgy curve k(R), corrected by the expectation value of the spin-orbit couplingHamiltonian. The second term in Eq. 3.31 is the centrifugal potential which isoccasioned by th e rotational motion in state jx,.. ,> = Yjm (0, 0); it reminds us that

th e rotational and vibrational motions canno t be t ruly indepen den t , s ince th erotational state inf luence s the e f fec t ive vibrational potential.




Needless to say, the R-dependence of U k k (R ) = e k (R) is not analytic. Forbound electronic states that exhibit local minima in k(R) , it can be usefullyexpanded in a Taylor se r ies about the equilibrium separation R = R e ,

U k

k2 kk

U kk(R) = U kk(Re) (dd R)

Re ( R

.) + 2 dR 2 ) R e ( R — R e ) 2 + • • •

3.32)

If we se t U k k (R e ) = 0 and recognize that dU k k /dR vanishes at the local minimumR = R e , then

U k ( R ) 2 e lk ( R — R e ) 23.33)

where the vibrational force constant k is defined as

k = (dd 2 U kkR 2 )iz3.34)

This is the familiar potential for a one-dimensional harmonic oscillator (Fig.

Figure 3.7 Harmonic oscillator approximation (dashed curve) to true potential

energy curve c(R) in a d iatomic m olecule.



VIBRATIONAL SPECTRO SCOPY IN DIATOMICS93.7). If we restrict ourselves to pure vibrational motion (J = 0 ), Eq. 3.30 thenbecomes

(

h 2 d 22/I d R2

+ 7 k (R — R e ) 2 )S (R) = ES k (R )N

(3.35)

From prior experience with the one-dimensional quantum mechanical har-monic oscillator [4], we know that its eigenfunctions and eigenvalues are

Sk( R) - = N y e x 2 1 2 11 v (X )v >

3 .36a)

Ev i b hco (v +1)3 .36b)

= 0, 1, 2 , .

where x = (R — R e ) .\/pN w/h,

./k/p N , and the H v are the Hermite poly-

nomials in x. To keep things in perspective, we recall that these are the diatomicvibrational states only in the harmonic approximation (ignoring higher thansecond-order terms in the Taylor ser ies expansion (3.32)) in the absence ofrotation (J = 0 ).

In the El approximation, absorption or emission of radiation can accom-pany a on e-photon t rans i t ion between vibrational states Iv> and Iv'> only if<z) 0O. Since the vibrational states depend on R , the value of this transitionmoment is affected by the R-dependence of p = p(R), the permane nt dipole

mom ent . In homonuclear diatomics, p(R ) -= 0 for all R. In heteronuclear

diatomics, the R-dependence of p must resemble the schematic behavior shownin Fig. 3.8. This i s so because p(R) 0 both in the united-atom limit (R = 0 ) andin the separated-atom limit (R = co) where the diatomic dissociates into neutralatoms. Thus p ( R ) generally is not given by a closed-form expression, but can beapproximated by a T aylor ser ies about R = R e :

1 (0 2„• •*R ) = R e ) +R — R e )

O R R.2 OR2)Re

(R — R e ) 2 +3.37)

This should not be confused with the Taylor se r i e s expansion of U k k (R), Eq. 3 .32.T he validity of terminating the expansion for U k k (R ) with the second-order term(i.e., the harmonic approximation) is not connected with the validity of breakingoff the expansion of p(R) with some low-order term—these two approximationsare unrelated. Textbooks commonly base their derivation of El vibrationalse lec t ion rules on the premise that the ser ies fo r p(R) can be terminated with theterm that is linear in (R — R e ) . This can be approximately valid over limited-Rregions , but is (strictly speaking) unphysical fo r general R . T he error of using thislinear approximation is mitigated by the fact that the vibrational eigenstates

tend to limit con tr ibutions o f p(R) in the matrix element <v1p(R)Iv'> to smallregions near R = R e (Fig. 3.8).



90OTATION ANDVIBRATION IN DIATOMICS

Figure 3.8 Qualitative behavior of the dipole moment function p(R) for a

heteronuclear d iatomic m olecule that dissociates into neutral atoms. The straight line

tangent to p(R) at R = R 0 represents the linear approximation to ti(R) which breaks

off the Taylor series (3.39) after the term proportional to (R — R e ). When th e

vibrational motion is limited to a small range of internuclear separationsR_ 5 R 5Rthe deviations between the true dipole moment function an d its linear

approximation remain sm all.

Using th e series expansion for p(R), th e vibrational transition dipole momentbecomes

Op<vip(R)10 = p (R e)(*) +

()e

(v i(R — R e ) I v ' >u rc

+ 1(le

211)< 1AR — R e ) 2 1 e > 3.38)2 OcWe now need a systematic way to evaluate matrix elements like

<OR — R e rle>. This is provided by the second quantization formulation [5 ] ofthe one -dimens ional harmonic o scillator problem , which parallels in some waysth e ladder operator treatme nt of angular mom en tum. The harmon ic osc illatorH a m i l t o n i a n is

h 2 d23 2

rivibe ) 2lkq

2it N dR 2/1 N



VIBRATIONAL SPECTROSCOPY IN DIATOMICS1with i3 = (hli)(010R) and q = (R — R e ) . Since the harmonic oscillator frequency is

= (k/p101 / 2 , this is the s a m e as

,^ 2

1nvib=iN a )2 1 1 2

z-tiN

We now define the destruct ion operator

aip N coq)I N /2/2 N hco

Its Hermitian conjugate, known as the creation operator, is

= +tiNweN/ 21tNhco

(3.39)

(3.40)

(3.41)

because both /5 and q are Hermitian. The inverses of Eqs. 3 .40 an d 3.41 are

/5 =2

(pNh11/2(a + a + )

q = i ) 112(a — a + )

2 1 2 10(3.42)

and these allow us to rewrite the Hamiltonian (3.39) compactly in terms of thecreation and de struct ion operators,

hcovib =—

2via • - - F a + a ) (3.43)

A useful commutat ion relationship for these operators is

[ a , al=2/iNhco

C / 5 — i t tNaq, f i + i m N o N 1

=q , /5 ] + -2

i

h[ 1 ' 7 ,1so that

[ a , al= 13.44)

Using this, the Hamiltonian becomes

hcov ib = —

2(a + a + 1 +a 1 - =co(a + a + (3.45)




because aa + — a+ a = 1 from Eq. 3.44 . If we call the operator product a + athe number operator, the Hamiltonian (3.45) can be rewritten as

fivib = hae7 +3 .46)

H owever, we know from prior experience that

fiviblv> = ha(v + 1 ) 1 v >3.47)

Comparison of Eqs . (3 .46) and (3.47) then shows that

RIO = vlv>3.48)

The state Iv> is then an eigenstate of the number operator with eigenvalue y ,

which shows the number of vibrational quanta in that state.W e n ow wish to de te rmine what k ind of state a+ Iv > is . W e c an do this bytesting the ne w state a+ Iv > with the number ope rator,

lq(a + Iv>)a + 1 s7 + [ N , a 4 ])1y>3.49)

But

[N, a + ] =[a+ a, a + ] =a + [a, a + ] =a+3 .5 0 )

using the commutator in Eq. 3.44. Then Eq . 3.49 becomes

N(a10) = (a+ + a ) ly> = (1) + 1 )(a ± 1 v > )3 .5 1 )

He n ce the new s tate (a + I v > ) is an eigenstate of N with e igenvalue (v + 1) . Sincethe eigenstates of a one-dimensional harmonic oscillator are nondegenerate, thisimplies that

ely> c c I v + 1>3.52)

W e can similarly show that

c t I v > ,xI v — 1 >3.53)

so that the operators a+ and a have the effect of creating and annihilating one

quantum respectively in the harmonic os cillator. It can further be dem ons tratedthat

<vlaa + Iv > (a + 1 0) + (a + 1 0)

= (v + 1)<v + 11v + 1> = (v + 1)3.54)



VIBRATIONAL SPECTROSCOPY IN DIATOMICS 93

and that

<vlealv>alv>) + (alv> ) = y3.55)

so that to within a phase factor

aly> = v+1 l y + 1 >

aly> = — 1 >3.56)

Then we finally have the matrix eleme nts useful for evaluating the contributions

to the transition moment in Eq. 3.38 ,

<vja + I v '> =

< O W > - = N /y + 1 6 v,v' -13.57)

W e n ow calculate the first few terms in the transition moment (3.38) explicitly.The leading term ii(12,)<yly'> vanishes by o r thogonali ty of the harmonicoscillator states . For the nex t (first-order) term, the substitution for q using Eq.3.42 leads to

h

(vI(R — R e )iv'> = i( 2/)

)1/2

[ N A

1]

3 . 58 )

so that the l ine ar term in the expansion (3.37) of p(R) yields the selection ruleAv = y' — y = + 1 fo r vibrational transitions. The seco nd-order t e rm i s pro-portional to

h<vi(R — R e ) 2 10 =vlaa — a + a — aa + +a + a + Iv ' ›3.59)

. 2 1 . 1 N c o

The respective integrands resulting from expansion of (a — a + ) 2 can be e asily beshown to give Ay = +2, 0, 0, and — 2 . H e n ce the secon d-order term allowsAv = ±2 over tone vibrational transitions to occur, and higher overtones canresult from the succeed ing terms in the expansion of p(R). In practice, thefundamental transitions A y = ±1 are usually the ones with the larges t proba-bilities, and the overtone transition probabilities for A y = + n fall off rapidlywith increas ing n, because the transition moment integral samples a com-paratively limited region of R near R e .

Typical vibrat ional frequencies in heteronuclear diatomics tend to bebracketed between those of IC 1 and those in hydrides (i n which low reduce d

masses are accompanied by high vibrational frequencies ). In wave numbers, thefundamental vibrational frequencies W e = col2irc are — 384 cm -1 and

2990 cm' C! and H C 1 , respectively. V ibrational frequencies are listed for




Table 3.2

cm-1

Vibrational constants of diatomic molecules,

Molecule We CO,X, WeYe

40Ar2 25.74138Ba160 669.76 2.02 -0.00312060 2169.81358 13.28831 0.0105111H2 4401.21 1 2 1 . 3 3 0.8121H35a 2990.946 52.8186 0.22431271 2 214.50 0.614127135a 384.29 1 . 5 0 123Na35C1 366.0 2.023Na 1 1 - 1 1172.2 1 9 . 7 2 0.160

23Na 2 159.12 0.7254 - 0.001091 4 1 ,42 2358.57 1 4 . 3 2 -0.0022614N160 1904.20 14.075 0.0111602 1580.19 1 1 . 9 8 0.0474

160ll 3737.76 84.881 0.540

Data taken from K. P. Huber and G . Herzberg , Molecular and

Molecular Structure: Constants of Diatomic Molecules, V anNostrand-Reinhold, New York, 1 9 7 9 .

several other diatomic molecules in Table 3.2. These frequencies lie in theinfrared to near-infrared regicr of the electromagnetic spectrum.

3.4 VIBRATION-ROTATION SPECTRA IN DIATOMICS

Molecular vibrational spectra exhibit fine structure in gases, because rotationaltransitions can occur simultaneously with vibrational transitions. Indiatomics with small vibration-rotation coupling, the selection rules on Av andAJ are exact ly as in the cases of pure vibrational and pure rotationalspectroscopy, respectively:

AJ = ±1

Av = +1, (+2, +3, ...)3 . 6 0 )

These rules hold provided the electronic state in question has no component oforbital angular momentum along the molecular axis (Chapter 4). When thiscomponent is nonvanishing, the selection rules are influenced by interactionbetween electronic and rotational angular momenta, and the AJ 0 transitionbecomes allowed:

AJ = 0, ±1

Av = +1, (+2, +3, ...)3 . 6 1 )



P

V I B R A T I O N-OTA TION

R branch

SPECTRA IN DIATOMICS5branch

6

5

4

2

V' 0

•

6

5

4

V"2

0

3.‘4

(Q)5f 6

y

Figure 3.9 Energy levels and transitions giving rise to vibration—rotation spectra in

a diatomic molecule. The upper state and lower state quantum numbers are (v', J')

and (y", J"), respectively. The schematic spectrum at bottom shows line intensitiesweighted by rotational state populations, which are proportional to

(2J" + 1) exp[—hcBJ"(J" +1)1k7]. The rotational consansandB. are as-sumed to be equal, resulting in equally spaced rotational lines. This assumption is

clearly not valid in the HCI vibration—rotation spectrum in Fig. 3.3.

The vibration-rotation spect rum can be unders tood on the basis of a partial

en ergy level diagram for the lowest vibrationa l and rotation al levels (Fig. 3.9).T he upper and lower level quantum numbers are denoted by (v'f) and (v"J"),respectively. Trans i tions for which AJ = + 1, AJ = — 1 , and AJ = 0 (the latteroccurring only in s tates with non zero orbital angular mome ntum along themolecular axis) are called R-, P-, and Q-branch t rans i tions . T he t rans i t ionfrequencies are derived from

hv = AEv i b +AE r „, = hco(v' + — ko(v" + D

+ hcB'J'(J' + 1) — hcB"J"(J" + 1)3.62)

or

-A cm -1 ) = i7 +B'J'(J' + 1) — B"J"(J" + 1)3.63)




where T , 0 = c o ( y ' — v")127rc i s the frequency for the pure vibrational transition. Inthe case of the P branch, letting J" = Jand J' = J — 1 gives

= —(B' + B")J + (B' — B")J2 vp(J)3.64)

whereas for the R branch in which J" = J, J' = J + 1,

= T i c ) + (3B' — B")J + (B' — B")J 2 + 2B'1 z ( . n3 .65 )

If B' = B" B (i.e., if the diatomic exhibits iden t ical rotational constants in bothvibrational states le> and Iv"»

i p(J) = - 3 0 — 2 B J

VR (J) =o + 2B(J + 1)

3 .66)

He n ce th e rotational fine structure lines are predicted to be equally spaced infrequency if B i s independent of the vibrational quantum number v. In fact, therotational constant, described earlier as B = h2 12hcp N R4 for a rigid rotor withseparation R o , becomes

h 2 By=2hcilly

< v iR2 I v > (3.67)

in a vibrating diatomic. B „ then acquires a y-dependence, largely because th eharmon ic oscillator potential is asym me tric about R = R e . Since U k k (R ) levelsoff to th e separated atom asymptote for large R (Fig . 3 .10 ) but falls rapidly fo rsmall R , <yi 1/R 2 iy> (and therefore B y ) decreases as y increases. This fact isaccommo dated e xperimental ly by f i t ting meas ured B y values to the expression[6]

B y = Be — Œ e ( 1 ' + 4 + y e (v + 4) 2 + (5,(v + 4) 3 ' • •3.68)

The y-dependence in the rotat ional constant B y is clearly visible in the HC1

ne ar-infrared spectrum shown in Fig. 3.3. This vibration—rotation spectrumconsists of the y' = 0 to y' = 1 absorptive transition with rotation al finestructure in a P branch (whose absorption lines at successive J values appear atever lower frequencies according to Eq. 3 .64) and an R branch (whoseabsorption l ines run to higher frequen cies as J i s increase d, Eq. 3.65). No Qbranch line o ccurs, because HClin its closed-shell electronic ground state has n oelectronic angular mom entum. The absorption lines are labeled according to thevalue of J" in the lower vibrational state : R(0) is the R-branch line from J" =0,

P(1) is th e P-branch l ine fro m J" = 1,etc. The rotational line spacings decreasea t higher f requen cies in both branches, in consequence of the quadratic



VIBRATION-ROTATION SPECTRA IN DIATOMICS7Figure 3.10 Potential energy curve for the electronic ground state of Na2 , with

effective internuclear separations R y indicated by d ots for y = 0 through 45. These R,

values are computed from the experimental rotational constants B y via

hcBy =h2 12p m fq. R y is close to R e in the lowest vibrational states, and increases

with v.

(if - B")J 2 terms in v(J) and v R (J). This implies that B" > B ' , or that the

rotat ional constant is larger in vibrationa l state y = 0 than in y = 1.

Each of the rotational lines in Fig. 3 .3 is split into doublets spaced by2 cm - because the isotopes H"Cl and 1 - 1 3 7 C1 have slightly differe nt reduce d

masses, and therefore differen t rotational constants according to Eq. 3.25.Vibration-rotation spectra like the one in Fig. 3.3 can be an alyzed to obtainaccurate values of the rotational constant in the upper and lower vibrationallevels for both isotopes.




3.5 CENTRIFUGAL DISTORTION

Since chemical bonds can be stretched, the cent r i fugal force accompanyingdiatomic rotation pushes the nuclei farther apart than they would be in anonrota t ing diatomic. This in turn reduces the rotational constant B„ whichthen depends on J as well as on v. To treat this effect quantitatively, we begin bydef ining F, as the centrifugal force pulling the nuclei apart, F, as the harmonicoscillator restoring force pulling the nuclei together, R e as the equilibriumseparation when J =0 , and R, as the separation when J O. Clea rly R, > Re ,

due to the centrifugal force . Classically, the centrifugal force is given by

F, = ,u0,02 Rc= .1 2 /PriR3.69)

using the fact that the rotational angular momentum J = 1 w = A IN R,2 c o . The

magnitude of the harmonic oscillator restoring force is

1 F r i = k (R, — R e )3.70)

because U k k (R ) = Ik(R , — R e ) 2 inthe harmonic approximation, andFr = —dU/dR,. Balancing the centrifugal and restoring forces gives

J2 /1 2 N R = k(R, — R e )3.71)

or

R,— R e = J2 / t i N RA.3.72)Then the rotational energy is

j2

Erot+ 1 k (R , — R e ) 22 . 1 . 1R (3.73)

where the latter term is included in the rotational energy because it reflects thechange in the diatomic potential due to the centrifugal displacement of R f romR e to R. Using twice the expression for R, from Eq. 3.72, the rotational energy

then becomes

j E ro t = (R e + J / 1 2 N Rk) 2 +1k (ft: 2;0

.1 2 1 (j222/I N (R e + J2//iNR0 + k2,u N Rk

)

j21 1 2.14± • +

2 p link2/2NR

J 4- 2,u N RjRk ± • • •(3.74)



CENTRIFUGAL DISTORTION9where we have used the identity (1 + x ) -2 1 - 2x + 0(x 2 ) for small x.Quantum me chanically, this express ion for the rotational energy becom es

E 1 0 , = J(J + 1)h /2/I N R - J2 (J + 1)2 0/214,Rk3.75)

and the rotational energy in wave numbe rs (with y-dependence of the rotationalconstants included) is now

Er o t (cm - 1 ) = B,J(J + 1) - D J (J + 1)23.76)

with

1D

v = 2hct ikk <vi —R61

40 (3.77)

Like B v , D „ decreases as y increases; its y-dependence has often bee n f i tted fo rdiatomics using expres sions of the form [6 ]

D v = De -13,(v +1) + "•

If the y-dependence in B „ and D„ is ignored , the fact that

h2

Band

h 4

2hcp, N R

implies that the cen trifugal distortion constant i s

D 4B 3

(3.78)

(3.79)

(3.80)

where W e is the widely used notation for the vibrational fundamental frequencyin cm' (which we have been calling -f / 0 ) . T he physical significance of Eq. 3.80 isthat bonds with stronge r force constants (and hence larger W e ) experience lesscen tr ifugal dis tortion. I t is inte res t ing that kno wledge of the rotational andvibrational constants B and We permits one to predict (with reasonableaccuracy) the more subtle quantity D .

When D é 0 , the rotational line s in a pure rotat ional s pectrum (or in a

vibration-rotation spectrum) are no longer equally spaced , but become moreclosely spaced at higher J. However, D „ is us ually n ot particularly large. In 1 2 ,

which has one of the weake r vibrat ional force cons tants amon g ordinary




diatomics (excluding van der Waals molecules!), B e = . 0374 cm -1 and

W e = 2 1 4 .6 cm -1 —yielding D e — 4 .5 5 x 1 0 cm -1 , which is seven orde rs ofmagnitude sma ller than B e . In H C1 (B e = 10.6 cm -1 ) , the distortion constant D e

is 5 . 3 x 1 0 cm -1 ; cen trifugal distortion is far too s mall to observe in the H C1far-infrared spectrum in Fig. 3.2.

3.6 THE AN HARMONIC OSCILLATOR

Most of our discussion of vibrational eigenstates and select ion rules has beencentered on the harmonic approximation. Of course, the effective vibrationalpotential U k k (R ) i s not we ll approximate d by a parabola for ene rgies corre-sponding to large vibrational quantum numbers y (cf. Fig. 3.7), and considerablework has been e xpend ed to f ind alternative expressions for either U k k (R ) or thevibrational ene rgy levels which are both compact and accurate. I t has becom e

conve ntional to fi t experimentally determined vibrational levels G y (cm -1 ) toexpressions of the form [6 ]

= co e (v + — coexe(v + 4 ) 2eYe(V 1 ) 3 +• • •3.81)

The first term in G, is the harmonic approximation to the vibrational ene rgy,and the remaining terms are the anharmonic corrections. The negative sign in theleading anharmonic te rm corresponds to the fact that rea l vibrational leve lsbecome more close ly space d with higher v. A typical value of c o e x e /co, is 0 . 0 0 2 8for 2, which indicates that its ground-state potential ene rgy curve is very nearly

harmonic near the bottom of it s well. For the A ll exci ted s tate in NaH(Chapter 4 ) , c o e x e / c o e is —0.0174 , which is anom alous because it is negative; thisoccurs because the A 1 E ± potential curve in Nall is pathologically misshapendue to perturbation by other ne arby exci ted states.

If the vibrat ional e nergy level expression (3.81) i s cut off after i ts lead inganharmonic term

G , =- coe (v + D — w e x e (v +3.82)

the resulting levels are the exact eigenvalues of an approximate vibrat ional

Hamiltonianin

whichthe

potential is given bythe

analytic functionUkk(R) De[l _ e - a(R — Re )] 23.83)

This is known as the Morse poten t ial . At R = Re , the first and second

derivatives of the Morse potential are [7 ]

d U k k (R ) 0

dR —

d 2 Ukk (R)

dR2 =2a 2 D e k3.84)



THE ANHARMONIC OSCILLATOR01with the consequence that

(hD e \- 1 2 0 ) e = a3.85)ncptp,7rc

u N

with D e in cm -1 . I t can also be shown that

ha'w e x e =3.86)

zincit N

The Morse potent ial ex pression for U k k (R) h'as the limits D e and 0 as R

approaches co and R e , respectively. W hen R 0 , Uk k (R ) D e exp(2aR e ) , which

is typically large. He n ce the Morse poten tial is capable of modeling thequalitative features of a realistic diatomic poten tial en ergy curve. If the latter iswell described by a Morse potential, then spectroscopically determine d values of(0, and w e x e can be combined using Eqs. 3 .85 and 3.86 to estimate the moleculardissociation energy [7]

D e = a)!/4coe x e3.87)

measured between the bottom of the potential well and the asymptote of Uk k (R )at large R. This can yield useful approximations to D e if the dissociation energy

is not available by o ther mean s. In more general cases where the vibrationalene rgy levels are given by Eq. 3.81 rather than 3.82, the dissociation ene rgy D o

= 0 is rigorously given in terms of the level differences

AG(v) G ( v + 1 ) — G ( v )

=20 )exe (V + 1 ) + weYe( 3 v 2 + 6v + .V)

+...

3.88)

as

D o = ,E0 A G(v )3.89)

if the vibrational level spacing vanishes between level v n i a x and level (v e ,,. + 1) atthe dissociation limit. The ar ea unde r an experimental plot of AG(v) versus v

then equals D o in principle (Fig. 3.11). M any of the terms G(v) for larger v will notbe known in practice, so that the experimental AG(v) curve is frequentlyex trapolated to AG = 0 to estimate the dissociation energy. The use of Eq. 3.87

to estimate D e amounts to making a linear extrapolation of AG(v), since for theMorse potential only the two leading terms in Eq. 3.88 contribute to AG(v). SuchBirge -Sponer extrapolations frequen tly yield dissociation ene rgies that are




00000V

Figure 3 .11 Birge-Sponer plot of AG( v) versus v for the electronic ground state of

Na 2 . The area under the AG(v) curve yields the ground state dissociation energy Do .

Linear extrapolation of the points for low v (straight line) would clearly yield a grossoverestimate of C. Nonlinear extrapolation of the points

exhibited for 0.,< „ y 45(P. Kusch and M. M. Hesse l , J. Chem. Phys. 68: 2591 (1978)) leads to an improved

estimate of Do . A still better approximation to Do can be made by analyzing

vibrational levels for v up to 55 from the more recent Na 2 fluorescence spectrumshown in Fig. 4.1.

— 3 0 % to o large for diatomics that disso ciate into ne utral atoms [6]. Diatomics

that dissociate into ions ex hibit potentials that behave as R 1 at long range.Such potentials ex hibit infinite n umbers of vibrational levels; plots of G(v) versusy then fun asym ptotically along the y axis fo r large y, and the Birge-Sponer

extrapolation becomes inapplicable.

REFERENCES

1. J. B. Marion, Classical Dynamics of Particles an d Sys tems , Academic , New York, 1 9 6 5 .

2 . J. C . Tully, Nonadiabatic processes in molecular collisions. In W. H . Miller (Ed.),Dynamics of Molecular Collisions, Part B, Plenum, New York, 1 9 7 6 .

3. L. D. Landau and E . M . Lifshitz, Q u an t u m Mechanics: Nonrelativistic Theory, 2d ed. ,Addison-Wesley, Reading, MA, 1 9 5 8 .



PROBLEMS 103

4 . E. Merzbacher , Q u an t u m Mechanics, Wiley , New York, 1961 .

5 . K. Gottfried, Q u an t u m Mechanics, Vol . 1 , W. A. Benjamin, New York, 1 9 6 6 .6 . G . Herzberg, Molecular Spectra and Molecular Structure, I. Spectra of Diatomic

Molecules, 2d ed. , Van Nos t r and , Princeton, NJ, 1950 .7 . G . W. King, Spectroscopy and Molecular Structure, Hol t , Rinehart, & Winston, New

York, 1964 .

PROBLEMS

1 . In the pure rotational spectrum of 1 1 - 1 3 5 C 1 , l ine s f rom the initial states J =2and J =3 are observed with equal intensity. Calculate the temperature of the

sample. (For 1 H 3 5 C 1 , the rotational constant is 1O.6 cm'.)

2 . Use the second quantization formalism to develop the selection rule on<y 'l(R - R e )" I y> for arbitrary n.

3 . For a heteronuclear diatomic molecule AB, the dipole moment function in

the neighborhood of R = R e is given by

ti(R) = a + b ( R — R e ) + c(R - R e ) 2 + d(R - Re) 3

in which a, b, c, and d are constants. Treating this molecule as a harmonicoscillator, calculate the relative intensities of the y = 0 -> 1 fundamental and

= 0 -+2 and 0 -> 3 overtone transitions in the El approximation in terms ofthese constants and the harmonic oscillator constants j and c o .

4 . S o m e of the frequencies and assignments of the 1 1 - 1 3 5 C 1 vibration-rotationl ines in Fig. 3.3 are given below.

ij(cm ') Assignment

2963 R(3)2944 R(2)

29 06 R(0)2 8 6 5 P ( 1 )2843 P(2)2821 P(3 )

Determine the values of the rotational constants (in cm -1 ) and the associatedbond lengths (in A) of 1 I - 3 5 C 1 in vibrational states y = 0 and y = 1 . The nuclearmasses of 1 } 1 and "Cl are 1 .007825 and 34.96885 amu, respectively.

5 . From an analysis of the B i ll. X 1 E: fluorescence bands of 23 Na2 (seeChapter 4), the vibrational energy levels in the electronic ground state can be



104 ROTATION AND VIBRATION IN DIATOMICS

represented by

G(v) = 159.12 ( y + — 0.725(y + 4) 2 — 0 .00 11 (y + 4)3

in cm'. Determine the vibrational quantum number y m a x at which thevibrat ional level s pacing vanishes , and estimate the dissociation energy D o .Compare this dissociation en ergy with tha t es t imated us ing a linear Birge-Sponer extrapolation, and with the directly measured value D o = 0 .7 3 e V .



ELECTRONIC STRUC TURE

AND SP ECTRA INDIATOMICS

Like the atomic spectra discussed in Chapter 2, electronic band spectra indiatomic molecules arise from transitions between differen t electronic s ta tes .Both types of spectra occur at wavelengths ranging from the vacuum ultraviolet

to the infrared regions of the electromagnet ic spectrum. A complication indiatomic band spe ctra is that changes in vibrational and /or rotational s tategenerally accompany electronic transitions in molecules, en dowing bandspectra with rich rovibrational structure . Analysis of this structure (which oftenlends a bandl ike appearance to diatomic spectra, in contras t to the discrete linespectra characteristic of atoms) can yield a wealth of information about groundand ex ci ted electronic s tate sym me tries , de tailed potent ial energy curves, andvibrational wave functions. An anthology of such information is presen ted for

seve ral diatomic molecules in this chapter.

An uncomm only lucid ex ample of an electronic band spec trum is the Na 2fluorescence spectrum shown in Fig. 4.1 , which was obtained by ex citing Na 2vapor in a 4 5 3 °C oven using ne arly mon ochromatic 5 6 8 2 A light from a Kr + ionlaser. At this temperature, Na 2 exis ts in i ts electronic ground s tate (the X ' E g+state, in notation to be developed later) with appreciable populations in severalvibrational and many rotational levels . H owever, the 5862-A excitation wave-length is unique ly matched in Na 2 by the ene rgy level difference between the

s: state (y" = 3, J" = 51 ) and an electron ically excited s tate (A lE:) withy' = 3 4 , J' =50 . The latter level then becom es s electively pumped by the laser. Itsubsequently relaxes by fluorescence transitions to X' E: Na 2 in vibrational

levels y " between 0 and 56 , producing the ex hibited spectrum. O nly transitionsto y" 4 are shown; a schematic energy level diagram showing some of thesetransitions is given in Fig. 4.2.

105



LASER L INE

ffr6209060

Na, A I Z:-... X'E + : Kr+ ( 5682 A )

_ , 505C).•50

.3414049) I

IA VR (49)

J'.50

20 25 30I

.:50 R (49).:4J

[P (51)404505 111111r

7407000mFigure 4 .1 Fluorescence spectrum (fluorescence intensity as a function of fluorescence wavelength) for N a 2 vapor

pumped by a 5682-A4 krypton ion laser. This wavelength excites N a 2 molecules from V' = 3, J" = 51 in the electronic

ground state to y' = 34, J' = 50 in the Al E u+ excited electronic state. The shown fluorescence lines result from transitions

from the laser-excited level down to y" = 4 through 56 in the electronic ground state. Reproduced by permission fromK. K. Verma, A. R. Rajaei-Rizi, W. C. Stwalley, and W. T. Zemke, J. Chem. Phys. 78: 3601 (1983).



ELECTRONIC STRUCTURE AND SPECTRA IN DIATOMICS 107

20000

10000

4 Figure 4.2 Energy level diagram for the fluorescence spectrum in Fig. 4.1.Absorption of the 5682-A laser photon pumps Na 2 molecules from V = 3, J' = 5 1 in

the X 1 1: electronic state to V ' = 34, J' = 50 in the A E „± electronic state. Subse-

quent fluorescence transitions_connect v'=34 in the A 1 .j state with v"=0 through56 in the electronic ground state. Only three of these fluorescence transitions are

shown for clarity. Rotational levels are not shown. Internuclear separations andenergies are in A and cm -1 , respectively.

Figure 4.1 illustrates some of the El selection rules on rotational andvibrational structure in electronic band spectra. The observed rotationalselection rule, AJ = + 1 (the laser-excited level with J' =5 0 fluorescences onlyto lower levels with J" =4 9 or 51 ) is reminiscent of that in vibration—rotationspectra of molecules with no component of electronic angular momentum along

the molecular axis (Chapter 3). The R-

and P-branch notations in Fig. 4.1 areidentical in meaning to those used in vibration—rotation spectra. In markedcontrast, there i s no apparent selection rule on Ay = y ' — y " : lines t e rm inat ing in



108LECTRONIC STRUCTURE AND SPECTRA IN DIATOMICS

y" = 43 have comparable intens i ty to l ine s terminat ing in y" = 5, and theintensity pattern of vibrational lines appears somewhat haphazard to theuneducated eye. The positions of the successive vibrational lines in a givenrotational branch (R or P ) accurately reflect the ene rgy spacings of theanharmonic oscillator levels in ground-state Na 2 ; thes e spacings grow narroweras the dissociation l imit is approached ne ar 8 0 0 n m . The absence of a restrictivevibrational se lect ion rule in electronic band spec tra vas t ly e nhance s thei rinformation content; the spectral line intensities and positions in Fig. 4.1 allowprecise construction of an em pirical potential-ene rgy curve for X'Eg+ Na 2 forvibrational ene rgies up to > 9 9 % of its dissociation energy.

Electronic band spectra may also be observed in absorption of continuumlight (e.g., from a high-pressure Hg lamp) by a gas-phase sample. The observedspectrum is then a superposition of absorption line s arising from excitation of all

levels (y", J") in the electronic ground s tate to levels (y', J') in the electronicallyex c i ted s ta te , weighted by the appropriate Boltzman factors of the initialrovibronic levels (y", J") . This lack of selectivity causes far greater spectralcongestion in band spectra, and high spectral reso lut ion is required to de tectsuccess ive rotational lines within a vibrational band. Such crowding of spectral

l ine s can be rel ieved by preparing the diatomic species in a supersonic jet, in

which very low vibrational and rotational temperatures are routinely attained.In this m anne r, molecules with predom inantly y" = 0 and low J" values areproduced.

It is customary to obtain fluorescence excitation spectra rather than ab-sorption spectra in jets, where the total fluorescence intensity is monitored as a

function of excitation laser wavelength. In cases where the fluorescence quantumefficiency (defined as fluorescence photons emitted/laser photons absorbed) isindependent of excitation wavelength, the fluorescence excitation spectrumcoincides with the absorption spectrum. Part of the fluorescence excitationspectrum of an Na 2 jet operate d with vibration al and rotationa l tempe ratures of— 50 and 3 0 K , respectively, is shown in Fig. 4.3. The four intense bands arisefrom electronic transitions from y" = 0 in the VI: ground s ta te to y' = 25through 2 8 in the A 1 E: excited state. The rotational fine structure in each bandconsists of a barely reso lved ser ies of narrow lines, creating an envelope thatpeaks asymm etrically toward the blue ed ge at the b an d h ead . Such rotat ional

envelopes (or contours) are said to be shaded to th e red. Electronic transitions areoccasionally characterized by rotational contours that are shaded to the blue,i.e., by contours in which the bandhead lies at the long-wavelength edge. Suchcontours are absent in the fluorescence spectrum in Fig. 4 .1 , where the selectivepreparation of J' =50 in the A E: state simplifies the rotational structure in thefluorescence spectrum. Analysis of absorption or fluorescence excitation spectrayields information about the vibrational structure and potential ene rgy curve ofthe upper (a s oppose d to lowe r) electron ic state.

This chapter begins with a t reatment of symmetry and electronic structure in

diatomic molecules . The symmetry selection rules for electronic transitions are

derived, and vibrational band intensities (cf. Fig. 4.1) are described in terms of



(no

28.o)

seSO44440,4*I * 4 1 1 1

17600cm 17400cm4

SYMMETRY AND ELECTRONIC STRUCTURE IN DIATOMICS 109

(25,0)

(26,0)

Figure 4.3 Fluorescence excitation spectrum (total fluorescence intensity versusexcitation wavelength) of Na2 molecules in a supersonic jet. The four intense groupsof barely resolved lines are due to electronic transitions from y" = 0 in the electronicground state to V = 25 through 28 in the Al E L; excited state. The individual lines aredue to rotational fine structure, which is discussed in Section 4.6. Reproduced by

permission from J. L . Gole, G. J. Green, S. A. Pace, and D. R. Preuss, J. Chem. Phys.

76: 2251 (1982).

Franck-Condon factors. The most common angular momentum coupling casesare di scus sed , and rotational fine structure in electronic transitions (cf. Fig. 4.3)is rationalized for heteronuclear and homonuclear diatomics using Herzbergdiagrams.

4.1 SYMMETRY AND ELECTRONIC STRUCTURE INDIATOMICS

All diatomic molecules belong to either the Cor p o h point group, and s omuch of their electronic structure and nomenclature is derived from theproperties of these two groups. In what follows, the Cartesian z axis is alwaystaken to be along the molecular axis (the line connecting the two nucle i). The xand y axes are both normal to the internuclear axis, as shown in Fig. 4.4.

The partial character table for the heteronuclear point group C which hasan infinite number

of classes and irreducible representations) is

C

2C 9

z + 1- 1zII cos 0 x, y) , (R x , R y )A co s 2 0 ( 1 )

cos 3 0



110L E C T R O N I C STRUCTURE AND SPECTRA IN DIATOMICS

Figure 4.4 Orientation of Cartesianaxes in a diatomic molecule AB.

The symmetry elements C o and c r 1 , are the rotation by an arbitrary angle 0 aboutthe principal ( z ) axis and a reflection plane containing the principal axis,respectively. The superscript in the notations for the ± and I irreduciblerepresentations (IRs) indicates th e behavior of the IR s under th e a„ operation.Since the characters of the E operation in all IR s of C are either 1 or 2, thisme ans that all diatomic electronic states are spatially either n o n d e ge n e r a t e ordoubly de generate .

The behavior of the vector components ( x , y, z ) and the rotations (R, R, R z )under the group operations proves to be important for determining th e El andM 1 select ion rules fo r electronic transitions. In particular, the vector z i s

obviously unaffected by all of the C soit t ransforms as the I +y m m eR . The rotation R z i s not changed by E or C o , but changes

sign (direction) under any o that it belongs to the IR . Some

of the group operations trans form th e vectors (x , y) into linear combinations of xand y, so that (x , y) form a basis for a two-dimensional JR of Cop t ,. If the (x , y)basis vectors are rotated by an angle 0 about the z axis , the resulting ne w basis(x', y') i s related to the former basis by

rx'l [cosLy'i [sin 4)

—sin 0 Tx]

co s 01y]

()[ (4 .1)

The character x(C 0 ) in this bas is is then 2 co s 0 . Similarly, suppose a reflectionplane a„ contains the x axis . Then under this u„ operation,

[ x'l [x1 _ 0 1 [ y l

p c i= v

Y(4.2)

Th en x(a„) = 0 ; i t can eas i ly be shown tha t this resul t is inde pend en t of the



E

E g+ 1

Ei 1

H g 2Ag 2E: 1

I. 1

1 1 . 2A u 2

2C...

1 112 cos 4 ) 2 cos 2 4 ) 1

12 co s 4 )

2 cos 2 4 )

1

0

0

SYMMETRY AND ELECTRONIC STRUCTURE IN DIATOMICS11

orien tation o f the o - , plane with respect to the x and y axes [1] . Finally, by thedefinition of the identity operation

Ex.Y1 xyl (4.3)

and x(E) = 2 . We conclude that the (x , y ) vectors together form a basis for theH JR of C., and this fact is reflec ted in the character table. It may similarly beshown that the rotations (R, R) about the x and y axes also form a basis for theII IR .

T he homonuclear point group D o c h can be generated from Cby adding theinversion operation i about a center of symmetry, (x , y, z) ( — x , — y, — z ) . Thiscreates two additional classes of group operations: the improper rotations

= iC„ and the two fold rotations C2 = i(7,. T he homonuclear point groupcharacter table is

D c o h

j 2S...

1 1 1

1 1 — 1 R z

2 —2 cos ( / ) 0 (R i, R y )

2 2 cos 2 4 ) 0— 1 — 1 — 1 z— 1 —1 1—2 2 cos 4 ) 0 , y )—2 —2 cos 2 4 ) 0

From our discuss ion of the C . point group, z transforms as either E g+ ors ince i z = — z, it must belong to E ut ince

XXi [Y [y i (4.4)

the character x (i) in the (x , y) basis is —2, and s o (x , y) forms a basis for the H.rather than the Hg IR . In general , the vector components always transform as u-type IR s in D c o h . The reverse is true of the rotat ions R x , R y , R z , which alwaystransform as g-type IRs.

W e are now prepared to discuss the relationships between symmetry and

elec t ronic structure in diatomics. In the absence of spin— orbi t coupl ing in atoms ,one has the comm utation relationships involving the electronic Hamiltonian



112LECTRONIC STRUCTURE AND SPECTRA IN DIATOM ICS

and angular momenta:

[ii, L ) = 0i, L z ] = 0

[ 1 ? , P ] = os ; ] = o

4.5)

[fl h2]0 [fi l]o

This mean s that L, S, J, M L , Ms , Mj can all be good quantum numbers , inaddition to the principal quantum number, in spherically sy mme tric potentials.(All six of these cannot simultaneously be good quantum numbers, fo r reasonsexplained i n s ec t ion 2 .2 an d Appendix E.) In the reduced cylindrical symmetry ofdiatomic molecules, however, two of these commutation relationships in theabsence of spin—orbit coupling.become modified [2],

[fi,

LI0

[R, P] 0 04.6)

so that L and J are no longer good quantum numbers. The projectionsL z = M L h, S z = Ms h, and Jz = Mj h of L , S, and J along the molecular axisremain conserved in the cylindrically sym me tric potential. Diatomic s tateswhich are eigenfunctions of L z (i.e. states in which M L is a good quantumnumber) must have the 0-dependence ex p( + iA0) with A I ML I , because suchfunctions are the only physica lly acceptable eigenfunctions of L z ,

exp( + iA0) = ± A h ex p( ± iA0)4.7)

For continuity of this function at 4) = 0 (2n), A must be integral,

A = 0, 1, 2, . . .4.8)

In analogy to L z = + Ah, one also has S z = +Eh,where E can be half-integralfor spin angular mome ntum,

E = 0, I , 1 ,. .4.9)

Wh en A = 0, th e electronic s tate i s obviously 0 - inde pende nt and unaffectedby the C o operation. H ence , a state with A = 0 exhibits x(C 0 ) = 1 and thereforeis some type of E state, which, according to the C nd D o o h character tables , isspatially nondegenerate. For A = 1, we have two diatomic states behaving asex p( + i0). These a re degenerate a priori ( i . e . , in the absence of spin—orbitcoupling), since the electronic ene rgy is inde pendent of whether L z points in the+z or — z direction. One may then take l inear combinat ions of these two states '



CORRELATION OF MOLECULAR STATES WITH SEPARATED-ATOM STATES 113

0 -dependen ce to form

cos ( /) = - 1 - (e 4 +lrs in 0 = — e -ick) ylr

2 i(4.10)

with r = (x 2 + y 2 ) 1 / 2 .

Clearly, these linear com binat ions t ransform as the vectors (x , y) under thegroup operations of C o o „ and D o h —so they form a basis for the MC.) andTI.(D,„ h ) IRs . It can similarly be shown that the functions exp( ±214)) for A = 2form a basis for the A(C) and A g (D o c h ) IRs. These are examples of the fact thatthe Gree k let ter notations for the diatomic point group IR s give the A values

associated with the electronic states d irectly:Aype o f state

012 31 : 1

Each of the / tates in molecules belonging to C or D c h exhibits a definitebehavior ( + or — ) under the a i , reflection operation, and this is always indicatedin the supersc ript that accompan ies the IR notation. For the doubly degene ratestates (H , A , (1), .) it is always possible to choose linear combinations analogousto those in Eq. 4 .10 in order that each of the two combinat ions i s e i therunaffected or changes sign with respect to a particular reflection plane. Thereflection symmetries of the cos 0 and s in 0 combinations in Eq. 4 .10 , forexample, are respectively ( +) and ( — ) with respect to a o - 1 , reflection planecontaining the x and z axes .

4.2 CORRELATION OF MOLECULAR STATES WITH

SEPARATED-ATOM STATES

The question of which diatomic term sym bols may be o btained by adiabaticallybringing together a toms A and B, initially in electronic s tates with angularmomentum quantum numbers (/A , S A ) and (/ B , sB ) , can be answe red wi thoutrecourse to electronic structure calculations. Since the electronic (orbital plus

spin) degeneracies on the respective atoms are ( 2 1 A + 1 ) (2 s A + 1) and(2 / B + 1 X 2 s B + 1), a total of (2 / A + 1 )(2s A + 1 )(21 B + 1 ) (2 s B + 1) diatomic statesmust correlate with the separa ted-atom s ta tes . According to one of the




commutation relationships (4.6), M L is a good quantum number for diatomics atall internuclear separat ions R (in the absence of spin— orbit coupling). ML mustthen be conserved as the atoms are pulled apart, and so A = IML 1 must equal theabsolute sum of magnetic quantum numbers m i A and m / B for the separatedatoms. A similar argument applies to the spin angular momenta: M s in any

diatomic s tate is nece ssarily the sum of the separated-atom quantum numbersm s A and m s B . The notation E 1M5 1 is commonly used to specify the projection

of spin angular mom entum along the molecular axi s ; i t should n ot be confuse dwith the unrelated use of the notation to denote d iatomic electronic stateswith A = O .

We now consider the hete ronuclear correlation problem for several examplesof increasing complexity. An example of the simplest case (both atom s in 2 S

states) is the ground sta te of NaH, which dissociates into the ground-state atomsNa(3 2 S ) + H ( 1 'S). Since / A = / B = 0 in the separated atom S states, the total z

component of orbital angular mom en tum is L z = hML, = h(miA + nim) = O .

H ence only dia tomic s ta tes (A = 0 ) can co rrelate with the S-state atoms. Theymust furthermore be E ± rather than E states , because the atomic S states are

even with respect to reflection in any c v plane containing the molecular axis (Fig.4.5). The number of diatomic states which correlate with the S-state atoms is(2 1 A + 1 )(2 / B + 1 )(2s A + 1 )(2s B + 1) = 4. He n ce the pertinent diatomic states are

+ (one s tate with M s = 0) and 3- E + (three states with M s = 0, + 1), which willla te r be s een to be bound and repulsive states , respectively.

The next few exci t ed s ta te s in NaH correlate with Na(3 2 P ) + H (1 2 S), fo rwhich / A = 1 and / 8 = O . The allowed m 1 values for the separated atoms are then

m iA= 0, + 1 and m / B = 0; these atomic states must give rise to

(2 1 A + 1 )(2s A + 1 )(21 B + 1 )(2s B + 1) = 12 diatomic s tates. It helps to tabulate the

1 2 possible combinations of m i A , m, B , n i s A , ins B as shown in Table 4.1, which alsolists the resultant A = Int L I and M s values. In this table, an upward (downward)arrow den otes the value + ( --1-) for either in s A or m s B . W e obtain one E stateeach with Ms = + , and tw o E states with Ms =0 ; this is possible only if thereis one 1 E state (with Ms = 0 ) and three 3 E states (with Ms =0 , + 1 ). These E

Figure 4.5 Reflection symmetry of NaH diatomic states correlating with (a)

H(1 2 S) + Na(32S) and (b) H(1 2 S) + Na( 321: '2).



CORRELATION O F MOLECULAR STATES WITH SEPARATED-ATOM STATES 115

Table 4 .1 Diatomic states correlating with 2 P + 2 Satomic states

M /A

Mai0 A Ms +1

1 1

1 0

0o

1

T T 1 1

T 1 1 0

0

1 — 1

1 o

1 o — 1

1 T 1

1 1 — 1

s tates mus t be ± s tates , s ince Table 4.1 shows they arise only from atomicstates with m iA m 3 = O. Such states (composed from the H s orbital and the N ap z orbital oriented along the molecular axis) are even with respect to any a,

reflection (Fig. 4.5). Table 4.1 similarly re veals two H states e ach with M s = +1,and four H states w ith M s = O. These are naturally grouped into one and

one 3n manifold o f states; th e inherent twofold spatial degeneracy of n states(Section 4.1) is asserted by the appearance of two, four, and two (rather than on e,two, a nd one) states, respectively, with M s = — 1 , 0 , and + 1 .

T hese corre la t ions are summarized fo r NaH in Fig. 4.6 , which showstheoretical potential en ergy curves for all diatomic states which dissociate intoeither Na(3 2 S) + H(1 2 S) or Na(3 2 P ) + H (1 2 S). T he ground atomic s ta tes aresplit into the X i E + bound and a 3 E ± repulsive diatomic state s as the atoms arebrought together; the atomic s tates corresponding to Na(3 2 P ) + H (1 2 S ) are split

into th e A i E + , b 3 H , B i ll, and c 3 E + diatomic states. W e will see in the discussionof Hund's coupling cases (Section 4.5) that when the spin— orbit coupling is s mallbut nonnegligible, the orbital and spin angular mome ntum componen ts A andcan couple to form a resultant electronic angular mome ntum Q with possiblevalues D = A + S, IA— S. In the 3 1 1 states (A = S = 1 ) , the poss ible Dresultants are (1 + 1), . , (1 — 1) = 2, 1, 0. T he D values are notated assubsc ripts to the diatomic term symbols (and are analogous to the quantumnumber J in atomic term s ymbols); th e 3 /1- states are then s aid to be s plit into3n 2 , 3ni , and 3 H 0 sublevels un der s pin— orbit coupling. T he spin—orbitcoupling is so s mall in NaH that these three subleve ls have indistinguishable

energies on the scale of Fig. 4.6 , which shows only one potential ene rgy curve fo rthe b 3 1 - 1 state. In 1 2 (Fig. 4.7), the reverse is true: the A3n 1 ., B3n o u , and 1 1-2u



-I6220 -

-1 6 2 .24

I S 1 0 4

H -No ( 2 P )

H

-o( 2 S)

H(S)

-162,4

-162.44I

4 260R(bohr)


Figure 4.6 Potential energy curves for six electronic states of NaH. The curve

labeled "ionic" is the function e 2 14ne0 R, approaching the energy of Na + + H at

infinite separation. From multiconfigurational self-consistent field calculations by E.

S. Sachs, J. Hinze, and N. H. Sabelli, J. Chem. Phys. 62: 3367 (1975); used with

permission.

states have such widely separated potential energy curves as a consequence oflarge s pin–orbit coupling that they be have dyn amically like se parate elec tronicstates. (The incorporation of the g and u labels in subscripts of 1 2 s ta te termsymbols is necessary because 1 2 , unlike N a H , belongs to the p o o h point group.)

A brief aside about diatomic electron ic state notation is necessary here . Amolecule can have many e lectronic s tates of a given s ymmetry and multiplicity( e . g . , i n ), and additional symbols are neede d to tell them apart. The electronicground sta te i s always de noted with an X — a s in VI fo r N a H , X 1 E: fo r 1 2 .T he lowes t ex ci ted s ta te wi th the same multiplicity as the ground s ta te i ssupposed to be den oted wi th A, the next one up is B , and so on. The lowestexcited state with different multiplicity than the ground state s hould be labeleda, the nex t is called b, etc. These good intentions have not always been followed

historically, partly be cause there are frequen tly "phantom s ta te s"—sta te s thatare difficult to observe spectroscopically because of selection rules, and areoverlooked— so that labels that have become well established in the literatureturn out to be incorrect when phantom states are flushed out by improve dtechniques. T he way we have labeled the N aH potential energy curves in Fig. 4.5is orthodox, be cause it is positively known that there are no other NaH stateswith comparable o r lower ene rgy than the one s pic tured (our preced ingdiscussion should convince yo u of that!). However, in 1 2 the A and B state

0 „abeling in Fig. 4.6 is clearly wrong (they should not be capitalized fo r 3 11 and3H when the ground state is X'Eg+ ; they are not the lowest two states of

their multiplicity, either). These A and B state notations in 1 2 are well entrenched



s+'s1 0 0

0 44290

1 0

R (A)

Figure 4.7 Potential energy curves for electronic states of 1 2 . The sets of numbers

2440, etc., give orbital occupancies of the a 9 5p, nu 5p, n;5p, and cf 5p molecular

orbitals, respectively (Section 4.3). The X 1 E g+ , B 3 n o ., and A311 1 u potential energy

curves have b een characterized spectroscopically. U sed with permission from R. S.

Mulliken, J. Chem. Phys. 55: 288 (1971).

117




anyway. Such state prefixes should generally be regarded as interesting relicsthat draw attention to the most easily observed excited states; their supposedstructural implications should not be taken at face value.

We consider next the heteronuclear diatomic states that correlate with twoatoms in 2 P s ta tes , a s ituation that introduces complications not anticipated inthe simpler cases. Since / A = / 1 3 = 1 and S A = s B = t combining two 2 P atomscreates 36 diatomic states. By compiling a table similar to Table 4.1 (butallowing n t /B as well as m iA to range from — 1 to + 1) for the 3 6 possiblecombinations of m b k , inn3, insA, and m s B , one readily finds 8 A states with A = 2(two each with M s = ±1, four with Ms =0) , 16 it states with A = 1 (4 each withMs = +1, 8 with Ms = 0 ), and 1 2 E states (3 each with Ms = + 1, 6 withMs = 0). O ne may thus conclude that there are one A and one 3 A manifold ofstates with 2 and 6 states, respectively (since states with A 0 0 are spatiallydoubly degenerate), and that there are tw o '1 1 and two 3 11 manifolds totaling 4

and1 2 s ta tes , res pect ively.

The ques t ionthen

arises asto whether

the 1 2 E stateshave E + or E - character. The configurations corresponding to these 12 statesare l is ted in T able 4.2. T he f i rs t four configurations (states 4), through 4) 4 )

involve only p z orbitals with m i A = M / B = O , which point along the molecular axisand are even under a„. Hence states 4 ) 1 through 04 (one each with Ms = + 1,two with Ms =0) represent one 'I + and one 3 E + manifold of s ta tes . States 05

through 4) 1 2 are composed of p orbitals with nu= ±1 and in / B = ±1. Using thenotat ions p + and p_ fo r p orbitals with n i l = + an d m i = — 1 , respect ively ,properly antisymmetrized expre s s ions fo r states 4) 5 through 012 in the

Table 4.2 Diatomic E states correlating with 2 P + 2 P atomic states

M/A1 1 3

—1 11 1 state

T 1T 2I ) 3

1 1) 4T 5I }4„ ,4,

I . ' 6 5 ( P7

I 1) 8

T 9T 1 io, wiiolI 1) 1 2



0 - (0 5av(06 ± 010)

a(4 7 ± 010

ai,(08 ± 4) 1 2)

=

=

=

±(05 09)

±(06 ± 010)

±(0 7 ± 4 ) 1 1 )

±(08 ± 4) 1 2 )

(4.14)

± 09) = ±

C O R R E L A T I O N OF M O L E C U L A R STATES WITH SEPARATED-ATOM STATES 119

separated-atom limit are

05 = CpA_(1)pB+(2) — pA_(2 )pBA-(1)1a(i)Œ(2)

06 = DA-(1 )pB4-(2) — P A-(2 ) 1 9 B+( 1 ) 1 [ Œ ( 1 ) )6 ( 2) + a(2 )/3 ( 1 )]

07 = EpA-( 1 )PB+ ( 2 ) + P A -(2 )Po +( 1 ) 1 E 0 4 1 ) )6 ( 2 ) — a( 2 )fl( 1 ) 1

0 8 = [PA -(1 )PB + (2) — P A -(2)PB +( 1 )1 /8( 1 ))6(2)

09 = [PA + ( 1 )PB - ( 2) — P A + ( 2)PB - (1)]a(1 )a(2)

0io = E P A + ( 1 )Po - ( 2 ) — P A + ( 2)Po -( 1 ) 1 E Œ ( 1 )fl(2) + a( 2 ) )9 ( 1 ) 1

= [PA + ( 1 )PB -( 2 ) + P A + ( 2)P B -( 1 )1Earnfl(2) — a( 2 )/3 ( 1 )]

4 ) 1 2 = [PA + ( 1 )PB - (2 ) — P A + ( 2 )PB - ( 1 ) ] /3 ( 1 ) 1 3 ( 2 )

Since the at , operation converts the function exp( + i 4 ) ) into the functionexp(-T- i4) )—a, reverses the sense of any rotat ion by an an gle 4 ) about the zaxis—we have

avPA+

0- vPB+ = PB T4 .12)

with the result that

C r v05 =rv0o = 05

( 7 v06 = 010v010 = 064.13)

4 3 - v 07 = 011v011 = 07

4 7 v08 = 012v012 = 08

O ne may thus form linear combinations of states 4) 5 through 012 exhibitingdefinite parity under a y :

Hence, these linear combinations may be classified as shown in T able 4.3, usingthe properties of states 05 through 4) 1 2 taken from Table 4 .2 and Eq s . 4 .14 . It isapparent that these states yield one manifold each of 1 E + , 3 E + , 1 E - , an d

ith one, three , one , andthree states, respectively.




Table 4.3

State Symmetry Ms

05 ± 09 /± +1

05 - 9 +106 + 010 X + 0

4) 6 - 010 0

07 + 011 I+ 0

07 - 6 1 1 0

08 ± 4) 1 2 X + — 1

4) 8 - 012 — 1

In summary, the diatomic s tates that correlate with two 2 P atoms include

(4.15)

In the presence of spin— orbit coupling, the triplet states with A 0 0 become splitin to three compone nts wi th .f2 = A + S , . . . , I A — S I = A + 1, A, A — 1 . Theresulting diatomic states then become

1 A, 3 6,3, 3 .6 s 2 , 3 A1

'n, 3 n2, 3 n 1 , 3 n oln3n3

n 3 n2 55 (4.16)

Fina lly, it is instructive to discuss the homonuclear analogs to two of thesimplest cases t rea ted above. These are furnished by Na 2 , a well-studieddiatomic (cf. Figs. 4.1 and 4.3) whose ground state dissociates into 2Na(3 2 S ) . Thebound )0E: and repulsive a 3 E: diatomic Na 2 states co rrelate with the ground-

state atoms, as shown in Fig. 4.8. Aside from the inclusion of the g and u labelsappropriate to the D o h point group, these are en tirely analogous to the X'Eand a 3 / NaH states formed from ground-state Na and H atoms (Fig. 4.6).However , the nex t f ew Na 2 excited states correlate with the degenerateconfigurations

Na(3 2 S) + Na(3 2 P)

Na(3 2 P) + Na(3 2 S )

for which I A = 0, /8 = 1 and / A = 1 , 1 1 3 = 0, respectively. T welve diatomic statesarise from the former configuration, and 1 2 additional states (of the same e nergy



L i 2

80 12 14

R ( B O H R )

471

. I

n9

LCAO-MO WAVE FUNCTIONS IN DIATOMICS21

Figure 4.8 Multiconfigurational self-consistent field potential energy curves for

low-lying electronic states of Li 2 and Na 2 . Reproduced with permission from D.

Konowalow, M. Rosenkrantz, and M. Olson, J. Chem. Phys. 72 : 2612 (1980).

in the separated-atom limit) arise from the latter. As a result, 24 Na 2 statescorrelate with one 2 S and one 2 P atom, in contrast to the 1 2 N a H states thatdissociate into Na(3 2 P) + H(1 2 S ) . Corresponding to each of the latter states inNaH (Fig. 4.6), namely

AlE + ,b 3 H , BIll, c 3 E+

there is a pair of conjugate states with opposite inversion symmetry N a 2 :

b 3 1 1 „ ,E„ "

3 llg, ing 3 E+

Theoretical potential energy curves are shown for six of these N a 2 states in Fig.4.8.

4.3 LCAO—MO WAVE FUNCTIONS IN DIATOMICS

In our discussion of the Born-Oppenheimer principle (Section 3.1) we pointedout that eigen functions ith(r; R ) > of the electronic Hamiltonian

— h 2 n 2N el ie I

i=i 2m e7 r g o i=1 N= ,r i — R N I



122 E L E C T R O N I C STRUCTURE AND SPECTRA IN DIATOMICS

12 Z A Z g e 2

• 472 0r i— ri l

▪

4 7 r 8 0 IR A — R B I(4.17)

may be found fo r fixed nuclear positions RA, R B by solving the clamped-nuclei

e igenvalue equationf le ikkkfr;

Elc(R)Itkar;3 . 1 0 )

T he eigenfunctions depend parametrically on the choice of internuclear sep-aration R I R A — R B I . T he R-dependent eigenvalues e k (R ) act as potentialene rgy funct ions for nuclear vibrat ional motion when the Born-Oppenheimerseparation of nuclear an d electronic motions is valid.

It is illumina ting to treat covalent bonding in the simples t diatomic species ,the H molecule-ion. T he electronic H amiltonian (4.17) for H reduces to

— h 2 22 ( 1A )V ±

em ,ore 0 R r A r (4.18)

with r A Ir —R A I and rB Ir —R B I; it ex hibits no electron—electron repulsionterms. T he corresponding lq Schrbdinger equation can be solved exactly fo rfixed R. The exact ground-state potential ene rgy curve d isplays a minimum atR e = 2 . 0 0 a 0 ( 1 . 0 6 A) with ene rgy —2 .79 eV relative to the energy of the separatedproton an d ground-state hydrogen atom (Fig. 4.9). T he ex act solut ions for HIprove to be us eful in assessing the accuracy of variational wave func tions in thisprototype diatomic.

A widely use d approximation to t rue molecular wave funct ions em ploysline ar combinations of atom ic orbitals (LCA0s) to s imulate the molecularorbitals (MO s). At the lowest level of approximation, an 11 .21 - MO may berepresented as a superpos i tion of two H atom states ce ntered on the res pectivenuclei,

1 0 > = c A lO A > + c B I O B >4 .19)

with expansion coefficients c A , C B to be de termined by sym metry, normalization,and (in MOs with larger basis se ts) th e variational principle. Sin ce fq belongs

to the D c o h point group, 1 0 1 2 mus t be unaffected by the inversion i . This requiresthat I c A l = IcBI, and that the AOs IOA> and 1 4 ) B ) be ide ntical apart from phaseand the fact that they are centered on different nuclei. T he M O (4.19) may thenbe rewritten (for real AOs and expansion coefficients)

1 0 + > = c (1 0 A > ± 1 4 )B > )4 .2 0 )

No rmalization then d eman ds that

1 = c 2 ( < 4 ) A 1 0 A> ± 2 < d ) A l O s > + < O B I O B >

c 2 (2 + 2S B )4 .21)



Exac t

LCAO-MO WAVE FUNCTIONS IN DIATOMICS 123

b.0

0LU

—2

1 Figure 4.9 I-2+ potential energy curves: trial energies e (R) an d E_ (R ) obtainedfrom LCAO—MOsf r ,> and 1 4 , _ > , re spect ive ly , an d the exact ground-state potentialenergy curve. Energies an d separations are in eV and in Bohr radii, respect ively.

where S A B -a O fi A l O B > is the R -d e pe n d e n t overlap integral between the opposite

A0s. The MO then assumes the explicit form

1+ > =

vj2(1 ± SAB)(14A> ± 100)4.22)

In a trial L C A O — M O wave function for ground-state 1 -1, which correlates with

proton and a is H atom as R oo, we may use the hydrogen Is states

l > = exp(—r A /a 0 ) / .\»rag an d 14 B > = exp(—r B /a0 )/\brag for A0s. The overlap

integral using these AO s becomes AB = 7 - 3dVexp[—(r A + rB )/a0 ]nao

(4.23)



1 2 4LECTRONIC STRUCTURE AND SPECTRA IN DIATOMICS

which is readily evaluated using ellipsoidal coordinates to yield

SAB = e"(1 + P + P I 3 )4.24)

with p = Rlao (Problem 4.8). As the two nuclei coalesce (R 0 ), the overlapintegral approaches unity according to Eq. 4.24. SAB decreases monotonically tozero as the nuclei are pulled apart (R co). In this limit, the MO (4.22)consequently approaches

Itfr+> = 1 (14)A> ±10B>)4.25)2

In the same limit, the Hi' electronic Hamiltonian (4.18) becomes

— h2V 22 ( 1

2rn ens o r r)

e 22= "A11 4ne o r B1 e o r A(4.26)

where fi A and 1 4 B are the hydrogen atom Hamiltonians for atoms A and B.

Since

1 1 A 1 0A > = Eisk5A>

and

11BI4B> = EislOB>

the trial ene rg i e s e + may be evaluated in the limit R -4 co using the asymptoticEqs. 4 .25 and 4.26 , respect ively , for i t / i ± > and fi e i :

<0±111010+> = 1 / 2 ( < 0 A 1 1 -leilOA> ± 2<4A11101 0B> + <0B111010 B>)

= 1/2 (<. _ 4 ) A i f i A l2zgo vp i HA> ± 2<0 I R A l O B >rB

e 22 G A I I OB > + <OBInBIOB>- r7CE0 AO B I

2neoB 19 B> )

= lE is(<0 A i0A > ± 2 (4)A10B> + <OBIOB>)

= Els4.27)

Equation 4.27 follows because the integrals <O A I 1 / r B i c k A >O B I 1 /r A I O B > ,

<0Alq5 B > SAB, and < 0 A 1 1 / r B I O B > all tend to zero as R -4 co . Hence both of the



8 +(R) =+ S A B= HAA HB

(4.28)

LCAO —MO WAVE FUNCTIONS IN DIATOMICS 125

asymptotic L C A O — M O s (4 .2 5 ) are exact eigenfunctions of the asymptoticHamiltonian (4.26) with eigenvalue E 1 5 (the total electronic energy of the

dissociated fragments H + +H(1s)) . This correct asym ptotic behavior provide s a

partial justification of the L C A O — M O method. The trial energies e ± (R ) maynow be obtained for general R using Eqs . 4 .18 and 4.22 , with the res ult that

where

HAA --= HBB = < OAIRell OA>

e 2 2A<0A1/1 A10A>0 l 1 0 A > ± 4 n E B Rr7r80se 2

= Eis + J +4 7 r E 0 R4.29)

and

H A B H B A = < O A I F l e d O H >e 2 12= <OAIIIBIOB> A

r£ 0

(0 l0 B > 4 7 r E 0 R <OAIOB>-t

,2G. kJAB

= Eis SAB K +4n E B R

Here we have define d two new integrals, the Coulomb integral

— e 2

47reo< O A I —

rB1 0 A >

and the exchange integral

— e 2 K =<OAI0 1 3>47rEo (4.30)

(4.31)

(4.32)

As written in - Eq. 4 . 3 1 , the Coulomb integral is simply the expectation value ofthe energy due to electrostatic attraction between nucleus B and a i s e lec t roncentered on nucleus A. The ex change integral is an intrinsically quantumphenomenon, and has no analogous class ical electrostat ic interpretat ion.Collating the results of Eqs. 4.28-4.30, we finally o btain the trial energies

e 2+ K= E1 5 +4irc0R

+1 + SAB

(4.33)



126 ELECTRONIC STRUCTURE AND SPECTRA IN DIATOMICS

Since the Coulomb and ex change inte grals (which are both ne gative-definite)approach zero as R oo, the first term E ls in 8 ± (R ) i s the dissociation limit ofthe trial energy, in agreement with Eq. 4.27 . The second term is the electrostaticinternuclear repuls ion en ergy. I t may be shown (Problem 4.8) tha t analyt ic

expressions for the Coulomb and exc hange integrals are

J=

K=

e 2 [1 _e _ 2 p (i + 1)1

P L I

e -P( 1 + p ) (4.34)

4 7 t E 0 a 0

e 2

4nEoao

with p = Rlao . It is then straightforward to evaluate the LCAO-MO trialenergies e + (R), which are compared with the exact FI 2 ground-state potential-

energy curve in Fig. 4.9. The LCAO > generated f rom the positivesuperposition of AO s (Fig. 4.10 ) is a bound state with potential en ergy minimumat R e = 2 .50a 0 , correspond ing to an energy — 1 .78 eV relative to H + + H(1s ).This underest imates the true dissociation ene rgy (2.79 e V ) by 36% , and E + (R) lieseverywhere above the true ground state potential, in accordance with thevariational principle. The LCAO >, which has a nodal plane bisecting theinternuclear axis (Fig. 4.10), is a purely repulsive state in which the nucleiexperience a force pushing them apart at all finite R .

In the LCAO-M0 perspective of Eq. 4 .33 , the chemical bonding (i .e. ,lowering of total energy relative to H + + H ( 1 s ) ) in state 1/1 +> originates in part

from the electrostat ic conse quences of concen trat ing electron den si ty betwee nthe nuclei (the Coulomb term) and in part from the exchange term. I t is e asy toshow that the presence of an electron in regions between the nuclei electrosta-tically tends to draw the nuclei together, whereas an electron in other regionsexerts a net repulsive e ffect between the nuclei (Fig. 4.11). This suggests a

tempting rationalization of the bonding and repulsive characters of M O s 1 1 11 +>and h p _ > , respectively, in terms of the relative de grees to which charge dens ity isconcen trated between the nuclei [3,4]: the repulsive characte r of Ili/ -> could beattributed to the presence of the nodarplane , which diminishes the internuclearcharge dens ity. Such an interpretation overlooks kine tic energy effects (electron

localization in a l imited internuclear region increases the expectation value ofkinetic ene rgy), and analyses of the physical origin of che mical bonding are

advisedly made on the basis of accurate ra ther than zeroth-order LC AO-M Owave functions . A detai led exam inat ion of contributions to the total energyusing an exact ground-state 112 1 - wave funct ion reve als that chemical bondingarises from a subtle balance between e lectrostatic and kinetic energy effects [5].

Zeroth-order approximations to higher ex cited states in Fq may be obtainedfrom linear combinations of higher-energy hydrogen atom A0 s, s ubject to thesymmetry and norm alizat ion con straints of Eqs. 4.20 and 4.22. Like the states10 _0 fo rmed f rom the is A0s, the higher lying LCAO s yield inaccurate trial

ene rgie s— but the i r nodal patterns do furnish useful illustrations of the



Figure 4.10 Contour plots of the LCAO-MOs It//_> (above) and Ii//,> (below).

Curves are surfaces on which the wavefunction exhibits constant values; solid anddashed curves correspond to positive and negative values, respectively. The out-ermost contours in both cases define surfaces containing —90% of the electronprobability density. The incremental change in wavefunction value between adjacent

contours is 0.04 bohr -3 / 2 . The border squares have sides 10 bohrs long; the

internuclear separation is 2 bohrs, the equilibrium distance in ground-state H

Dashed straight line in Ilk_ > plot shows _location of nodal plane.

127



128 E L E C T R O N I C STRUCTURE AND SPECTRA IN DIATOMICS

-e

+Ze

e

-e_ 0- -

+Ze

Figure 4 .11 Electrostatic forces experienced by nuclei in a homonuclear moleculed ue to presence of an electron in region between th e nuclei (top) and outsid e thisregion (middle). The electron— nuclear attraction draws the nuclear together

in theformer case, and pulls them apart in the latter case. The region in which th e electron'spresence tends to stabilize the molecule is shaded in the diagram at bottom.

symmetry properties of one -electron orbitals in diatomic molecules. T he partialhierarchy of LC A O — M O s in F2 is illustrate d in Fig. 4.12, which shows contourplots of LC A O — M O s formed from the is, 2 s, and 2p A0s, and in Fig . 4.13,which gives schematic energy correlations between the AO s and the diatomicL CAO—MOs in H. In ana logy to the M O s formed from the is A0s, thepositive l inear combinations in Fig. 4.12 yield bound states that exhibit lowerene rgies than those of the se parated atoms with which they correlate (Fig. 4.13).T he negative l inear com binat ions , which all show nod al planes bisecting theinternuclear axis, yield repulsive states that are unstable with respect todissociation into the correlating atom ic states .

Since L z is conserved in each of these on e-electron orbitals , the wavefunctions mus t exhibit a 0-dependence of the form exp( + iy10 ) with A =0 , 1 , 2 ,... (W e use the lower-case notation A ra ther than A when discuss ing one -electron orbi tals; A is rese rved fo r characterizing the total component L z oforbital angular mo men tum in many-electron diatomics.) One-electron orbitalswith A = 0 , 1 , 2 , . . . are

denoted(7, it , ( 5 , . . .

orbitals, re spectively; the subscripts gand u are appended to indicate the behavior of the homonuclea r LCAO— M Osunder inversion. To differentiate between M O s having the same point group




1 7 r

2o -

lo-

B O N D I N G

NTIBONDING

Figure 4.12 Contour plots of LCAO-MOs formed from 1s, 2s, 2p2 , and 2p, AOs

(bottom to top) in the homonuclear diatomic molecule F 2 . Distance scale is in bohrs.

Solid and dashed contours correspond to positive and negative wavefunction values,respectively. Increments are 0.20 and 0.05 bohr -3 / 2 for inner-shell and valenceorbitals, respectively. Reproduced by permission from W. England, L. S. Salmon, andK. Ruedenberg, Topics in Current Chemistry 23 , 31 (1971).

symmetry, the quantum numbers ni of the AOs from which the LCAO s areformed are used as suff ixes in Fig . 4 .1 3. T he lowest two HI states 10_0

•

are both crls s tates , s ince these LC AO— M Os have no 0-dependence and wereformed from linear combinations of i s A0s . Since i tk ,> and Itt/ _> have g and uinversion symm etry, respectively, their orbital designations are ads and a u ls .W e s imilarly obtain bound 0 g 2s and repulsive o - u 2s M Os f rom the 2 s A0 s , and



130L E C T R O N I C STURCTURE AND SPECTRA IN DIATOMICS

Figure 4.13 Qualitative energy ordering of

MOs in I - 1 2+ . This figure should be compared withFig. 4.14.

bound ag 2p z and repulsive a u 2p z MOs from the 2p, A0s. Since the LC A Osformed from the 2p x and 2p y A0s, which are oriented perpend icular to the

molecular axis , behave as cos 0 ( = [exp(i0) + exp( — i0)]/2 ) and sin 0( = [exp(i0 ) — e xp( — i0)]/20, respectively, they are n2p orbitals. By inspection ofthe contour plots in Fig. 4.12 , it is apparent that the bonding and repulsive norbitals have u and g symm etry, respectively.

T hese one -elec t ron orbi ta ls may be use d to conce ptually bui ld up many-electron configurations in heavier diatomics using the Aufbau prescription ofplacing e lectrons in orbitals according to the Pa uli principle. The energyordering of M O s varies with the diatomic [6]. For the diatomics with highernuclear charge (e.g., 02 and F2 in the first row of the periodic table) the orbitalsare ordered as shown in Fig. 4.14 . In contrast to HI, the a toms in the semolecules exhibit large splittings between their 2 s and 2p AOs as a consequenceof configuration interaction (Chapter 2 ) . T here is thus co mparatively li tt le



LCAO-MO WAVE FUNCTIONS IN DIATOM ICS 131

as

cru

i s

Figure 4.14 Energy ordering of MOs in 02 andF 2 .r I s

mixing between the widely spearated o - .2 s and a g 2 p z MOs in 02 an d F2, so thatpartial hybridization endows them with relatively little o - g 2 p z and a g 2 s character,respect ively. In N2, C2, B2, B e 2 , and Li 2 the 2s-2p atomic splitting is smaller (itvanishes in H 2 ), and thus the energy di f fe rence be tween the pure ag 2 s and a g 2 p z

s smaller. Mutual mixing of the c ;r g 2 s and a . g 2 p z MOs in these lightermolecules then increases the splitting between them, inverting the ene rgyorder of the o - g 2 p and nu 2 p levels (Fig. 4 .15) . The electron configuration infirst-row diatomics can be read off from Figs. 4 . 1 4 and 4 . 1 5 by inspection.Ground-state N2 (which has 1 4 electrons) has the configuration(o - g 1 s) 2 (a t , 1 s ) 2 (a g 2s) 2 (a u 2 s) 2(7tu2p)4(0-g2p s 2 .) Since all of the MOs are fullyoccupied, ground-state N2 is a totally symmetric state with zero n et orbital andspin angular momentum. Hence A = 0 and S = 0 , and the pertinent termsymbol is VE:. Ground-state F2 (1 8 electrons) has the configuration(0.gis)2(0..is)2(ag2s)2(0..2s)2(ag2p)2(n.2p)4(ng2p)4; it is also an VI: s ta te , for thesame reasons.

Evaluating the poss ible term symbols for diatomics with partially filled MOsis slightly more involved. A 5 configuration (e.g., HI ( o - g ls ) i or Lq-(ag ls)2 (a u ls ) 2 (o g 2s) 1 ) has a single valence electron with 2 = 0 and s Since




Li 2 , B e2 , B 2 , C 2 , N 2

* 2 p

cr9

2p

vu 2 P

au 2s

a 2

a*

Is

Figure 4.15 Energy ordering of MOs in Li2 ,ll

Is

Be2 , 1 3 2 , C 2 , and N 2 : (a) prior to mixing of ag 2swith a g 2p and a: ; 2s with a 2p; and (b) after

(b )ixing.

filled orbitals do not contribute to orbital or spin angular mom entum, on eobtains A = 0 and S = 1 . Hence a c' configuration gives rise to a 2 E state. Bysimilar reasoning the + / — and g/u classif icat ions depend on the reflection an dinversion symmetries of the partially occupied orbital. The 1 - 1 2 ' and Li . grounds tates men tioned above are therefore both 2 E ; states; th e Li excited s tates withconfigurations (a g ls) 2 (o - u 1 s) 2 (o - u 2 s ) 1 and (o - g ls) 2 (o - u 1 s ) 2 (n u 2 p) 1 are 2 E: and 2 1-1 , 4

states, respectively.In a nonequ iva l en t a 2 configuration (in which two valence electrons occupy

differe nt a orbitals a a and a b ), the four possible distributions of electron spins areshown in Table 4.4 . W e find that we have a singlet state (M s = 0) and a triplet

state (Ms = 0, + 1). The sym me try d es ignat ion will be given by the directproduct of point groups to which MOs a a and a b belong. For example, if o- a and

a b are c g and a u orbitals with positive refle -cTion symmetry (e.g., LCAOs of s or pz

A0s), th e resultant states are l Eu+ , 3 Eu+ states because Eg+ E+ = Eu+.

The interesting and important equ iva l en t 7 r 2 case is typified by ground-state

02 , which has the configuration (agi s)2(o-u1s)2(0.g2s)2(7.2s)2(0.g2p)2(i.c.2p)4_

(ng 2 p) 2 according to th e level ordering in Fig. 4 . 1 4 . Since th e partially filled r t g

(a )




Table 4.4iatomic states from nonequivalent o-2

configuration

aa ab A Ms

I. 1

T4 .

T41T

oooo

+ 1— 1

oo

Table 4.5 Diatomic states from equivalent n2

configuration

A = — 1 = +1 A Ms State

Ti 2 o1 4 . 2 0 4) 2

T T o + 1T . 1 o o

4 T o o 9 5 54 I o — 1 4) 6

orbitals are doubly de gene rate , we count the ways of placing two e lectrons in

two n g orbitals with A = + 1 as shown in Table 4.5. We obviously obtain a 'Amanifold of states (A = 2 , M s = 0 ), and 3 E and 1 I manifolds as well. W e alsokno w that since n g 7r = g±g A, the latter three states s hould evolvefrom an equivalent n: configuration. Then the possible states are either1 5+ 3 E - A or 3 I +-g . To choose between these alternatives, we9 5 inspect some of the wave functions for the s tates counted in Table 4.5. Let n + bethe orbital for an electron with 2 = + 1, and n_ be that for an electron with

= —. Then the ant isymmetr ized state 4 ) 3 i s

4 ) 3 = [n + (1)n_(2) —4.35)

while the ant isymmetr ized state 4 ) 6 is

95 6 =( 1 ) 7 r- (2 ) —1 ) 7 r+ (2 )ifi( 1 ) f l ( 2 )4.36)

U n d e r the CT , operation, R z changes sign and the z component of spatial angularmomentum becomes reversed . This implies that

avir + = -

= it+4.37)




and so au 0 3 = — 49 3 , C r v 4 9 6 = —46 . Then 03 and 06, which are clearly M s = +

and Ms = — 1 components of a triplet state, are E rather than I + states. W emay take linear combinations of 4 ) 4 and 05 to form th e Ms = 0 tripletcomponent , [n ,(1)n _(2) — _(1 )n,(2)][a(1 )#(2) + a(2) ) 3(1)]. The l inear com-bination of 04 and 05 orthogonal to this forms th e 1 I + state,[n + (O n _(2) + n_(1 )7r + (2)][a(1)fl(2) — a(2)fl(1)]. Note that the latter linear com -bina t ion i s e ven unde r a, as required for a + s ta te . W e con clude that th epossible term symbols in an equivalent it configuration are 1 E:,

and A g • The lowes t three s ta tes in 02 are in fact X 3 E g- , a l A g and1 3 1 E:- ; the triplet state has the lowest e nergy according to Hund ' s rule.

The nonequ iva l en t 7 t 2 configuration can arise when one of two valenceelectrons is in a n u orbital and the other is in a n g orbital ( e . g . , in excited states o fC 2 ) . Finding the term s ymbols a r i s ing f rom this configuration is left as anexercise for the reader.

The atomic SC F calculations described in Section 2.3 may be ex tended inprinciple to diatom ic molecules with close d-shell electron configurations. Thediatomic electronic Hamiltonian in the clamped-nuclei approximation (Eqs. 3.7and 3.9) may be b roken down in to a sum o f one-electron o perators 11 andelectron repulsion terms e 2 14nv 0 rii ,

n r2 /lei = E [ — —2 ? - 7 -- V q,i= 1n v 0 N=1 Iri - R NI

n 1 2i=17rE0 i<.; Ir i- ri l

12+E4ne 0 i < ; Ir i- ri l

(4.38)

(For s ta t ionary nucle i , the nuclear repulsion term Z A Z Be 2 1 47reoIRA — R 8 1becomes a constant that may be added to the total energy a t the end of thecalculation.) The diatomic H amiltonian (4.38) is identical in form to th e many-e lec tron a tomic H amiltonian given in Eq. 2.62. One ma y write a single-determinant m any-electron wave funct ion an alogous to E q. 2.61 , with eac h ofthe spatial wave functions ( / ) i (with 1 j n/2) representing a doubly occupiedmolecular orbital. A set of equations an alogous to th e Har t ree-Fock equation(2.67) may then be solved nume rically to de termine the best obtainable single-

determ inant wave funct ion in the form of the Slater de terminant (2.61). T hediscrepancy between the resulting S C F electronic e ne rgy (given by Eq. 2.70, withH. denoting the diagonal matr ix e lem en t of the one -elect ron operator iîdefined in Eq. 4.38) and the true electronic ene rgy is the correlation error. Such aprocedure is so cumbersom e in diatomics and polyatomics that relatively fe wsuch calculations have been performed in spec ies o ther than a toms [7]. Inatoms, the centrosymmetr ic one-electron Hamiltonian fl i (Eq. 2.48) allowsfactorization of the atomic o rbitals into radial and angular parts. The latter canbe represen ted in atoms by spherical harmonics Y i ,,(0, 0, and the numericaloptimization of the H artree-Fock wave function becomes confined to the radial

coordinate. In diatomics, the angular depende nce of molecular orbitals und er




the cylindrically symmetric Hamiltonian is e 1 4 (Section 4.1), so that numericalSC F calculations must be performed over the two remaining coordinates. In

nonlinear polyatomics, the molecular orbitals must gene rally be optimized over

a full three-dime ns ional grid. Compoun ding this problem is the fact that SC Fenergies must also be calculated o ver a grid of molecular ge ome tries (bond

lengths and bond angles) to search for an equilibrium geome try.M ore commonly , the mo lecular orbitals in a single-determinant wave

function are expressed as l inear combinations of atomic orbitals (LCA0s),

I d ) i > = E IXi>ai i4.39)

where the I x ; > are atomic orbitals (A0s) ce ntered on the nuclei and the aii are

expansion coefficients. A min imal basis set of AO s includes a ll AO s that areoccupied in the separated constituent atoms. The coefficients may be variedto minimize the ene rgy. Parameters in the AO s thems elves may also be varied,but these are frequently fixed at values es tablished by prior expe rience with thesame atoms in similar molecules. ST Os (Eq. 2.58 ) may be used for the AO s in Eq.4.39. H owever, nume rical calculation of many of the resulting matrix elements

< W U > , and <ijiji> is slow u s in g S T 0 s , and the use of Gaussian typeorbitals (GT0s) of the form

iGn t .> = N r 7exp (— 07 ?)Y E . ( 1 9 , 4))4.40)

is far more econom ical [7 ,8]. For this reason, LCAO— MO — SCF calculationshave some t ime s em ployed AO s obtained by expres s ing STO s wi th knownparameters as l ine ar com binat ions of seve ra l GTO s . The inconvenience ofevaluating the resulting larger num ber of H amiltonian matrix e lem en ts i s morethan offse t by their efficiency of calculat ion us ing GT O s.

Improved accuracy may be obtained by using ex panded basis sets in single-determinant wave funct ions, but such calculations still do not remove thecorrelation e rror asso ciated with represe nt ing the electron— electron repulsion1 /r ii by the time-averaged expectation value <1/ri ; > . The configuration inter-action technique, which is an alogous to that de scribed in Section 2.3 for atoms,begins with a many-electron wave function cons is t ing of a superposition of theclosed-shell de terminant and addit ional determ inants in which electron s arepromoted to unoccupied orbitals,

q i ( 1 , 2,n) = E CNAN4.41)

The coefficients CN are obtained in a variational calculation. In a multicon-

figurational self-consistent field (MCSCF) calculation, these expansion coeffi-cients are simultaneo usly varied with the parameters a ii of the basis functions1 4 ) i > in Eq. 4.39. For detailed discussions of electronic structure calculations in



13 6LECTRONIC STRUCTURE AND SPECTRA IN DIATOMICS

molecules, the reade r is r e fer red to W. H . Flygare, Molecular Structure andDynamics (Prentice-Hall, Englewood Cliffs , NJ, 1978) and references therein.

4.4 ELECTRONIC SPECTRA OF DIATOM ICS

I t was em phasized in the introduction to this chapter that molecular electronictransitions are gene rally accompanied by simultaneo us changes in vibrationaland rotational states . A calculation of the transit ion energy of a particularspectrosco pic line thus requires kno wledge of the rovibrational energy for adiatom ic with vibration al and rotational quantum numbers v" and rn thelower diatomic state

E(v", J") = ol;(v" + — co'»c;(v" +• -

+ B1"(J" + 1) + D',;„.1"(J" + 1)2 + • • •

G "(v") + F (r)4.42)

along with the total energy

E(v% J') = G'(v') + Fc (J') + T 4.43)

in the upper electronic state. T is the en ergy se paration (conventionally in c m -1 )between the minima in the two electronic state potential en ergy curves. (If the

upper electronic state is purely repulsive, i ts se parated-atom asymptote is usedto ca lculate 'r .) T he transi t ion frequency in cm is then

= Te + [G'(v') — G "(v" )] + [F ( Y ) — F(J")]4.44)

which shows that numerous combinations of (1)% J') and (v" , J") can add richstructure to e lect ronic spec t ra in molecules . W e will se e la te r that the Elrotat ional select ion rules are reminiscent of the one s we der ived for purerotational and vibration— rotation spectra (although electron ic state symm etrymust be carefully cons ide red , us ing the H erzberg diagrams introduced in

Section 4.6). Since the upper and lower e lec t ronic s ta tes have unrelatedvibrationa l poten tials, it will turn out that all A v = — v" are El-allowed apriori in electronic transitions.

In the B orn-Oppenheimer approximation, the total diatomic wave functionsin the upper and lower states are

I V(1 " ,itVei(r; R)>IX(R)X. r (k )>4.45)

and

71" ( r , )>IXAR)XJ-(ii»4.46)



ELECTRONIC SPECTRA O F DIATOMICS37

Since changes in rotat ional and vibrational as well as electron ic state areposs ible , we must consider both the r- and R - dependence in the total dipolemoment operator when calculating El transition probabilities,

p = -E er i + E eZ N RN = p l ur nucl

For El transitions we must then have

<i" › = < 1 1 / 0 X 1 / 1 1 14 / :el +nucl.

=0 0

(4.47)

(4.48)

These expressions ignore rotation, which is considere d in Section 4 .6 . The termproportiona l to <1/4 1 I t k e I > vanishes due to orthogon ality of the electronic states.Note that < C i Z e ' I P e i I tfrav- > does not factor into < C I I I s e i l i K e ' i > < Z e ' I Z e , ), since theelectronic s tates IC I > and w > depend parametrically on R as well as on r.Instead, this m atrix e lem en t i s

f dRx: , (R)x v „(R) J driKet(r; R)PeitKi(r • R)Rx:,K4R)z,(R)M e (R ) (4.49)

where Me (R ) is th e (R-dependent) electronic transition moment f unc t ion . IfM e (R ) varies slowly over the range of R for which x(R) and z„, , (R) a resubstantial, i t becom es me aningful to write

< T '(r , R)Ifil 1 " ( 1 " ,Me(R) dR;C(R)X„-(R)

= M e(R )<Ov ">4 . 50 )

where M e (R ) is an averaged value of the electronic transition moment function.The probability of the electronic transition is then proport ion al to

R ) I pi w" ( r , R ) > 1 2 = Ime( R ) 1 2 1 < c l v " > 1 2

4 . 5 1 )

where K el y ">I 2 is called the Franck-Condon factor for the y" — > y ' vibrationalband of the electronic transition (we allude to i t as a band, since it exhibitsrotational structure in the ga s phase as shown fo r Na 2 in Fig. 4.3). Since th evibrational states Iv'> and le> belong to poten t ia l en ergy funct ions havingdifferent shape, the Franck-Condon factor Kely">1 2 can a s s u m e any value 1

regardless of y', y" • This is typified by th e see mingly random vibrat ional bandintensities in the Na 2 fluorescence spectrum of Fig. 4 .1 . It is only in the speciallimit where the upper and lower electronic states have iden t ical potential energy




curves that one obtains Kely">1 2 =ince it is only then that th e electronicstates have ident ical orthonorm al sets of vibrational states .

The Franck-Condon factors do obey a sum rule, however. If one sums th eFranck-Condon factors for transitions from a particular vibrational leve l y" in

the lower electronic state to the complete set of levels y' in the upper electronicstate, o n e obtains

E K e l v " > 1 2 = E <v"Iv'Xv'iv"> =

= 1

4.52)

by closure. This implies that summ ing th e vibrational band intensities over anelectronic band spectrum allows direct measurement of the averaged e lectronic

transition moment function Me (R ) according to Eq. 4.51 .W e now turn to the El selection rules embodied in the electronic transitionmoment

Me(R) = <VeliPeliK>4 . 5 3 )

T o have a nonvanishing matrix eleme nt (4.53 ) , it i s neces sary for the directproduct of irreducible representations

F(l/Iei) 0 Roe) 0I(

t k ' 1 )

to contain th e totally symm etric irreducible repres en tation (E+ in CD O E , h ) . Since th e components (1 2 0 ) „ , , , and (p e l ) z of the dipole mom ent operatortransform as H u and E U F in D u ,h , the direct products

r e l ) 0 ( 1 - 1D 0 TWOz

must contain th e Ig+ representa t ion for El transitions in homonuclear dia-tomics. (The g, u subscripts m ay be d ropped to gene ralize this discussion toheteronuclear diatomics.) In many homonuclear dia tomics , the electronicground state Iti/u ' i > is X l E g+ . Then e lectronic transitions from th e ground state tostate 11,fr> a re El-allowed if

0 (Hu) E + containswhich happen s on ly when pc) is e i ther n u or E t 1 - . Working out

similar direct products fo r electronic states>fh_ arb itrary symmetry



ELECTRONIC SPECTRA O F DIATOMICS39

would show that the E l selection rules fo r electronic transit ions in diatomics are

A A = 0, +1

+ 4 - - x - -* in E — E transit ions

u 4—> g , g x g, u < x >uLaporte rule)

Since p e , does not operate on the spin coordinates , we also have the El spinse lection rules

fo r small It o . When the spin—orbit coupling is large, the selection rule becom esAQ = 0 , + 1 . Examples of El-allowed transit ions under these se lection rules are2 1 1 1 / 2 _ 21 1 1 1 2 , 3 1 - 1 _lli, 2 5 /2 _ 2A 5 1 2 , 3 1 1 2 _ 3.6. 3; a pair of El-forbiddentransit ions would be 3 1 1 1 — 3H 2 , 2 1 1 3 , 2 _ 21 1 1 / 2 . Wh en A A = 0 , the z compo-nen t of <friIP e iIÇi> is nonvan i shing , and the electronic t ransi t ion is said to beparallel (i.e., polarized along the molecular axis). When AA = + 1, the x and/or y

components of the t ransi tion mo me nt are nonzero , and the t ransi t ion is termedperpendicular. The formalism de veloped in Chapter 1 implies that to effec t a n E lparallel absorptive transition in a diatomic molecule, the electric field vector E

of the incident electromagnet ic wave must have a componen t a long themolecular axis (the El transition probability amplitude is proportional toE • <00101 1 k1A Perpendicular transit ions require the presence of an E compo-nent normal to the molecular axis.

For M 1 electronic transit ions in diatomics , we inspect the matrix elem ents<CilLiVe'i>, where L is the orbital angular mom en tum opera tor . S ince thecomponents of L transform as the rotat ions (R x , R y , R z ) , the direct product

TWei) (F (Rx , R)"\

rWel) F(i1) (rigt f r e l )4.54)F(R z )»

must contain the totally symm etric irreducible repres en tation fo r M 1-allowedtransitions. This is satisfied only if

A A = 0, +1

u, g 4--> g , u x > ganti-Laporte)

+, + x > +, in E — E transit ions

AS = 0

It may similarly be shown that fo r E 2 electronic transitions in diatomics one has




the selection rules

A A = 0, ±1, ±2

u, g, u <ganti-Laporte)

A S =

T he A'I t ,+ an d B i ll states in N a 2 (Fig. 4.8) are two of the best-characterizeddiatomic states in the literature because the A'Eu+ X'Eg+ and B l rl u X'Eg+t ransi t ions are El-allowed, because these transitions occur at visible wave -lengths easily generated by tunable lasers, and because N a 2 vapor is easily made.T he 3 E: an d 'ri g Na 2 bound states correlating with Na(3 2 S) + N a(3 2 P) are lesswell known, because the E l trans i tions to these states from the ground state aresymmetry forbidden and because the M 1 and E 2 transit ions (symmetry-allowedfo r l E g+ ) are far weaker than El trans i t ion s (Chapter 1 ) . T he best-knownelectronic t ransi t ions in 1 2 are the A 3 1 - 1 1 . 4 - - X' 1g+ (red-IR) and 1 3 3 1 - 1 0 „4—X'E g+(green) transitions (Fig. 4.7); the latter is responsible for the purple color in 1 2 .They are El-forbidden according to the selection rules (AS 0 0 ), bu t they gainsmall E l intensity because the large spin—orbit coupling in 1 2 cause s con-siderable admixtures of singlet character into the A and B states and tr ipletcharacter into the X state [9]. These transit ions do obey the selection rule on A S 2

(= + 1and 0 , respectively). T he 3n 2 30E: transition (AS2 = +2) is extremelyweak, on the other hand, because it violates the AS2 = 0 , + 1 selection rule.

1 1 J 1 1 1 i1 L 1 L i 1 t 1 1Observedspectrum

I = 04

I I II =

0 56

v" = 2

v i = I 6 V

Figure 4.16 The observed X' Z. —> A' E L; electronic band absorption spectrum of ahigh-temperature Na, vapor is a superposition of spectra originating from y" = 0,1, 2, ... ; only the first three series are shown. While the band intensities arising froma particular y" level showsystematic variations (in fact, their intensity envelopes

exhibit the same number of nodes as the vibrational state Iv">), this regularity is not

apparent in the composite absorption spectrum.



ANGULAR MOMENTUM COUPLING CASES 1 41

The large number of 1 / 4-y" vibrational bands observed in an electronictransition can com plicate the analysis of electronic band spectra, particularly if

several of the initial y" levels have appreciable populations (e.g., when

c o : kTIhc atthermal

equilibrium). N eglecting rotat ional fine structure,becomes equal to Te + G ' (v ') — G " (v " ). If we should superimpose the spectraarising from, say, y " = 0 , 1 , and 2 and re call that the y' 4- y" band intensities areweighted by (seemingly) random Franck-Condon factors, the resultingelectronic band spectrum (Fig. 4 . 1 6 ) shows no comprehensible patterns, eventhough the regularity would be apparent if only a single progression of bands

(e.g., from y" = 0) were present. The Ritz combination principle o ffers a brute-force method of assigning (y' , y") com binations to the bands: one obtains allpossible difference frequencies between pairs of band frequencies i,"(y', y") todetermine if particular difference frequencies crop up repeatedly. For example,

[ V ,(3, 0) —0)] and U(3 , 1 ) — 1 3 (2, 1)] and U7(3, 2) — 2)] must all equa l G '(3) — G '(2), and this gives a start in organizing the assignment of the spectrum.This approach is kn own as a Deslandres analysis (Problem 4.6) . A preferableapproach is to simplify the spectrum ex perimentally: the use of supersonic jetsproduce s gase s with very low vibrational tempe ratures, e ss en tially populatingonly y" = 0 (Fig. 4.3).

4.5 ANGULAR MOMENTUM COUPLING CASES

The expressions we derived for diatomic rotational ene rgy leve ls in Chapter 3are applicable only to molecules in- 1 E s tates . In such molecules, the magnitude1 J1 of the rotational angular mom entum is cons erved, and the rotor ene rgy levels(E, = BJ(J + 1) in the absence of cen trifugal distortion and vibration—rotationinteraction) are those of a f reely rota ting molecule whos e nuclear angularmome ntum is uncoupled to an y other angular momenta. In more general cas e swhere A and/or S i s non zero, i t is the magnitude of the total (electronic plusrota tional) ra ther than jus t rota tional angular mom en tum tha t i s con se rved.Seve ral coupling scheme s for spin, orbital, and rotational angular momenta maybe identif ied, de pending on the magnitudes of the magnetic field generated bythe electrons ' orbital motion and the spin— orbit coupling.

Hand's case (a ) describes the majority of molecules with A 0 0 that exhibitsma ll spin— orbit coupling. Since the electronic orbital and spin angularmomenta L and S are not mutually strongly coupled, they precess independentlyabout the quantization axis (the mo lecular axis ) established by the magneticf ields arising from e lectronic motion. The projections Ah and E h of L and Srespectively along the molecular axis are then conserved, as is their sum Oh. Incontrast, the parts of L and S normal to the molecular axis oscillate rapidly; theyare denoted L1 and S 1 (Fig. 4 . 17) .

For the total angular momentum J , the quantity 1 J 1 2 =)h 2 is

necessarily a constant of motion. (J was previously used for rotational angularmomentum in I s ta te molecules in Chapter 3; i t i s now res erved for total



142 ELECTRONIC STRUCTURE AND SPECTRA IN DIATOMICS

Figure 4.17 Hund's case (a) coupling. The orbital and spin angular momenta L

and S precess rapidly about the molecular axis with fixed projections Ah and Eh. The

total angular momentum normal to the molecular axis is in effect the rotational

angular momentum N, since the normal components 11 andS 1 of L and S fluctuaterapidly and their expectation values are zero. The total angular momentum J is thevector sum of N and 0, the projection of (L + S) upon the molecular axis.

Interactions between L i, S iand the rotational angular momentum N give rise to A-

doubling (see text).

(electronic plus rotational) angular mome ntum in diatomics with A 0 O.) Sincethe rotat ional angular mom en tum vector N must be normal to the molecularaxis , it has no z compone nt— which implies that the projection of J along th e

molecular axis must be (L + S) z = f2h (Fig. 4.17). According to th e inequality1 J 1 2 J!,we must then have J(J + 1 )h 2 12 h 2 , or JQH e n c e fo r given /2 ,the allowed J quantum numbers for the total angular mom entum are

J = f2, 12 + 1, 2 +2, ...4.55)

W e finally compute th e rotational energy E r o t = 1 N 1 2 /2 1 for a case (a) molecule.Since the portion J1 of Jwhich lies in a plane perpen dicular to the molecularaxi s is

= (L + S) 1 + N4.56)

we have

j2

= (L 1 + S 1 ) 2 + N 2 + 2N (L1 + S 1 ) + J4.57)

The classical rotational en ergy is then

E r o , = J — (L1 + 5 1 ) 2 — 2 N - (L 1 +4.58)



ANGULAR MOMENTUM COUPLING CAS ES 143

The quantum analog of this expression for the rotational energy in cm is (inthe absence of cen trifugal distortion)

k = 13,[J(J +1)— (2 — < ( L 1 + S 1 )2

> — 2<N • (L 1 + SI)>] (4.59)where the brackets < > denote expectation values. The termsf -2 2 <LI + S1) 2 > are often lumped with the vibronic energy, in which case therotational energy is

k = BAJ +1)— 2B<N • (L I + SI ) >

4.60)

The first term resembles the rotational energy o f aolecule, but is subject tothe restriction J D . The second t e rm de scr ibes the electronic-rotational

in teractions, which constitute a sm all perturbation to the rotational energy inmolecular states with A O. In the absence of electronic-rotational inter-actions, such states are doubly degene rate with compone nts ex hibiting + and

— behavior unde r cr, (Section 4.1). This degen eracy is s plit by the perturbation. 8 ,1 N T • ( L 1 + S1 ) , producing closely spaced doublets of rotational en ergy levelswith opposite reflection sym metry (Figure 4.18). This phenome non i s known asA-doubl ing .

For atoms with Russ ell-Saun ders c oupling (those in which IL can be treate das a perturbation) the spin-orbital correction to the ene rgy was see n to behave

J

+

J

5

4

3

2

I

+

J

4

3

2

-

+

6 _

_

-

5 +

+

7 1 :

+_

37

2

4

3

-

: 1 ' -

+_

+3

7 1 - i

2

I

-

-1 7

0 +

3Tr0

Figure 4.18 Rotational energy levels in Hund's case (a) for a 3n0 state, a 3 n I

state, and a 3n 2 state. No levels appear with J < O. Each J level is split by A-

doubling into sublevels with opposite reflection symm etry. The 0-doubling is greatly

exaggerated.




as A[J(J + 1) — L(L + 1 ) — S (S + 1 )]. The corresponding spin—orbi tal energyin diatomics ha s the form A S V , where A is a spin—orbital constant [ 10 ] . Thetotal energy in excess of the vibronic en ergy (cf. Eq. 4.59) then becomes

E r o t = 1 3 1 1 ( J + 1 ) — OL I + S 1 ) 2 > — 2<N ' (Li + SI)>]

+ (A — 1 3, ) .0 24.61)

The differences arising from the (A — 1 3 , ) 1 2 2 term in the rotational energies ofspin—orbital compone nts with differen t 1 2 are reflected in the case (a) energylevel diagram in Fig. 4.18.

In Hund's case (b) the magnetic fields due to orbiting e lectrons are so weakthat the electron spin angular mome ntum S does not precess about themolecular axi s . M olecules with A = 0 (E states) and S 0 belong to cas e (a) bydefault, but light molecules (with few e lectrons) having A 0 0 may also e xhibit

case (b) coupling. For simplicity, we res trict our discussion to case (b) moleculeswith A = 0 . The ro ta tional angular mom en tum N is directed normal to themolecule axis. In the absence of orbital angular mom entum, the local magneticfield is dominated by molecular rotation, with the result that S precesse s about J(Fig. 4.19). The rotat ional ene rgies to zeroth order are

E,.., = 1 3 „ N ( N + 1 )4.62)

with N = 0, 1, 2, ... Eac h rotat ional level N may be s pli t by spin—rotationcoupling into sublevels with total angular mom entum quantum number

J = N + S, N + S — 1, , —4.63)

A schematic en ergy level diagram for rotationa l states in 2 E and 3 E molecules isshown in Fig. 4.20 .

In Hund's case (c) the spin— orbit coupling is so large that L and S aremutually coupled to form a resultant J. which precesses about the molecularaxis with fixe d projection (Fig. 4.21). The rotational e nergy levels are givenby the same expression, (4.61), as in case (a). An important distinction betweencases ( a ) and ( c ) is that while the spin—orbit contribution A S 2 2 to Eq. 4.61 i s

Figure 4.19 Hund's case (b) for A =0. The spin

angular momentum S and the rotational angular mom-

entum N precess about the total angular momentum J.



ANGULAR MOMENTUM COUPLING CASES 145

11/12 55 6/2

9/2

54

44 7/2 4 5

7/2

3

33 5/2 3

5/2

42

22I

3/2

3/2

2

1

3,1

1,2,01/20 1/2 0 I

Figure 4.20 Rotational energy levels in Hund's case (b) for a 2 1state and for a 3 E

state. Each N level (Eq. 4.62) is split into 2S + 1 sublevels through the interaction

between the rotational angular momentum N and the spin angular momentum S.

generally the order of the rotat iona l level spacing or smaller in case (a), itbecomes large enough in case (c) to endow the different D component s of 2 S + 'As tates with individual potential ene rgy curves . This is what happens in the

B 3 Hou, A 3 H1„, and 3 1 - 1 2 „ states of 1 2 (Fig. 4.7), and these are good exam ples ofcase (c) coupling in a heavy molec ule with large spin—orbit coupling. Since L

and S do not precess independently about the molecular axis in case (c)

Figure 4 .21 Hund's case (c) coupling. The orbital and spin angular momenta L

and S are strongly coupled to form a resultant J.. The projection of J. on the

molecular axis and the rotational angular momentum N combine to form the total

angular momentum J. L and S clearly do not have well-defined projections on the

molecular axis, and so case (c) molecular states cannot be strictly characterized interms of the quantum numbers A and E .




molecules , A and E (unlike fl) are not good quantum numbers, and thedesignat ion of these 12 states as 3 H states i s n ot rigorous.

4.6 ROTATIONAL FINE STRUCTURE IN ELECTRONIC BAND

SPECTRA

In Se ction 4 .4 we worked out the El electronic and vibrational selec tion rulesfor electronic band spectra , and it rem ains for us to determine the selection rulesthat gove rn the rotational fine structure. W e have se e n that no symm etryse lect ion rule ex is ts fo r Av, but that the vibrational band intensities areproport ional to Franck -Condo n factors in the B orn-Oppenheime r approxi-mation. T o understand the selection rules for simultaneous changes in electronicand rotational state, we must find how Ilk i x r „,> = kiei>IJM> t ransforms underthe symme try operations of the molecule. A complication arises here because we

have derived our e arlier pure rotational and vibration—rotation selection rulesusing space-fixed coordinates (i.e., the rotational state IJM> depends on angles(0 , ( ¢ ) relative to Cartesian coordinates (x , y, z) , which are fixed in space and donot rotate with the mo lecule), where as our group theore t ical E l elec tronicselection rules fo rere de rived using electronic s tates W ei> ,which depen d on molecule- f i xed coordinates (r A i , rB i , 4)) fo r each e lectron i ; thesecoordinates do rotate with th e molecule . T o con sider s imultaneous rotationaland electronic state changes, we must use one set of coordinates consistently.

In heteronuclear diatomics It i / e ao t > can be classified according to its behaviorunder molecule-f ixed ref lection o r , . W e a lread y know tha t the molecule-fixeds tates M e l > obey

( 7 „ 1 0 0 > = 1 0 0 >or E + states

= — Mel>ortates

and that states with A 0 0 are doubly dege ne rate (at least prior to A-doubling)with com pone nts that can be chos en to exhibit + and — behavior unde rWhile the rotational states IJM> are space -fixe d, i t can be s hown that [11]

a t , (molecule-fixe d) = i (space-fixed)4.64)

with the res ult that [12]

o - „ (molecule-fixed)IJM > = (space-fixed)Y, m (0, 4)) = 17,m (n — 0, + n) (4.65)

= ( — 1 ) 1 J M >

Consequently,

av(molecule -fixed)lOarot> = (±)( TItifelXrot>4.66)



ROTATIONAL FINE STRUCTURE IN ELECTRONIC BAND SPECTRA 147

where the upper ( +) sign applies to /- states and the Inert—) signapplies toI states. In the remainder of this sec t ion we us e ( + ) and ( — ) te • indicate th ebehavior of 1 1 / / e ao t > under a„ (molecule-fixed); this s hould n ot be c onfused withth e superscripts in the electronic state notations X +, . We finally determinehow p is influence d by ay (molecule-fixed). Since this ope ration is equivalent to i

(space-fixed), and the latter operat ion converts p into —0,

v (molecule-fixed )p =

4.67)

It follows that fo r < C a r 'o t l f i l l i/ e 'O C o t > to be nonvanishing, th e s tates and

KIX:!cd> must exhibit opposite reflection behavior under a, (molecule-fixed), and( +) s tates can combine only with ( — ) states in the El approximation. T o k e e ptrack of all rotationa l transitions that are consistent with this rule for a given

electronic transition, we se t up a Herzberg diagram [ 1 0 ] in which the rotationalJ levels in the relevant electronic states are labeled with their behavior unde r o - „

(molecule-fixed). For definiteness, suppose we are interes ted in a l Eelectronic transition (which is El-allowed according to the se lect ion rulesobtained in Section 4.4). Using Eq. 4 .66 , the pertinent Herzberg diagram can begenerated as shown in Fig. 4.22, where the arrows show all allowe d transitionssubject to AJ = 0, + 1 connecting levels that are ( + ) under a t , (molecule-fixed )

- with levels that a re (—). This diagram e xhibits on ly R- and P-branch transitions.N o Q 7 branch (AJ = 0 ) transitions are possible, since they would conne ct ( + )with ( +) or ( — ) with ( — ) levels. A more interest ing example i s the case of a

1 E + 111 transition. The-pertinent Herzberg diagram (Fig. 4.23) reflects the factthat doubly de gene rate In s ta tes can have both ( + ) and ( — ) component sunder u„ (molecule-fixed) for any J, but cannot have J =0 . (Re call that werequire J 12, and . 0 = 1 in a 1 1 1 state.) In this case LIJ = 0 transitions arepossible as well as AJ = + , transitions, so Fig. 4.23 provides an example of thefact that Q branches can appear in electronic transitions involving s tates withA 0 .

In homonuc l ear diatomics an add i tiona l symm etry e lemen t i s nee ded toclassify state symmetries in the p o o h point group, and we can us e i (molecule-fixed) for this purpose . A new complication, peculiar to homo nuclear molecules

J" o

Figure 4.22 Herzberg diagram for allowed rotational transitions accompanying a

'ZVelectronic transition in a heteronuclear diatomic molecule.



1 4 8LECTRONIC STRUCTURE AND SPECTRA IN DIATOMICS

2 Figure4.23 Allowed rotational transitions in a 1 E + -*Ill electronic transition.

Note the absence of the Q(0) an d P(1) transitions, since J' 1 in a F l state.

in which both nuclei are the s a m e isotope, is the effect of nuclear exchange

symmetry on rotational state populations. Under the space-fixed nuclear

exchange operation XN, the total wave function I W lth YY=.7ucl.spin>

must behave as [ 1 0 ]

X N I W > = + I W >or boson nuclei ( s )

= -- 1 W >or fermion nucle i (a )4.68)

Since the vibrational state I Z v i b > depends only on IRI = R , it is unaffected byexchange of the nuc le i . To dec ide the effect of XN on 10 , ,z r , „ > , we use the identi ty[ 1 0 ]

XN =4.69)

where j 1 represents inversion of all particles through the origin and i 2 representsinversion of electrons only through the origin. We have already seen that

I JM> = ( — 1) - 1 1 z r 0 t > . For the effect of j 1 on I t f r e i > , we note thatC 2 o - „ in D h , where the latter two operations are applied to all particles in

the molecule . C2 then has no effect on N i e l > , s i n c e N i e l > depends only on theelectronic coordinates relative to the positions of the nuclei. Then

ii1 0 0 > = a „ 1 0 0 > = 1 0 . 1 >

or + states

= —1 1 / 0 >or E - states4.70)

because a, reflects the positions of the elec t rons , but not of the nuclei. Theinversion i 2 does not affect 'L o t > because it inverts only the electioniccoordinates. By definition,

1211Pei > = ItPei>n g s ta tes

= — 1 1 1 / 0 >

n u states

4.71)



ROTATIONAL FINE STURCTURE IN ELECTRONIC BAND S PECTRA 149

W e finally consider the effect ofXN on the nuclear spin function I t h nucl . spin>• Eachnucleus has spin I with (2 1 + 1) magnetic sublevels, giving a total of (2 1 + 1) 2

diatomic nuclear spin states. O f these, (2 1 + 1)(/ + 1) states are always sym-metric (s) with respect to XN, and (2 1 + 1)1 are antisymmetric (a). Forconcreteness , consider the H2 molecule in which both nuclei are 1 H with spinI =4. From the nuclear ma gnetic sublevels a (m 1 = - ) and fi (m i = —4 ) on theindividual nuclei, we m ay form the symmetric (s) and antisymmetric ( a ) nuclearspin states

a(1)a(2)s)

a ( 03(2) + 42)13(1 )s)

/3 ( 1 )$(2 )s)

a(1)fi(2) — a(2)fl(1 )

a)

for the diatomic molecule . For I = I-, there are (2 1 + 1)(/ + 1) = 3 and(2 1 + 1) 1 = 1 nuclear spin states that are symmetric (s) and antisymmetric (a),respectively, unde r XN.

The possible combinations of electron ic, rotation al, an d nuclear spin statescan no w be compiled [ 1 0 ] as shown in T able 4 .6 . T he total wave function IT >must be (s) and (a) under the nuclear exchange XN for boson an d fermion nuclei,respectively. Fermion nuclei ex hibit half-integral spin: I =-4 (1 F 1 , ' H e , ' 3 C),I =4(23a), I =4(17), etc. Bosons include inte gral-spin n uclei, such as 'He,

1 6 0 (I =0 ) and ' 4 N (I = 1 ) .Fo r 1 1 1 2 in it s X l E g+ ground s ta te , the nuclei are fermions with I =4. T he

exchange symme try o r i t f r Yr

X N 1 1 fr elZ rot> = i 1 i2 itkelXrot >

= R-1Y(±)][-F]ltPeixrot>

= (-1)JliPearot>4.72)

so that I d / el IL r o t is (s) for even J and (a) for odd J. T he total wave function I W>T

must be (a) for these fermion nuclei . According to Table 4. -6 , the X'tate

Table 4.6

Statistical

l earot>

VI>eight

a a s ( 2 1 + 1)/s a a ( 2 1 + 1)/

a s a ( 2 1 + 1 )(/ + 1 )s s s (2 1 + 1 )(/ + 1 )

under XN is




rotational leve l pupulations in a thermal gas gain extra factors (beyond th eBoltzmann factors in Eq. 3 .28) of (2 1 + 1)1 = 1 fo r even J and(2 1 + 1)(1 + 1) = 3 fo r odd J. This leads to a nearly 3: 1 intens ity alternation inrotat ional s tate populations of adjacent levels in X l E g+ 1 H 2 . For 1 6 02 in itsX 3 E, T ground state, the nuclei are bosons with I = O . The exchange symmetry of10e1Xrot> in this cas e becomes

iii2liPearot > = — 1 Y ( ) JE + 1 1 1 f r o x . >

= (— l)J'llPearot>4.73)

so that l i f r e i x ,„> is (a) for even J and (s) for odd J. Since I T> must be (s) in thismolecule, the rotational state populations gain statistical factors of(2 1 + 1)1 = 0 for even J and (2 1 + 1)(/ + 1) = 1 fo r odd J. Consequently,X36kJ cannot exis t in even - Jlevels at all. Such levels can be populated in

other electronic states (e.g. , a Ag) of 1 6 02 , however.Prior to digressing on the subject of nuclear exchange symm etry, wemen t ioned tha t a new sym met ry e leme nt be s ide s o - , (molecule-fixed) wasrequired to class ify e lectronic-rotat ional s tates in homonuclear diatomics. Alogical choice is i (molecule-fixed), an operation which belongs to D c o h but notC. I t may be s hown that î (molecule-fixe d) is equivalen t to XN (space-fixed),and so the procedures worked out in the foregoing discussion may be used toclassify I t P e l x,..,> as either (s) or (a) unde r XN in lieu of dete rmining their behaviorunder m olecule-fixed inversion . The d ipole moment operator p in homonuclearmolecules is (s ) under XN [1 1 ]. This leads to th e conclusion that only states10eIXrot> with l ike sy mm etry under XN can be connec t ed by El transitions inelectronic band spectra:

The Herzberg diagrams for homonuclear diatomics can now be augme nted with

(s) and ( a ) labels deno ting behavior under XN. For a 1 E: 'Eu+ transition, thediagram is shown in Fig. 4.24, where a l l of the El-allowed transitionss imultaneously obey ( + )4-4 ( - ), (s)4-*(s ), and (a)4-* ( a ) . As in 1 E 1 E +

transitions for heteronuclear molecules, only R and P branches can a ppear. It iseasy to show that the Herzberg diagrams for l Eg+lEg+ and l E u+lE u+

orbid any transitions from occurring at all (i.e., the Herzbergdiagrams have the Laporte rule built into them). For a 'E g+ transition(Fig. 4.25), all three rotation al branches appear. Note that the P(1) a n d Q ( 0 ) linesare absent, since the J =0 level cannot occur in a state. A wealth ofadditional Herzberg diagrams m ay be found in G. H erzberg's classic Spectra of

Diatomic Molecules [10].



o 2 3

2 3

a

J" = o 2

2

a

3 Liu

ROTA TIONA L FINE STRUCTURE I N E L E C T R O N IC BAND SPECTRA51

Figure 4.24 Allowed rotational transitions in a 1 E: -+ 1 E u+ electronic transition in a

heteronuclear diatomic molecule. Such Herzberg diagrams automatically incorporate

the Laporte g 4-4 u selection rule, since no allowed rotational transitions can be drawn

for a 1 E: 4-+ 1 E: or a1 Eu+ 4-0 1

1u+ electronic transition.

Figure 4.25 Allowed rotational transitions in a 1 I+ — > 1 n u transition.

The Na 2 fluorescence spectra in Fig. 4.26 are excellent example s of therotational selection rules predicted by the 1 E , . 4 . 'H Herzberg diagram in Fig.4.25. The first of these spectra was excited using the 4727-A line from an Ar +laser, which matches the energy difference between l E g+ (y" = t, J" =37) and1 1 1 u (y ' = 9, J' =38) in Na 2 . The laser l inewidth was considerably broader thanthe energy separation (due to A doubling) between the (a) and (s) 'H. levelsbelonging to J' =38 in Na 2 . Since the initial J" =37 level is an (a) level in a E g+

state, however, the se lec t ion rules in Fig. 4.25 show that only the (a) sublevel in= 38 can be reached from J" =37 in El transition. Subsequent fluorescence

transitions from 'Hu (I/ = 9, J' =38) to the l E state in various y ' l evels can onlyterminate in J" =37 or 39 according to Fig. 4.25. H e n c e , the 4 1 „ l E g+fluorescence spectrum excited at 4 7 2 7 A exhibits only P and R branches. Thesecond spectrum was excited by the 480 0-A Ar + laser line, which matches theenergy separation between 1 E: (y " = 3, J" =43) and 'H u (y' = 6 , J" = 43). Sucha transition can populate only the (s) sublevel in J' =43; reasoning similar tothat outlined above shows that fluorescence f rom y' = 6, J' =43 can then



1 0 1 5 1 09

A , I ILl1Loser

10 31

1 0 1 4 1

IQ 1 3 1

01bl 107j IQ 5 1 fo o l

1 0 1 2 1

717

LI( A . 1 1-

1 0 1 7 1

01611 C ) 1 1 1

lo

1R,P21

Loser

1R,P51 1R,7 1R,P9 1 [R, P 1 4 1

1 R , P 1 8 1 1R, P17,

1 1

P 1 2 1flII

1R,P01

4 7 0 0

R,P 31 1 [R ,P 41

1 1

4 8 0 0

9 0 0

15 0 0 01 0 0

Wavelength,

1R,P131I II

L tlt5003 0 0

1R, P19 I

1 R , P 2 0 1

H IR, P211

1

5 4 0 0

1R, P151

R , P 1 61R,P61

1 11R,P61

1 R , P 1 0 11 1

LJ

(a)

5 4 0 03 0 02 0 01 0 00 0 09 0 08 0 07 0 0

Wavelength, , 4

(b )

Figure 4.26 Bl n u -÷ E: fluorescence spectra of Na 2 vapor following excitationby an argon ion laser at (a) 4727 A and (b) 4880A . The first spectrum is due to the

transitions V = 9, J' = 38 J" = 37, 39; its rotational fine structure exhibits only

P and R branches. The second spectrum arises from the transitions v' = 6,

J' = 43 V, J" = 43; only the Q branch appears in its rotational structure. These

spectra are excellent examples of the selection rules in Fig. 4.25. Monochromatic

laser excitation of an X' E: molecule creates a Fn u molecule in either an s or an a

level, but not both. According to Fig. 4.25, the !eve: can consequently fluoresce with

either (P , R) or Q branches, but not both. The numbers in boxes give y" for the lower

level in fluorescence transitions. Vibrational band intensities are proportional to the

Franck-Condon factors Kylv">1 2 . Reproduced with permission from W. Demtrbder ,

M. McClintock, and R . N. Zare, J. Chem. Phys. 51: 5495 (1969).

152



ROTATIONAL FINE STRUCTURE IN ELECTRONIC BAND SPECTRA 153

exhibit only a Q branch. Hence, the (y', y " ) fluoresce nce bands are do ublets in thefirst fluorescence spectrum, but are singlets in the second one. The bandintensities are proportional to the pertinent Franck-Condon factors Ky'ly">J 2 in

both spectra.Rotational l ine ass ignments are easily made by inspe ction in pure rotationaland vibration-rotation spe ctra (Chapter 3). In the former case , the rotationalenergies and transi t ion frequencies are

Eret(J)/he = BAJ + 1) - DJ 2 (J ± 1 ) 24.74)

and

= 2BJ - 4DJ 34.75)

where J denotes the upper level in absorptive transit ions ( A T = + 1). Since thecen trifugal dis tort ion constant D is generally small compared to B

(D/B 1O), the rotational line s are very nearly equally spaced in frequency.For vibration-rotation spectra, the P-, Q -, and R-branch l ine f requencies(ignoring c en trifugal distortion) are

= -Ç ' 4 3")J (13' - B")J 2

i ) , z ) =B' - B")J (B' - B")J 2

4.76)

yR =i5 0 2-B + (3B' - B")J (B' - B")J 2

where J pertains to the lower level in the t rans i t ion ands the vibrationalene rgy difference . Since B' and B" are nearly the same in vibration-rotationspectra (because B varies weakly with y within a given electronic state), Eqs. 4 .76imply that rotational line s in such spe ctra will also be roughly equally space d infrequency fo r low J (Fig. 3.3).

For rotat ional fine structure in e lectronic band s pectra, the P- and R-branchl ine pos i t ions are still given by Eqs. 4.76, ex cept that i 5 0 now becomes

+ G'(//) - G"(v"). The important physical differen ce here is that B' and B" areoften grossly different in trans i t ion s between differen t electronic states. Forexample, H and B e are 0 . 0 2 9 and 0 . 0 3 7 cm -1 , respectively, for the1 3 3 11., , <- X 1 E: transition in 1 2 (Fig. 4.7). T he rotational line positions in the P

and R branches a r e n o longer even approximately equally s paced. For thatmatter, i " , p and 1 3 R both vary as (B' - B")J 2 for large J—s ince they are thendominated by terms quadrat ic in J—and hence they run in the same directionas functions of J. In contrast, and T/ R run in opposite directions as functions ofJ fo r small J, where the linear term s dominate. He nce , e i ther the P or the Rbranch turns around as a function of J (depending on whether B' <B" or

> B" ) at the bandhead as shown in Fig. 4.27. T he approximate value o f J atwhich one o f these branches turns around can be found by differen t iat ing the



I C I

-

— 6030(a)

NO

000(b )

Figure 417 Rotational fine structures at 300 K for the y" = 0 y' = 0 absorptive

transitions in (a) the X 1 E+ B3n 0 transition in ICI, B'L = 0.1142 cm -1 ,

B 'e = 0.0872 cm -1 ; an d (b) the x2n r , 2 E + transition in NO , B 'e' = 1.67195 cm -1 ,

B 'e = 1.9965 cm -1 . The rotational line sp acings are smaller for ICI, owing to its larger

reduced mass. The ICI spectrum is shaded to the red , and typifies the usual situation

in which B'L > B'e•The reverse is true in the x2n AE + transition of NO , in which

the rotational structure is shad ed to the blue. Horizontal energy scale is in cm -1 for

both spectra.

154



POTENTIAL ENERGY CURVES FROM ELECTRONIC BAND SPECTRA 155

equations fornd flik with respect to J,

e N p—(B' + B") + 2(B' — B")J = 0

dJ

= 3B' — B" + 2(B' — B")J = 0dJ

(4.77)

Then the J values at which thes e branches turn around are

J; = (B' + B")/2(B' — B")

J ; = (B " — 3 B ) 1 2 (B ' — B " )4.78)

Only one of these values is physical (positive). Since the upper electronic state iscommo nly more we akly bound than the lower s tate (R; >Re") , one usuallyobserves B 'e < Be"—in which case the R branch is the one that turns around(J; >0 ). Bo th the P and R branches then run to lower frequencies for large J(since (B ' —B " )<O) , and the band is said to be shade d to th e red. Wh en R 'e <R,the band is shade d to t he b lue . The rotational fine structure is barely resolved inthe N a 2 fluorescence excitation spectrum of Fig. 4.3, but the asymmetric shadingof the vibrational bands to the red is clearly asser ted. I t occurs because theequilibrium se paration of N a 2 i s cons iderably larger in the A' / 4 " than in the

X 1 E g+ state, as is apparent in Fig. 4.8.

4.7 POTENTIAL ENERGY CURVES FROM ELECTRONIC BAND

SPECTRA

The spectroscopic techniques described in this and the preceding chapters yieldan impressive array of molecular constants, in terms of which the rovibrationalene rgy levels may be ex panded to de si red accuracy via

= coe (v + 4) — w e x e ( v + . 1 ) 2 + e Y e ( v + D 3 + • • •

+ Be J(J + 1) — DeJ2 (J + 1)2 — c x e J(.1 + 1)(v + + • • (4.79)

For example, spectral lines in vibration—rotation spectra of heteronucleardiatomics are read ily ass igned to part icular (v7) (v" J") transitions by in-spection. Their positions may be analyzed using Eqs. 3.64 , 3.65 , 3.68 , and 3.81 toextract the ground-stateconsansInomonu-clear diatomics (which are infrared-inactive), analysis of rotational fine structurein fluorescence spectra yields the rotat ional constants B, D e" , 4, . and if ,

for the upper and lower s tates . The vibrational band positions in suchspectra (e.g., the N a 2 fluorescence spectrum in Fig. 4.1) may be analyzed toobtain the vibrational constants co, ntheower state . Similar




treatment of the absorption or fluorescence excitation spectrum (cf. Fig. 4.3)yields the corresponding upper state vibrational constants aye , w e x/e , oyey ,' ,

Conclusive vibrational assignments in electronic band spe ctra can be d iff icult toestablish when th e electronic energy separation Te between upper and lowerstates is not independently known. Failure to observe the y" = 0 —> y' = 0 banddue to a small Franck-Condon factor Kely">1 2 , fo r example, can result inmisnumbering of the true leve ls y" = 1 , 2 , . .. , as y" = 0 , 1 , . . . [13 ] . A strategemfo r confirming vibrational assignments of electronic band spectra is describedlater in this sec t ion. Huber and Herzberg [14] have compiled molecularconstants for over 90 0 diatomic molecules and molecule-ions, based o n criticalexamination of the literature up to 1978.

In many applications, it is desirable to know the detailed potential energycurves U k k (R) for the upper and lower states. Such information would berequired to predict spect ra l line intensities of heretofore unobserved vibronic

transitions (e.g., for investigation as possible laser transitions). W e have alreadyshown that the radial Schrödinger equation (3.30) can be solved under a givenpotential U k k (R ) to obtain the vibrational eigenvalues fo r J =0 . W e are n owconcerned with the reverse procedure: Given a set of spectroscopically deter-mined vibrational levels, can the detailed potential energy curve U k k (R ) bereconstructed? For nonpathological potentials, the intuitive ans wer i s yes . Whenthe vibrational levels are equally spaced in ene rgy , U k k (R ) is a parabola withcurvature determined by the level spacing. Nonuniformities in level spacing(manifested by nonvanish ing anharmonic constants w e x e , clue , ...) shouldde form the parabola in predictable directions.

The current technique for derivation of experimental potential energy curves,developed in the 1 9 3 0 s but popularized only with th e advent of high-speedcomputers, is based on the Sommerfeld condition [ 15 ] fo r quantization ofaction in systems undergoing periodic motion,

pdg = (y + = 0 , 1 , 24.80)

The generalized coordinate g and its conjugate momentum p vary periodically,and the line integral is evaluated over one cycle. (A similar quantizationcondition on angular momentum led to the famous Bohr postulate L = nh fo r

the hydrogen atom in the o ld quantum theory [1 6 ]. ) For vibrational motionsubject to a potential U ( R ) in a diatomic, th e classical energy isE =p2 /2/4 + U(R) . T he integral in (4.80) then translates into

R +2— U(R)dR = (y + 1)h4.81)R_

fo r periodic motion be tween th e classical turning Points R_ and R, (Fig. 4.28).Rydberg, Klein, and Rees demonstrated that this semiclassical procedure fo rderiving th e allowed vibrational energies E f rom a given potential U(R) may be

inver ted. If the rovibrational ene rgy (4.79) is rewritten in terms of the vibrational



POTENTIAL ENERGY CURVES FROM ELECTRONIC BAND SPECTRA 157

Figure 4.28 The classical turning points R_ and R ± for a given vibrational level.

For this vibrational energy, nuclear motion is classically forbidden for R <R_ and for

R > R + .

and rotational action variables I = h(v + 1 ) and ic = J (J + 1 )h 2 /2,u as

E (I , K) /hc = Ico e /h — I 2 w e x e /h 2 + 1 3 coeYe/h 3

it may be shownstate with known

with

and

+ K/hclq

[17] that the innerene rgy E v are

=

hf

— K 2 D e /h 2 c 2 BR: — I K o c e /h 2 cB e R

and outer turning points R ± of a vibrationalgiven by

(f/g +f2 )" 2 ± f

fE — E ( I , K ) 1 1 -112 d I0

fr OE [Ev-) ] -112 d I

(4.82)

(4.83)

(4.84)

(4.85)

- =1 2742p)

hg =

2744)1 / 2E ( I ,

0




The upper integration limit I' is the value of the vibrational action variable forwhich E(I', K) =E. These integrals are evaluated fo r K = O to compute thevibrational turnin g points R A _ for the nonrotating ( .1 = 0 ) molecule. Ree sshowed that these in tegrals are expressible in close d form [18] when thevibrational en ergy E(I,0) is quadratic in I, but this level of approximation doesnot yield accurate turning points over a broad range of vibrational ene rgies .He nce, these integrals are calculated n ume rically to yield two points R±(E) onthe potential ene rgy curve U k k (R) at each spectroscopic energy E. The fullpotential U k k (R) i s then cons t ructed by conne ct ing the turning points with asmooth curve. Figure 4.29 shows the experimental Rydberg-Klein-Rees (RKR)turning points for the X 1 E g+ state in Na 2 . Such RKR calculations furnish themost accurate e xperimental potent ial energy curves for diatomics. Their facileexecution using established computer codes has largely superseded character-izations of U k k (R ) by analytic f it ted pote ntials such as the Morse potential(Section 3.6).

Once RKR potential energy curves have been genera ted for the lower and

4000

Lu000

0

N a 2

4 ftFigure 4.29 RKR points for the X' 1 g+ state of Na 2 . These are the classical turningpoints R_, R + calculated for V' = 0 through 45 using spectroscopically determinedrovibrational energy levels from Na 2 fluorescence spectra. Separations are given in A .

Data are taken from P. Kusch and M. M. Hessel, J. Chem. Phys. 68: 2591 (1978).



24

22

20

1 8

0( 3 P ) + 0+ ( 2 D° )

2/:122..1(1 D) + 0 + ( 4 S ° )

- --......---..--......_v),...----1 + , 6/, 6 n6p ng. . . . . . .g .•••''''--

- -:--- z z . -b 4 1i 0 + ( 4 S ° )

1 5.,--;ss2X - t, and 4 Iu -

0+21 6

0( 1 D) + 0( 1 S)

0( 3 P) + OeSi-

0( 1 D) + 0( 1 D)

0( 3 P) + 0( 1 D)

1 A u B3X,o —

ingo. . . .. . z . 5,A g an d 5 X 1ll . . . . . . _n - _ - _ 1 _ ! --- - - - - - - - - - -s 5 1 3 1 4 -i and lr A 31+u=;:io

c - /u \ 0(3 P) + 0( P)

a g

X 3ti/oi

0( 3 P ) + 0-( 2 P°)

Estimated error:

<0.02 e V

-- <02 e V

---- <1.0 e V

I 04

.8.2.6.0.4.8.2.6Internuclear distance (A) .

Figure 4.30 Potential energy curves for 02+ , 02 , and 0;. Used with permission

from F. R. Gilmore, RAND Corporation Memorandum R-4034-PR (June 1964).

-4

159



14 1 -4an d 4 1 1 g

N( 2 D°) + N 4 - ( 3 P )

..r: ---•1+,••• ........ ...•.., ......•I ...".... .•••......-. \ \\ ,....."1:f...,.6.........' v.„ ‘:N.. ,

'S • • . -'kA - '..... n 4/-

N ( 4 S°) + N + ( iD)

s - ' , . . . . %,..

. .. .. . - .. - - ....- --. . . . . .-7 .. .7. . . . -.'--.--"--'• • • "-....... ... " _- --_-3- . .....,----

/ ,,,,,,--------( 4 3°) + N+( 3 P )

"i.-1 _ 45--',, IL'dmilill' rig3 /

/ /

4 A / 5/ 4nj((),/ //5 6I+ IU\ 6ll\ u

6E+\611 2H U

20

9/

Ye

C.,07

3/-an d 3 11„g

-(2Do) N (2Do)

a 1 1 1 g -

5

C , 3 1 -1 "yr'Yy,-- ----s.---B ' 3 1,-

I faso) + N (2Do )

--i-

5774

_ , 7 - - : : — - - - - z - - - : I k..---

/

:137 3 1

15 /.'-X211/ 5/+ ----- - " ` \ ,(4so) + N(3P)

H i / N(so) + N(4so)ii3/----;._-------

. . .21..---

/ 5.eV / 10 / /

Estimated error0.02 e V

—<0.2 e V1.0 eV

0.4.8.2.6.0.4.8.2.6Internuclear distance (A )

Figure 4 . 31 Potential energy curves for N 2+ , N2 , and N. Used with permissionfrom F. R. Gilmore, RAND Corporation Memorandum R-4034-PR (June 1964).

10

N ( 4 S°) + N ( 2 P°)

160

28

26

C

24

22B 2 /+-,



R E F E R E N C E S6 1

upper s tates (e .g., the X'E g+ and 1 3 3 1 1 0 „ states in 1 2 , Fig. 4.7), the radialSchrâclinger equation (3 .30) can be solved nume rically to obtain the correspond-ing vibrational states IC > and le > in the respective electron ic states. It is then

straightforward to calculate the predicted Franck-Condon factors Ky'le>1 2 forall vibrationa l bands in the electronic transition. The band intensities observedin an absorption or fluorescence spectrum s hould be proport ional to thesecalculated Franck-Condon factors (Section 4.4). If they are not, the vibrationalbands in the electronic spectrum m ay have been incorrectly assigned . In thismanner , Zare found it nece ssary to reassign the y' quantum numbers previouslyattr ibuted to vibration al bands in the )0E: - -> WIT. spectrum [13]. R KR

calculations are, of course, possible only for spectroscopically accessible

electronic states for which a reasonable number of the molecular constants inEq. 4 .82 are known. For this reas on, o nly the VI:, A 3 171 1 „, and B 3 1-1 0 4 . 0

potent ial en ergy curves for 1 2 in Fig. 4.7 are well known. The lowest-lyingX 1 E g+ 11. transition (d12 = 2) is so weak that no R KR curve has bee ngenerated for that ex cited state to the author's knowledge. R KR calculations are

not applicable to purely repulsive states, and the shapes (but not the asymptotes)of the repulsive poten t ial ene rgy curves de picted in this c hapter are largelyconjectural. Methods for predicting inte nsities of transitions to repulsive s tates("bound-continuum transitions”) have recently been de veloped [19].

A number of potential en ergy curves is shown for 0 2 , 02- , and 02' in Fig.4.30, and for N2, N2- , and N -2 - in Fig. 4.31. The lowest three boun d s tates in

neutral 02 (X3 /, alA , and b 1 E:) arise from the equivalent n 2 configuration

(Section 4.3). El transitions among these three s t a te s are forbidden, in ac-cordance with the general rule that no El t rans i t ions e xis t among s ta tesgenerated f rom the same e lect ron configuration. The better-characterizedexcited s tates (i.e., the ones with more labeled vibrational levels) tend to be one swhich are El-connec ted to the ground-state molecule or molecule-ion. Asexamples, the B 3 E u- excited-state potential in 02 is known up to nearly itsdissociation limit, but the C 3 A„ sta te i s no t (Fig. 4.30). In N2, the X'E g+ grounds tate is a closed-shell configuration; i ts e xci ted s ta tes are among the mostthoroughly studied in any dia tomic . Since i t is no t conne cted to any lowerelectronic state by El transition, the A 3 E: state is metastable, and El

absorptive transitions from this state to higher s tates (e.g., C 3 1-1 „ ) can e asily beobserved. The well-known N2 laser, which operates at 3 3 7 n m , stems from the

y' = O -* y" = 0 transition between the C 3 1 I u and B 3 1 I g s tates .

REFERENCES

1 . D. S. Schonland, Molecular Symmetry, Van Nostrand, London, 1965.2 . L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory,

Addison-Wesley, Reading, MA, 1958.

3 . B. H . M a h a n , College Chemistry, Addison-Wesley, Reading, MA, 1966.4 . R. S. Berry, S. A. Rice, and J. R o s s , Physical Chemistry, Wiley, New York, 1980.




5 . K. Ruedenberg , R e v . Mod. Phys. 34: 326 (1962 ) ; M . J. Feinberg, K. Ruedenberg , andE. L. Mehler , Adv. Quantum C h e m . 5: 27 (1970 ) .

6 . J. G. V e rk ad e , A Pictorial Approach to Molecular Bonding, Springer-Verlag, N e wYork, 1986 .

7 . W . H . Flygare, Molecular Structure and Dynamics, Prentice-Hall, Englewood Cliffs,NJ, 1978.

8 . H . F. Schaefe r III , The Electronic Structure of Atoms and Molecules: A survey ofRigorous Quantum Mechanical Results, Addison-Wesley, Reading, MA, 1972.

9 . R. S. Mulliken, J. Chem. Phys. 55: 289 (1971 ) .

1 0 . G . Herzberg, Molecular Spectra and Molecular Structure; I. Spectra of Diatomic

Molecules, Van No s t ran d , Princeton, NJ, 1950 .

1 1 . J. H o uge n , The Calculation of Rotational Energy Levels and Rotational LineIntensities in Diatomic Molecules, National Bureau of Standards Monograph 115 ,U.S . Government Printing Office, Washington, DC, 1970 .

1 2 . K. Gottfried, Quantum Mechamics, V ol . 1 , W. A. Benjamin, New York, 1966 .1 3 . R . N . Za r e , J. Chem. Phys. 40 : 1934 (196 4) .

1 4 . K. P. Huber and G. Herzberg, Molecular Spectra andMolecular Structure: Constants

of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979.

1 5 . A. Sommerfeld, A n n . Phys. 57: 1 (1916).

1 6 . M . Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill,New York, 1 9 6 6 .

1 7 . R. Rydberg, A n n . Phys. 73: 3 76 ( 193 1 ) ; 0 . Klein, Z. Phys. 76: 226 (1932) .1 8 . A. L. G. Rees, Proc. Phys. Soc. (London) A 59: 998 (1947) .1 9 . W . T . Zemke, K. K. V e rm a , and W. C. Stwalley, Proc. Intern. Conf. on Lasers '81 , 21 6

(1981).

PROBLEMS

1. For a heteronuclear diatomic molecule , determine th e electronic termsym bols that correlate with 2 D + 2 D atoms .

2 . W rite down th e electron configurations for the ground states of B2, C2, N2,

and 0 2 . W hat te rm symbol corresponds to the ground s tate in each of these

molecules?3. For a homonuclear diatomic molecule, what term sym bols 2 S + 'A n can arise(a) from an equivalen t 6 2 configuration and (b) from a nonequivalent 7 r 2 (e.g.,7 c uirc) configuration?

4 . Determine the relative statistical weights of the even - and odd , / leve ls in85 Rb 2 and 4 0 K 2 , which have nuclear spin I = and 4, respectively.

5 . Using the potential energy curves in Fig. 4.31 fo r neutral N2, decide which ofthe shown s ta tes are El-metastable . Are all of the diatomic s tates which

correlate with N ( 4S ) + N ( 4 S) shown? W hich of the shown ex ci ted s tates aretheoretically accessible by M 1 transitions from ground-state N 2?



P R O B L E M S 163

6 . The C i f < — X 1 E: absorption spectrum of Na 2 in the near-ultravioletregion of the spectrum e xhibits bands at the following frequencies in cm -

288719 1 8 2933894799 6 8 2289869 2 1 39 3 6 89 5 2 497452 9 0 2 69 2 5 59 4 1 19 5 6 997932 9 1 0 09297943495899 9 0 32 9 1 4 19 3 2 494559 6 3 5

Starting from the ass umption that one of these frequencies corresponds to they" = O —) = 0 transition, make the vibrationa l ass ignm en ts by ordering theobserved band frequencies i )(y" , y ' ) into a Des l andr es table:

A'0 10 , 1 )6'1 2-)(0, 2)

V , (1 , 0 )o3 (1 , 1 )123(1, 2)

A IL1 11 272

3 ( 2 , 0 )'0 17 ( 2 , 1 )125 (2, 2)

The numbers S„,„ and An , represent the numerical difference s between adjacent

frequencies in the table. (In a correctly ordered table, all differences deno ted bythe same label (e.g., L I 2 ) must exhibit the same value to within expe rimentalerror, in accorda nce with the Ritz combination principle. The value found for

„ ) is the observed e ne rgy separation betwee n vibrational levels m and n in

the upper (lower) electronic state .) Dete rmine the vibrational constants c o : , co',co:x:, oldc' and (if possible) ( 1 ) ; Y : , o.1/e from these data.e

7 . The 0-3 absorption band of an allowed El electronic transition originates

from the X 3 I- s tate in 0. Rotat ional fine structure lines occur at the followingwave numbers:

51354 .37(1 )

51353 .99

51353 .50(2 )

5 1 3 5 1 . 3 7

51349 .4651348 .01(4 )

51345 .31(2 )51343 .38(5)

51339 .92




(a ) Complete the assignments of the four unlabeled lines. Is this band shadedto the re d or violet? Is B' greater than or less than B"?

(b ) Determine the numerical values of the rotational constants B' and B ".

(c ) Recognizing that this is an allowed transition in the relatively lightmolecule 02 , specify the term symbol of the upper state.

8. The analytic expressions 4.24 and 4.34 for the overlap, Coulomb, andexchange integrals in H I may be obtained using the ellipsoidal co ordinates A , p,and 0, where

=(r A + r B )/R

P = (r A —rB)/R

and the angle 0 has its usual mea ning in a diatomic molecule. T hese coordinates

exhibit the range

and the volume e lement i s le(2 2 — p2 ) d A d p 418 . Use this information tocon firm that S A B , J, and K are correctly given by Eqs . 4 .24 and 4.34 .



POLYATOMIC ROTATIONS

In this chapter, we exten d our t reatment of rotat ion in diatomic molecules tonon line ar polyatomic mo lecules . A traditional motivation for treatingpolyatomic ro ta tions quantum mechanically is that they form a basis fo rexperime ntal determinat ion for bond lengths and bond angles in gas-phasemolecules . M icrowave spectroscopy, a prolific area in chemical physics s ince1 946 , has provided the most accurate available e quilibrium geom etries fo r manypolar molecules. A background in polyatomic rotations is also a prerequisite fo runde rs tanding rota t ional fine structure in polyatomic vibrational spe ctra(Chapter 6). The shapes of rotational contours (i.e. , unres olved rotationa l finestructure) in polyatomic electronic band spectra are s ens itive to the relativeorientations of the principal rotational axes and the electronic transitionmoment (Chapter 7). Rotational contour analysis has thus provided an

invaluable means of assigning sym me tries to the electronic states involved insuch spectra.W e be gin this chapter with a derivation of the classical Hamiltonian for a

rigid, freely rotat ing polyatomic molecule . Such a polya tomic rotor may beclassified ac cording to its point group symmetry (Section 5.1 ). We formulate thequantum me chanical H amil tonian and rota tional ene rgy e igenvalues fo rmolecules having at least one C„ axis with n 3. Molecules with lowersymm etry (e.g., H 2 0) are far more difficult to treat, and readers are referred tospecialized sources on rotational spectroscopy for these cases. El selection rulesare obtained for pure rotational t ransi t ions, and examples are given of structural

information yielded by m icrowave spectra.

165



16 6OLYATOMIC ROTATIONS

5 .1 CLASSICAL HAM ILTONIAN AND SYMMETRY

CLASSIFICATION OF RIGID ROTORS

Consider a rigid body consisting of a collection of point masses m N located atpositions R N 7-e (XN, Y N , Z N)n a Cartesian coordinate s ystem which rotates withthe molecule. The origin of this coordinate sys tem coincides with the body'scenter of mass (Fig. 5.1) . If the body is cause d to rotate with angular velocities(0 x , (0 y, c o z about the molecule-fixed X, Y, and Z axes , respec tively, its rotationalkinetic energy becom es [1]

1

T=—•I • •co • lo) • I • c o2 (5 .1)

where the indices i, j ru n over- X, Y, and Z. The matrix I i s the rotational inertia

t ensor [1]

— E MNX NYN

E inN(x?,, + Z;1)

— E MNZNYN

— E MNX NZ N

1= — E MNYNXN

— E inNZNX N

— E MNYNZ N

E niNtxk +

IXx

I x

I X

I y

I y

I y

IXZ

IYZ

IZZ

(5 .2)

Wy

Figure 5.1 The rotational kinetic energy of a rigid body may be specified in terms of

angular velocities wx , wy , toz about arbitrary orthonormal axes X, Y , Z intersecting at

the center of mass.



CLASSICAL HAMILTONIAN AND SYMMETRY CLASSIFICATION OF RIGID ROTORS67

In general, the classical rotational kinetic en ergy (5.1) contains three diagonalterms (proportional to col, w?„ and 4). I t also co ntains six off-diagonal terms(proportional to obc c o y , etc.). Since the inert ia tens or is ne cessarily a symmetric

matrix according to Eq. 5.2, only three of these off-diagonal terms areindependent.I t always proves possible to rotate the XY Z axes into a new body-f ixed

coordinate system with orthonorm al axes a, b, and c (called the principal axes)

which diagonalize the inertia tensor,

(5.5)

(5.6)

E MM +

E mAak + cD

0

O

i„2E M/sIltiN m "NI

a 0

0

0

lb

0

0

0

I _

(5.3)

B y virtue of Eq. 5.1, these new coo rdinates a lso diagonalize the rotationalkinetic energy,

T =W a w a'+I c o c2 )5.4)

where wa,(24, W c are the rigid body's angular velocities about the principal axes(Fig. 5.2). Since the body-f ixed components of ro ta tional angular mom en tumabout the principal axe s are

J = la w a

= 4°6

J c = I c w c

the classical Ham iltonian for a rigid free rotor becomes

J! , 1 c2H =

2 / a 2/ 2/

= 4 7 r 2 c (A.1! + BJ +h



168 P O L Y A T O M I C ROTATIONS

coo

Figure 5.2 For a special choice of body-fixed axes (X , Y , Z) = (a , b, c), the rotational

kinetic energy assumes the diagonalform in Eq. 5.4. In the ellipsoid illustrated, one of

the principal axes a lies along the C c o axis, while the other two principal axes b and c

are any pair of mutually orthogonal axes which are perpendicular to a.

Equation 5 .6 defines three rotational constants having units of cm - 1 - ,

A = W8n 2 c1 0

B = h 187c 2 c1 b5.7)

C = hl8n2 c1,

T he principal axe s a, b, c are conventionally labeled so that the rotationalcons tants are ordered in decreasing magnitude, ABC Four classes ofnonlinear rigid rotors may then be dist inguished:

A = B = Cpherical to p

A> B = Crolate symmetric top

A = B > Cblate symmetric to p

A > B > C

symmetric to p

T he orientat ions of the principal axes are easily obtained by inspection inmolecules with sufficient symmetry. Any C„ symmetry axis (n 2) coincideswith a principal axis. In the D2h ethylene molecule, for example, the C 2 (x), C 2 (y),and C 2 (z) twofold axes are principal axes. All of the principal mome nts of inertiain ethylene are different in magnitude, so this m olecule is an asymmetric top. Allmolecules with one C3 or higher-order axis are symmetric tops, because theprincipal mome nts about two axes normal to a C„ axis with 3 are necessarilyequal. Such molecules include B F 3 (which belongs to the D3h point group), NH 3

and CH 3 I (C 3 , , ) , and benzene (D 6 *h ) . M olecules with two or more C3 or higher



CLASSICAL HAMILTONIAN AND SYMMETRY CLASSIFICATION OF RIGID ROTORS 169

order rotation axes are spherical tops; examples of these are CC1 4 (T ) and SF 6

(O h ). It is possible in principle for a molecule lacking a C h axis (n 3) to exhibittwo equal principal momen ts; such a molecule is terme d an accidental symmetric

top. The frequency reso lut ion afforded by microwave technology can de tectsuch small differences between two moments of inertia that very few m oleculespass fo r accidental symme tric tops under m icrowave spectroscopy.

In symmetric tops, tw o of the rotat ional moments of inertia are equal. T hethird moment i s ass ociated with ro ta t ion about the axis o f highest symmetry(called the figure axis). In prolate symmetric tops, the f igure axis exhibits thelarges t rotational constant (and the smallest principal moment of ine rtia). Suchmolecules con centrate most of their nuclear mass along the figure ax is (the a

axis), and tend to be cigar-shaped. Eclipsed and staggered e thane molecules areboth prolate s ymm etric tops, as is 2-butyne (Fig. 5.3). In oblate symmetric tops,

the nuclear mass tends to conce ntrate at an appreciable distance from the figureaxis (the c axis), end owing the figure axis with the smallest rotational constan tand the largest pr incipal mome nt of inertia. O blate tops (e.g., benze ne ) thusrese mble disks rather than cigars. Linear polyatomic molecules are a special caseof prolate s ymme tric tops, in which the rotational mom en t about the figure ax is(the C. axis) is negligible. For such specie s (e.g., O CS , H CN ) , the rotat ionalHamiltonian is formally identical to that for a diatomic molecule.

T he classical Hamiltonian (5.6) for a rigid rotor becomes simplified when twoor more of the pr inc ipal mome nts a re equal. In the case o f the spherical top

(A =B = C) , it is

4 7 r 2 c 41c2cH=

B(J2 ++

=

J 2

a5.8)

Figure 5.3 Examples of a prolate symm etric top, 2-butyne (left) and an oblatesymmetric top, benzene (right). B y convention, the figure axis is labeled the a axis and

the c axis in prolate and oblate tops, respectively.



170 POLYATOMIC ROTATIONS

For the prolate top (A > B = C), the H amiltonian become s

4n 2 cH =AJ2 . + B(fi, + J,2)]

h

=4n2c [ (A — B )J 42 , + B J 2 ]h

(5.9)

while for the oblate top (A =B>C) it is

4n 2 cH =BJ2 + (C — B)fi]

h(5 .10)

N o such simplification is possible for the asymme tric top, where A >B>C. We

will see that this fact co ns iderably complicates the derivation of the quantummechanical ene rgy levels for an asymme tric top.

5.2 RIGID ROTOR ANGULAR MOMENTA

Simple as they may appear, the classical Hamiltonians developed fo r rigid ro tOrsin the preceding sect ion are conceptually ne w. In our discussion of diatomicrotations in Chapter 3, the rotational states I JM> were obta ined aseigenfunctions of the space-fixed angular momentum operators j and

P =P + P +f. T he space-fixed an gular mome ntum componen ts f ,obey the familiar commutat ion rules

[i, : T y ] = ihf

[f ,05 .11)

and cyclic permutat ions o f x , y, z

In con trast , the rotor Ham iltonian (5.6) is expressed as a function of the body-fixed angular momen tum components Ja,1,9 Jssociated with rotations about

the principal axes a, b, an d c, which thems elves rotate with the molecule. It isunclear a priori what commutat ion relationships are obeyed by a,J1 ,5 J and J2 =, J a2 ± ±J. It would also be useful to know whether any space-f ixed components of J commute with particular body-fixed compon en ts of J, sothat com muting sets o f observables may be constructed as an aid in visualizingthe physical significance of rigid rotor wave functions.

An arbitrary three-dimensional rotation of a rigid body may be described [1 ]using the Euler angles 4) , 0 , x (Fig. 5.4). T he body-fixed a, b, and c axes areinitially aligned with the space-fixed x , y, and z axes, respectively. T he body isfirst rotated co unterclockwise by the angle 4 ) about the z axis; this rotation does

not affect the orien tation o f the c axis, but rotates the a and b axes in the xy



RIGID ROTOR ANGULAR MOMENTA 171

Figure 5.4 Euler angles (0, 8, X) describing the orientation of body-fixed axes(a, b, c) with respect to space-fixed axes (x , y, z ) . The molecule is first rotated by anangle 0 about the space-fixed z axis, causing molecule-fixed axes initially pointingalong x and y to be rotated into x ' and y', respectively. The molecule is then rotated bythe angle 9 about the new x ' axis, displacing the body-fixed axis initially pointingalong z into the direction c and the y' axis into direction y". The molecule is finallyrotated by the angle x about the c axis, rotating the x ' and y" axes into the a and b

axes, respectively.

plane. The second rotation is a counterclockwise rotation by the angle 0 aboutthe new orientation of the a axis ; i t rotates th e body-fixed c axis by the angle 0from th e space-fixed z axis. The body is f inally rotated counterclockwise by theangle x about the new c axis. The components of the r igid ro tor 's angularmomentum may be specified as projections of J along either th e space-fixed a x e s(Jx ,J , .1 z ) or body-fixed ax es (Ja ,J ,J ) . They may also be expressed in terms ofth e three independent angular mome ntum components Jo , Jo , Jx associa tedwith the Euler rotation axes,

h af -

0 4 )

h aJ = —

i 00

hjx=i a x

(5.12)




which are directed along the unit vectors (r), ÉÏ, and i, respec tively. I t isno t difficult to show geom etrically (Problem 5 .1) that the space-fixed angularmomentum components in terms of the Euler angles are

, h [sin 0 a s in 0 cos 0 a

acos 0 ---

1

/ s in 6 4 O zin eooh [ cos 0 a cos 0 cos o a l1 =in 0 —5.13)

1 ii n 0 Oz sin 6 4oo, h aJ.=–i 041

and that the body-fixed an gular mome ntum comp ±nces onn stsz

[s ini= h sin x ain x cos 0 0

sins in 0 00 0 e x

f. [cos x ao s x c o s 0 0 1sin5 . 1 4 )i Lsin e ao x, h a

. 1 , = —i a x

The well-known space-fixed commutat ion relations hips (5.1 1 ) may re adily beconfirmed using Eqs. 5.1 3 for fx , f, f in terms of the Euler angles. Us e of Eqs.

5.1 4 for the body-f ixed components f , 4, f quickly lea ds to the somewhatsurprising commutat ion rules:

fb] =and cyclic permutat ions o f a, b, c

In analogy to space-fixed an gular momenta , one conse quently obtains

and cyclic permutat ions o f a, b, c5 . 1 6 )

I t is man ifest from E qs. 5 .13 and 5 . 14 that

( 5 . 1 5 )

( 5 . 1 7 )

because the Euler angles 0 and z asso cia ted with f and fc respectively areindependent variables. It may also be shown [2] that each space-fixedcomponent of J commutes with any body-fixed compone nt of J,

i = x, y , or z

c c = a, b , or c ( 5 . 1 8 )f„

0



RIGID ROTOR STATES AND ENERGY LEVELS 173

5.3 RIGID ROTOR STATES AND ENERGY LEVELS

W e are now in a position to determine the rigid rotor energy levels for tops in

order of decreasing symmetry. The Hamiltonian (5.8) for a spherical top i sfunctionally indistinguishable from that of a diatomic rotor,

4 7 c 2 cBP

4 7 r 2 cH = =P - + + .11

h (5.19)

with the conse quence that spherical top energy levels in cm" are given by (cfEq. 3.24)

E j = BJ(J + 1)=0, 1, 2, ...5 .20)

Owing to the spherical symmetry of its inertia tensor (/ a = Ib = IA the sphericaltop's rotational leve ls depe nd on ly on the single quantum number J.

In the oblate symmetric top, the rotational H amiltonian is given by Eq. 5 .10 .The c axis i s denoted the figure axis. According to the commutation rulesobtained in Section 5.3, one possible commuting set of observables is Jz , J , andJ 2 . It is then possible to formulate rotational s tates I J K M > which simulta-neously obey the e igenvalue equations

PIJKM> = J(J + 1)h2 I JKM >=0, 1, 2, ...5 .21)

fIJKM> = MhIJKM>J M J5.22)

iIJKM> = K h I J K M >J KJ5.23)

This implies that the eigenstates IJKM> must behave as

IJKM> = ei m d'f (1 11 (0)e` K x5.24)

where M and K are both intege rs ranging from — J to J. The 0-dependentfunctions fa(0) prove to be hypergeometric functions of s in 2 (0 /2) [3 ,4] .

Theseare rarely given ex plicitly in modern spectroscopy texts, because no knowledgeof these functions is nece ssary to obtain e i ther the energy levels or the selectionrules fo r spectroscopic transitions in symmetric tops. By com bining E q s . 5 .1 0 .5 . 21 , and 5.23 , we obtain the en ergy levels for the oblate symmetric top in cm',

EJK = B J ( J + 1 ) + (C — B )K 25.25)

A schem atic ene rgy level diagram is given for the oblate top in Fig. 5.5.The quantum number M measures the projection of the rotat iona l an gular

mome ntum di upon the space-fixed z axis (Eq. 5.22). Since the rotational energyis independen t of the orientation of J in isotropic space, E JK is independe nt of M(Eq. 5.25). The projection Kh of J on the body-fixed c axis does influence the




K=O O 6 6

5

4

3

2

J = 0

OBLATE

ROLATE

Figure 5.5 Rotational energy levels for oblate and prolate symmetric tops. Notethat levels do not exist for W I > J (dashed lines). For given J, the rotational energy is

a decreasing (increasing) function of IK 1 for oblate (prolate) tops.

Figure 5.6 Rotational motion in an oblate top for (a) IKI = J and (b) K = O.

According to Eq. 5.25, the rotational energy for given J is smaller in case (a) than in

case (b). This is in agreement with the classical result, where the rotational energyJ2 /2/ depends on the total angular momentum J and the moment of inertia / aboutthe axis of rotation: / is clearly larger in case (a) than in case (b).

rotational energy, as is illustrated in Fig. 5.6 for an oblate top. Wh en I K I attainsits maximum value ( =J) fo r given J, the molecule is rotating about an axis thatis n early parallel to the f igure ax is . (It cann ot rotate purely about the c axis : In

such a state, one would have J. = = 0 , which would violate the uncertaintyprinciple embodied in Eq. 5.15.) Wh en K is sm all, the molecule unde rgoes atumbling motion dominated by rotation about the a and/or b axis . Since

nume rically differe nt rotational constants are associated with rotations about



RIGID ROTOR STATES AND ENERGY LEVELS75the figure axis and the normal axes, the rotational energy must depend on IKI. Itis insensitive to the sign of K, which only controls the direction of rotation aboutthe figure axis.

In a prola te symmetric top, the a axis rather than the c axis becomes the figureaxis. The body-fixed a, b, c axes depicted in Fig. 5 .4 are re placed by the b, c, and a

axes, respectively. T he mutually com muting se t of observables becomes Jz , f a ,

and J . W ith these modifications, the o blate top eigenvalue equations and wavefunctions (5.21)—(5.24) become applicable to the prolate top as well. Thepertinent rotational energies in cm -1 are

E JK =BJ(J + 1) + (A — B )K 25 . 2 6 )

T he ene rgy levels for the prolate top are contraste d with those for the o blate top

in Fig. 5.5 . Since (C — B) < O in the obla te top and ( A — B ) > 0 in the prolate

2, 2

2, I2,0

J K Kp o

10 1

0, 0J, 11%1

OBLATE

A=B>C

000, 0

J, IKp l

PROLATE

A>B = C

Figure 5.7 Correlation diagram for asymmetric rotor state energies between the

oblate (A = B > C) and prolate (A > B = C) limits. Here the rotational constants A

and C are fixed , while B (the horizontal coord inate) is varied continously between A

and C. The oblate and prolate to p energies are given by Eqs. 5.25 and 5.26,respectively; the asymmetric top energies for intermed iate rotational constants B

must be found by explicit d iagonalization of the asymmetric top Hamiltonian [2, 6].




Table 5.1 Rotational constants in cm- ' for some asymmetric, symmetric, and

spherical top molecules

Species Poin t group A

H 2 0 C 2v 27.877 14 .512 9 .285H 2 C0 C2 v 9.4053 1.2953 1 .1342HCO 2 H C s 2.58548 0 . 40 21 1 2 0 .347447C H 3 3 5 C 1 C3 v 5.09 0 .443401C H 3 1 2 7 I C3v 5.11 0 . 25 0 21 7C 2 H 6 D3d 2.681 0 .6621NH 3 C3v 9.4443 6 .196C 6 H 6 D oh 0 . 18 9 6 0 . 09 4 8C H 4 T d 5.2412

Data taken from G. Herzberg, Molecular Structure and Molecular Spectra, III. Electronic Spectra

and Electronic Structure of Polyatomic Molecules, V an Nostrand Reinhold, New York, 1 9 6 6 .

top, increasing K decreases the rotational energy in an o blate top, but increasesthe rotational energy in a prolate top.

In the asymmetric top, the rotat iona l H amiltonian (5.6) is proport ional toAJ + BJ + Cj? with A > B> C. Since no two body-f ixed compone nts of J

commute (Eq. 5.15), no wave function can s imultane ously be an eigenfunction ofany two (let alone all three) of the operators R„ :4, and J. It is possible to findthe rotat ional eigenstates and energies of an asymmetric to p by diagonalizing

the Hamiltonian (5.6) in the IJKM> basis. This has been done in several texts[2], and it will not be repeate d here . An asymmetric to p (in which A > B> C)may be considered an intermediate case in which the value o f the rotationalconstant B lies between the extremes exhibited in a prolate top (A >B = C) andin an oblate top (A =B > C ) . In this spirit, the energies of the asymmetric rotormay be visualized on a qualitative ene rgy level diagram showing the correlationsbetween the prolate and oblate limits (Fig. 5.7). T he vast m ajority of moleculesare asymmetric tops.

Some represe nta tive ro tat ional cons tants are listed fo r asymmetric, sym -metric, and spherical to p molecules in T able 5.1.

5.4 SELECTION RULES FOR PURE ROTATIONAL TRANSITIONS

W e begin this section by de riving the El selection rules fo r rotat ional transitionsin symmetric tops, since spherical and linear molecules m ay be cons ideredspecial cases of symm etric tops. Fat El-llowed transit ions from state IJKM> tostate 1J 'K'Pl '>, we require a nonzero electric dipole transition moment

<JKAilitifK'M'> = <JKMI tt y'M'> 0 05.27)



SELECTION RULES FOR PURE ROTATIONAL TRANSITIONS 177

Here p is the permanen t mo lecular electr ic dipole mo me nt. An intuit iveargument will suffice for the selection rule on A K. Since p must be parallel to thefigure ax is by symme try in any prolate or oblate top, and since K controls the

velocity of rotation about the figure axis, changing K ha s no effect on the motionof the molecule's permanent dipole moment. Accordingly, the presence of anoscillating ex ternal e lectric field cannot influence K, and we have the selectionrule AK = O .

T o obta in the selection rules on AJ and A M , we ex ploit the properties ofvector operators. All quantities that transform like vectors under three-dimensional rotations have operators ex hibiting commutation rules that areidentical to those shown by the space-fixed angular mom entum operators f , , :Tv

. Such operators, which we will denote V (17„ , i'/;„ ), exhibit the commuta-tion rules

[fi, [fx, Py ] = ih = -Cf y , 17x l

5.28)

and cyclic permutations of x , y, z

These operators are termed vector operators with respect to 3 . T he space-fixedangular momen tum components f , fy , .tz are obviously vector operators withrespect to themselves ( c f . Eq. 5.11). The position and linea r mom entumr = (x, y, z) and p = ( p „ , pz ) are also vector operators, as is the electric dipole

moment operator p . Since we have from Eq. 5.28 that fo r any ve ctor operator V

[Z, f/] M P ,

[f 1 1 V x5.29)

it follows that

[f ,i] = h(1x + iP ]5.30)

Usingheotation. , E E Px i P ,hismplieshat4,- P + f2 = h P+ . W e con sequen tly f ind that fo r any vector operator

<J KMIZT 7+ — 7+ 4 — hP+ IJ'K'M'>

= ( M — M ' — 1 ) h < JK M I P + IJ'K'm'> = 05.31)

Proceeding similarly with the operator V _ a . Px — iP y , we find thatP-] = — h P_ , with the res ult that

“Kmlfz i7_ - Ï f +

=(M — M' + 1)h<JKMI V _ 05.32)




Since [4, P] 0we also have

<JKM14/1 .

(M — M')h<JKMIPIfK'M'> = 05.33)

Re cognizing that th e dipole moment operator p i s a vector operator, we se e thatEqs. 5 .31 and 5.32 give the selection rule AM = + 1 for the x and y componentsof the El transition moment. From Eq. 5 .33 , it is clear that <JKMIp z IJ 'K'M'> i s

zero unless A M = O .For the se lection rule on A J, we may use the more complicated vector

operator iden tity [5]

[f2, [f 2 , V]] = 2h 2 [PV + f — ( .V ) 3 ]5.34)

This leads to the condition

cmi[P , [P, V ] — 2 h 2 [f 2 V + 'f -)j]IfK r Air>

= h4 [(./ - f) 2 -IN(J + J' + 1)2 — 1]<JKMIVIJ'K'M')

=05.35)

Thus, ei ther <JKMIVIJ'K'M'> vanishes, or J' = J + 1 (the quantity in thesecond set of square brackets cannot vanish when J J'nd J, J' 0 ). Hencewe f ind the se lect ion rule AJ = +1 for pure rotational transitions in a

symmetric top. The symmetric top El se lect ion rules can therefore be sum -marized as AJ = +1,A K = 0 . In the presence o f strong exte rnal electric fields,the rotational ene rgy levels de pend on M as well as on J and K (via the Starkaffect); the selection rules AM = 0 (t t z ) and A M = +1 (p , I c ) , ) then becom erelevant. For freely rotat ing s pherical tops and linear polyatomics, the Kdependence drops out of the expressions for the rotat ional H amiltonian an den ergy levels ; thei r se lec t ion rules reduce to AJ = +1, A M =0, + 1 as indiatomics (Chapter 3). It is worth remarking that all of the selection rulesderived in this sec t ion emerged as consequences of vector operator properties;considerat ion of the specific rotational wave functions (5.24) was unnecessary.

5.5 MICROWAVE SPECTROSCOPY OF POLYATOMIC

MOLECULES

Rotat ional transitions in most polyatomic molecules occur in the microwaveregion of the electromagnetic spectrum ( A = 1 mm to 1 cm). In a prototypemicrowave absorption experiment, microwaves are generated in a specializedoscillator (called a klystron) , propagated through a hollow re ctangular me tal



MICROWAVE SPECTROSCOPY O F POLYATOMIC MOLECULES 179

pipe (a waveguide) containing the sample gas at low pressure, and finallydetected at a silicon crystal. At gas pressures low enough to rende r collisionalbroadening negligible (Chapter 8 ) , the frequency widths of rotational absorptionlines are narrowed to the point where the i r f requen cy positions may bedete rmine d with high accuracy. This fact, coupled with the available frequencystability of klystron sources (better than one part in 10 6 ) , endo ws microwavespectroscopy with the capability of determining the rotational constantsextre mely to high precision. A drawback of obtaining microwave s pectra at suchlow gas pressures (1 0 -3 to 10 ' torr) is loss of se nsitivity, since the absorbance(Appendix D) is proportiona l to the molecule number density. This problem isaddressed by the technique of Stark-modulated microwave s pectroscopy [6].The waveguide used for microwave propagation through the sample gas i sbisected by an insulated, conducting septum running d own its length. Applica-

tion of a square-wave voltage (typical frequenc y — 1 0 0 kH z) to the septumgenerates a mod ulated e lectr ic field in the gas, modifying its rotationalene rgies by the Stark effect. The rotational line frequen cies in the presence of the

field are shifted from their zero-field positions, because the level Stark shiftsdepend on J, K, and M. He n ce the absorption lines may be switched in and outof resonance with the incident microwave frequency from the klystron, allowingselective detection of the molecule-specific absorption signal with a lock-inamplifier. This widely used method greatly enhances the sensitivity and

resolution of microwave spectroscopy.As examples of the use of microwave spectroscopy to determine equilibrium

geometry, consider the molecules NF3 and C H 3 C 1 , which both have C3 v

symmetry. If the N— F bond length is den oted land the bond angle is 0, the use ofEq. 5.3 leads to the expression

= 2m12 (1 — cos 0)5.36)

for the moment of inertia about the figure (C 3 ) axi s . Similarly, the tw o momentsof inertia about axes normal to the figure axis are both

= MF1 2 (1 - cos B ) +

, M N M F 1 2

(1 + 2 cos 0)a m F + MN (5.37)

Since the symmetric top selection rules are AJ = +1, A K = 0 , absorptivetransitions will be observed from state If — 1 , KM> to s ta tes IJKM> andIJKM + 1 > , occurring at frequencies 2BJ according to Eq. 5.25. Hence , themicrowave spectrum of NF 3 yields no measurement of the rotational constantC. Since two structural parameters (land 0) enter in the rotational moment a

single microwave s pectrum cann ot determine the geometry of NF 3 . However ,spectra of the isotopic species 1 4NF 3 and 1 5 1 \ T F 3 may be co mbined to give twoequations s imilar to Eq. 5.37, assuming the isotopes have identical geometry. In

this mann er, land 0 were determine d to be 1.71 A and 102°9 ' , respectively [6]. In




a similar vein, the perpendicular moments of inertia in C3 , C H 3 C 1 are given by

lb = MH1?(1 — cos 0) ++ mc + mci

m i i (m c + mo)/?(1 + 2 cos 0 )

m c 1 1 2 [(Inc 3m02 &nisi (1 + 2 co s ( N J 2 ]}(5.38)

3 where / 1 is the C— H bond length, 1 2 is the C—Cl bond length, and 0 is th eH—C—H bond angle. (The expression for the moment about the figure axis isanalogous to that for NF3 in E q. 5.36; the AK =0 se lec t ion rate prevents itsmeasurement by microwave spectroscopy.) In this cas e , microwave spectra o f

three isotopic species must be made in principle to determine l , 1 2 , and O . Theassumption that molecular geometries are insensitive to isotopic substitution isnot always justified; the C— H bond distance in C H 3 C 1 is in fact some 0 . 0 0 9 Alarger than the C— D bond distance in the deuterated compound [6]. Hydrogenatom positions appear to be especially prone to isotopic geometry variation.

REFERENCES

1 . J. B. Marion, Classical Dynamics of Particles and Systems, Academic , New York, 1965.

2. W. H . Flygare, Molecular Structure and Dynamics, Prentice-Hall , Englewood Cliffs,NJ, 1978.

3 . G. Herzberg, Molecular Spectra and Molecular Structure, II. Infrared and Raman

Spectra of Polyatomic Molecules, Van No s t ran d , Princeton, NJ, 1945.

4 . L. Pauling and E . B. W ilson, Introduction to Quantum Mechanics, McGraw-Hill, NewYork, 1935 .

5 . E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge Univ.Pre s s , London , 1935 .

6. C. H . Townes and A. L. Schawlow, Microwave Spectroscopy, Dover, New York, 1975.

PROBLEMS

1 . In this problem, we derive the expressions given in Eqs. 5.1 3 and 5.14 for thespace-f ixed and molecule-fixed angular momentum components in terms of theEuler angles 4 ) , 0, and x . It is already clear from Fig. 5.4 that f =f (0)0/00and f =f = (koa/a x . Hence, the problem reduces to finding :ix , fy , f , and f bin terms of the Euler angles.

(a) Show geometrically that the projection of the unit vector X in the xy plane is

—-4hen show that this projection, normalized to unit length,

1



PROBLEMS81

becomes

—" i s co sXi - = [ 1

2 . )2]1/2

=in 0

(b ) N oting that 2 1 and are orthogonal unit vectors in the xy plane, show that

5 c = (2 1 3c)2 i + ( O • .Z ) O = s in 0 5 +os 0 •

and sim ilarly that

= —cos 0 s in 0.

(c ) Using the fact that

'x•( 0 -1-13-w5?.. . a)

h(ä fy = = 4 X c )

derive Eqs. 5 .13 for J y .

(d ) Using an analogous procedure, de rive Eqs . 5 .14 fo r la and . 1 6 . (Hint: T heunit vectors a and f i s ' l ie in a tilted plane perpen dicular to 5 ( . E x p r e s s a andin terms of t j ' and the projection $1 of ' 0 upon this plane. The desiredquantit ies are given by Ja = a J and Jb =b- J.)

2 . Obtain the angular momentum commutation relations (5.15) and (5.17)directly from Eqs. 5.13-5.14 for the space-fixed and body-fixed an gularmome ntum component s .

3 . Classify each of the following molecules by point group and by rotor type

(spherical, symm etric, or asymm etric). W hich of them will exhibit a microwavespectrum?

(a ) SF 6(b ) Allene , C 3 H 4

(c ) Fluorobenzene, C 6 H 5 F(d) N H 3(e ) C H 2 C 1 2

4. The microwave spectroscopic bond lengths land bond angles 0 for the C 3 ,molecules PF3 and P 3 5 C 1 3 are given below. Determine whether each molecule is




a prolate or an oblate symmetric top.

/(A)(° )

PF 3.5 50 2°P 3 5 C 1 3. 0430 0 ° 6 '

5. The microwave spe c trum of NH 3 yields th e rotational constant

B = 9.933 cm'. If the N-H bond length is independe ntly kn own to be 1 . 0 1 4 Ain NH 3 , what is the H-N-H bond angle?

6 . Cons ider a hypothetical molecule AB s C which i s kno wn to e xhibit C4,

geom etry (it would be an Oh molecule if atoms B and C were identical and a l l

bond lengths were equal).

(a ) Obtain expressions for the principal moments of inertia.(b) In view of the symmetric top selection rules, fo r how many isotopic species

must microwave spectra be obtaine d to s pecify i ts ge ome try?

7. For what H-N-H bond angle would C3 . NH 3 become an accidentalspherical top?8.(a ) S ho w that the moment of inertia in the linear OCS molecule is given by

lb = [MOMC IL MCMA MOMS(lCO 1 CS) 2 ]/1 "

where m is the total mass m o + mc + ms .

(b ) The pure rotational spectrum of 16012,-132S exhibits adjacent lines at thefrequencies 4 8 6 5 1 . 7 and 60814.1 MHz. Assign the J values for thesetransitions and calculate the rotat ional constant B fo r this i sotope .

(c ) The frequencies of one of the lines in the spectrum of 16012c34s i s

2 3 7 3 1 .Hz. Assign this line, and compute th e experimental values of / c oand / O E .

9. The FNO molecule is a bent triatomic with / N F = 1.52 A, /N O = 1.13A; theF-N-0 bond angle is 1 1 0 ° .

(a ) Locate the center of mass, and evaluate the moment of inert ia tens or in ar ight-handed Cartes ian coordinate sy stem in which points along the N-Fbond, .j) lies in the molecular plane, and is perpen dicular to the plane.

(b ) Diagonalize the inertia tens or to obtain the principal moments of inertia la ,

Ih, le . Determine the directions a ,f the principal axes.



POLYATOMIC VIBRATIONS

In a polyatomic mo lecule with N nucle i , 3N independen t coord ina te s arerequired to specify all of the nuclear positions in space. We have already seen inth e preced ing chapter that rotations of nonlinear polyatomics about their center

of mass may be des cribed in terms of the three Euler angles 4 ) , 0 , and x . Threeaddi t ional coordinates are required to describe spatial translation of amolecule's cente r of mas s . He nce, there will be 3 N — 6 independent vibrationalcoordinates in a non linear polyatomic molecule . In a l ine ar polyatom icmolecule, the orientation may be given in terms of two independent angles 0 and4 ) . Linear polyatomics therefore ex hibit 3N — 5 rather than 3N — 6 inde penden tvibration al coordinates .

By their nature, such vibrational coord inates involve collect ive, oscillatorynuclear motions that le ave the molecule 's cen ter of mass undisplaced. It is ofinterest to know the relative nuclear displacem en ts and vibrational' frequencie sassociated with these coordinates , because they are instrumental in predictingband positions and intensities in vibrat ional spectrosco py. An importantquestion arises as to whether vibrat iona l motion occurs in modes that aredynam ica ll y uncoupled . If i t does , an isolated m olecule init ially having s eve ralquanta of vibrational energy in a particular mode will not spontaneouslyredistribute this e nergy into som e of its other m o d e s , even though such a processmay conserve energy. He nce, the form of the modes ha s implications not only forvibrational structure (as manifes ted by ene rgy levels and selection rules ininfrared spectra), but also fo r unde rstanding dynam ical process es l ike vibra-

t ional ene rgy trans fer in collisions and intramolecular vibrational relaxation(IVR). These phenomena are currently well-pursued research areas , and canonly be unde rs tood with a se ason ed physical appreciation of vibrational m o d e s .

183



184OLYATOMIC VIBRATIONS

This chapter begins with a classical treatment of vibrational motion, becausemost of the important concepts that are specific to vibrations in polyatomicscarry over naturally from the classical to the quantum mechanical descr ip t ion.In molecules with harmonic potential energy functions, vibrational motionoccurs in normal modes that a re mutually uncoupled. Coupling betweenvibrational modes inevitably occurs in the pre sence of anharmonic potentials(potentials exhibiting cubic and/or higher order terms in the nuclear coor-dinates) . In molecules with sufficient symmetry, the use of grouptheory sim plifies the procedure of obtaining the normal mode f requencies andcoordinates. We obtain El selection rules fo r vibrational transitions inpolyatomics, and consider th e rotational fine structure of vibrational bands. W efinally treat breakdown of the normal mode approximation in real molecules,and discuss the local mode formulation of vibrational motion in polyatomics.

6.1 CLASSICAL TREATMENT OF VIBRATIONS INPOLYATOM IC S

The positions of all N nuclei in a polyatomic molecule may be spe ci f ied us ing th e3 N Cartesian coordinates , 3N. In t e rms o f these , the nuclear kineticenergy is given by

1 3N

T = E7 = T(4 ..., (6.1)

where m i is the mass of the nucleus with which coordinate is associated. In theBorn-Oppenheimer approximation, the eigenvalues of the electronic Hamil-tonian will act as a conservative potential energy function V ( 1,•, 3N) fo rvibrational motion (in reali ty, V will only depend on 3 N — 6 independentcoordinates in nonlinear polyatomics). If one forms the Lagrangian functionL = T — V , the nuclear Cartesian coordinates will obey th e equations of motion

which lead to

d aL aL

d ti = 1 ,3N6.2)

— ( m . ) +

0= 1, 3N6.3)

dt "

These equations may be solved for a specified potential energy function V . Inanalogy to what was done fo r diatomics (Eq. 3.32), we may expand th evibrational potential V as a Taylor se r ies about the equilibrium geometry,

e f f ,

v. vo

Y ) + 1

2 av 3o (6.4)



CLASSICAL TREATMENT OF VIBRATIONS IN POLYATOMICS 185

Arbitrarily se tting 1 70 = 0 and recognizing that (a V/5 1 ) = 0 fo r all coordinatesat the equilibrium geome try, we obtain the general potential energy function

1 3N

= - E• •2

where we have introduced the notation c i i = (0 2 V /a i 0 ; ) 0 . If we now make th eharmonic approximat ion by ignoring third- and higher o rder term s in Eq. 6 .5 , th eequations of motion become

d — (rn . ) + E c = oJ a t

(6 .6)

In terms of themass-weighted coordinates

=6.7)

th e kinetic energy assumes the simpler form

1 3 NT = — E

2 i

The equations of motion then be come t ransforme d into

d( 11.) +b41•=0

d t " '

with b u = cu /(m 1 nti ) 1 1 2 . Note that motion in any mas s-weighted coordinate th i s

coupled to motion in all other coordinates j , when b u 0 0, so that polyatomicvibrations gene rally involve all of the nuclei moving simultaneously in collect ivemotions. Since the solution to th e classical equation of motion for an undampedone-dimensional harmonic oscillator is a sinusoidal function of time [1 ], it isphysically reaso nable to try s olutions of the form

=j sin(t ( 5 )= 1, . , 3N6 . 1 0 )

for the coupled homogeneous secon d-order differential equations (6.9). Differ-entiating this twice with r e s pec t to time gives

= — 2 sin(t 6 . 1 1 )

and then substituting E q s . 6 . 1 0 a n d 6 .1 1 into the coupled equations (6.9) yields

(6.5)

(6.8)

(6.9)

+ E b o = o,j = 1, ..., 3 N6 . 1 2)



18 6OLYATOMIC VIBRATIONS

because the function sin(t6) cance ls throughout. W riting the trial solu-tions n i in the form of Eq. 6 . 1 0 is tantamoun t, by the way, to assum ing that allof the nuclear motions in a given vibrational mode oscillate with the same

frequency y =\/7112n and at the same phase 6 . T he s imultaneous equations (6.12)for the 3 N oscillation amplitudes le can have non trivial solutions only when [1]

b

A

b21

b3N,1

b1 2

b22—

b 1 , 3 N

b3N,3N Â

=0 (6.13)

T he 3 N roots Ai, A2, /13N of this secular determinant are related to theallowed vibrational frequencies v i by A i = 4n 2 q. They depend on the coefficientsb i i , which are in turn governed by the potent ia l energy funct ion (viac i; = (a2vI8 1 ô i ) ) and the nuclear masse s (via b u = cu l( m i tn i ) 1 1 2 ) . It turns outthat six of the e igenvalues of (6 .13) will be zero in gene ral (five in linearmolecules) , correspond ing to the nonoscillating center-of-mass translationaland rotational motions. The only approximation we have used in this treatmentwas to break off the Taylor series (6.4) at the seco nd-order (harmonic)approximation; otherwise it is classically exa ct. The principal difficulty with thisapproach i s tha t exce pt in molecules with special symmetry, the coefficients

= (82v ika) are not easily evaluated in terms of the Cartes ian coordinates

W e will now apply this formalism tothe example

of a linear AB Cmolecule

inwhich the nuclear motion is artificially res tricted to motion along the internu-clear axis for simplicity. In a full three-dimen sional t reatme nt , such a moleculewould have three translations, tw o rotations, and four vibrational modes.Constraining the nuclei to one-dimensional motion along the axis will eliminatetwo of the trans lations , both ro ta tions , an d two ben ding vibrat ional modes(which involve nuc lear displaceme nts perpe ndicular to the axis). The nuclearmasses and Cartesian coordinates are shown in Fig. 6 .1 . For this system , thenuclear kine tic en ergy is rigorous ly given by

2 T = m,k ? + rra l 3 3 + rr i c 36 .14)• 222W e m a y use a harmonic approximation for the potential energy function

2V = k1g2k2 ( 3 — 2) 26 . 1 5 )

Figure 6 .1 Nuclear masses and Cartesian

displacement coordinates for linear ABC.

c 2 .

m ABC



CLASSICAL TREATMENT OF VIBRATIONS IN POLYATOMICS 187

which attributes Hooke's law forces to the A—B and B—C bonds with forceconstants k 1 and k 2 , respect ively. (A more accurate potential could also includes o m e interaction between the end atoms, and so depend on ( 3 —) as well). In

mass-weighted coordinates, the potential becomes

2V = k i rd/m A + rii(k 1 + k 2 )/m B + k 2 /m c

0/1 1 /2A/mAmB — 2k 2 n 2 n 3 /,1n/ B in c

The secular determinant (6.13) then becomes

(6 .16)

k l Im A — 2k i lN /m A m B — k i t„ ./m

Am B

k 1 k)/m B — 2

k 2 1 .v/ m B m c

0k 2 /N /m B m c2 /flic — A

(1—1

— A [ k i k2(1+ —)1

m A MBB MC

+ k i k 2 (m A + m B + mc )/m A m B mc = 06 .17)

Hence the three roots of the secular determinant are given by

(i + 2 2 = kl. 1 1-+k21+1MA MBB MC

2122 = (MA ± + Mdkik2/MAMBMC

23 = 0

(6.18)

These roots A i are related to the allowed vibrational frequencies v i by A i =4n 2 q.The third root 23 = 0 is associated with the zero-frequency translation of

molecule's center of mass along the molecular axis; the nonzero roots A i and 2 2correspond to vibrational modes involving stretching of the A—B and B—Cbonds.

We can gain considerably more insight by recasting this treatment in theform of a matrix eigenvalue problem, because then we can exploit several well-established theorems from matrix algebra [2]. The coeff ic ien ts b i i in thepotential energy function (6.9) may be organized into the matrix

=A {22

B =

b1 1

b21

b 1 2• b1,3N

(6 .19)

b3N 1 b 3 N , 3 N -




and the set of mass-weighted coordinates r i can be written as the column vector

r 1

(6.20)

213N _

with transpose

T it = [iin2 •••3N]

The secular determinant equation (6.13) then becomes

I B —0 (6.21)

(6.22)

where E is the unit matrix. Each of the roots 2k of this secular determinant can besubstituted back into Eqs. 6.12 to give a n e w set of equations. These give therelative amplitudes la of oscillation in the coordinates q i corresponding to the

kth allowed vibrational frequency v k =2 n :

blin7k + bi2n3k + ••• — kri ? k 0(6.23)

b3Niek +173N,2n3k + --• — /11 13N,k = °

For a given vibrational frequency v k , these relative displacements may be

expre s s ed in a column vector

,,0 -' 1 1 k

'1=

O'I2k

13N,k _

These column vectors may be normalized,

likk?ek

(6.24)

(6.25)

so that

(6 .26)



CLASSICAL TREATMENT O F VIBRATIONS IN POLYATOMICS 189

In matrix notation, the normalized column vectors

ilk

lk = 1 2 k (6.27)

1 3N,k _

obey the condition Pk .1 k = 1. All of the information about the form of thevibrational motions i s containe d in the normalized eigen vectors l k , whoseeleme nts reflect the relative displacements in each of the mass-weightedcoordinates. The number Ck in Eq. 6.25 is just a constant fixing the vibrationalenergy in the mode with frequen cy Vk; Ck- 2 is proportion al to the total energyT + V in mode k, and forms an allowe d vibrationa l ene rgy continuum in the

classical picture.Finding the roots 2 1 , 23 N of the secular determinant (6.13) i s mathema-

tically equivalent to findin g a similarity transformation which diagonalizes the

B matrix,

L .1• B • L =

-2 1

0

0•

•

0

0

22

0

0

0

0

23

0

0

0

0

0

0

0

1 1 3 N _

A (6.28)

where A is a diagonalized matrix of eigenvalues whose e leme nts are the roots.Each of the column vectors l k (Eq. 6 .27) is an e igenvec tor of B with eigenvalue Ak,

B' = Akik

According to the theory of the matrix eigenvalue problem [2] , the matrix L

required for the similarity transformation (6.28) is equivalen t to the matrix of

eigenvectors

L

1 1 1121,3N

1 2 1222,3N (6.29)

3 N , 1 3N,3N_

obtained by stacking up the eigenvectors 1 1 , 1 2 ,1 3 N side by s i de . Wh en B is a

symmetric matr ix (b i; = b.)—which it must be in our problem because= (mimi) 1 / 2 0 2 V/(g i O i — the e igenvec tor s l k form an or thonormal s e t

1 1 , • le = (5 k i e6 .30)



190O L Y A T O M I C VIBRATIONS

and the inverse L -1 of the matrix of e igenvectors is given by

V't6 .31)

where the elements (Ct) u of the transpose Ct of any m atrix C are defined as(C) i i =

T o i l lus tra te how we obta in the normalized e igenvec to rs l k for molecularvibrations, we resume our discussion of the linear ABC molecule . W henspecialized to the more symme trical case of linear ABA in which the end masses

and force constants are equal (mA = mc and k = k 2 = k) , the secular determi-nant roots 2k in Eq. 6 . 18 become

= kinIA

22 — kiMA 2k/MB

23 = 0

Substitution of the root 23 = 0 into one of Eqs . 6 . 23 then yields

(6.32)

(k /MA)/173WMAMB)/13 3 = (6.33)

since b 1 3 = 0 and 23 = O. Similarly,

1- .

301

0

3 4- b32 1 123 + b33r/33 — 2303 = 0 (6.34)

leads to

( — k/N/mAmB) 1 1 23 + (k /mA)03 = 06 .35)

Equations 6 .34 and 6 .35 imply that

7 3 / 1 2 3 = (mA/mB) " 2 and 03 /03 = 1

n 3 3 / 7 13 3 = (n2A / m 1

nd 03 /03 = 1

6 .36)

H ence i t i s apparent that 03 =03 , so that the zero-frequency mode(2 3 = 0) i s as sociated with overall molecular translation parallel to the mole-cular axis (Fig. 6.2). The normalized e igenvector for mode 3 can no w easi ly beshown to be

[01,001/2 -

1 3 = (MB/M) "2

(MA/M) 1 /2 -

(6.37)



NORMAL COORDINATES91

A

0- -0 -0

C 1 ) - 1 3

Figure 6.2 Eigenvectors 1 1 , 1 2 ,1 3 of the matrix B for linear Al3 2 . Lengths of arrows

give relative displacements in mass-weighted (not Cartesian) coordinates.

where m = 2 m A + m B is the total nuclear mas s. In a similar way, the normalizedeigenvectors for the other two modes can be derived as

11= (6.38)

1 2 =

[ (mB/20/ 2 -

— 2 (m A 1 2 m ) 1 / 2( m B /201/2

-

(6.39)

I t may be verif ied that these norma lized e igenvectors obey the normalizationand orthogonality conditions (6.26 ) and (6 .30 ). It is apparent f rom Eq. 6 .38 thatmode 1 is a symmetric stretching mode (Fig. 6 .2), because the coordinate 1 7 2 of

atom B has zero coefficient in eigenvector 1 1 , and the e igen value 2 1 has nodependence on m 5 — as would be expected if nucleus B were motionles s. I t i sclear from the form of eigenvector 1 2 that mode 2 is an asymmetric stretchingmode (Fig. 6.2 ) in which the A nuclei m ove together in a d irect ion opposi te tothat of nucleus B . Since all of the nuclei move i n mode 2 , 1 1 2 depends on m B aswell as on m A .

6.2 NORMAL COORDINATES

T o a good approximation , the res toring forces responsible for molecularvibrations are directed along chemical bonds. The potential energy function (6.4)is therefore not e asily ex presse d in Cartesian coordinates ex cept for molecules




having bond angles of 9 0 0 and/or 1 8 0 ° (e.g. , linear molecules, o ctahedral S F 6 ,and the hypothetical square-planar A4 molec ule). It may be more naturally de -scribed using 3N generalized coordinates g i which are re la ted by a lineartransformation to the mass-weighted coordinates,

or

q = M q6 .40 )

The en ormo us utili ty of the Lagrangian equations (6.2) lies in the fact that if q isrelated to t i by such a transformation, they beco me t rue in the generalized aswell as in Cartes ian coordinates :

= 06 .4 1 )dt agi ag

The potential function in the harmonic approximation now becomes

2 V a- - if - Bi1 = (M • q)t B -(M q)

= ( i t - B ' • q6 .42)

where the matrix B in the Cartesian basis has become transforme d into

B' = M t • B • M6.43)

in the basis of gene ralized coordinates. U nde r this linear transformation, 2 Vremains harmonic, because Eq. 6 .42 contains no higher than second-order termsin q i . The kinetic energy now become s

2 T i1ti1 = (M • 4)` • (M • 4)

6.44)

_ t i t m) . 4

W ith this change of basis, the se cular determinan t equation I B — A E I = 0becomes replaced by

I B' AML MI = 06 .45)

with B' = M t • B • M. However , the linear transformation (6.40) leaves the

e igenvalues )Lk and the physical eigenvectors unchanged [2 ] , so that the secularequations (6.22) and (6.45) yield precisely the same re sults; the transformation

simply creates lee way for choosing conve nient coo rdinates with which toexpress the potential energy function.

d aL aL



NORMAL COORDINATES 193

W e now ass e r t tha t for harmonic potentials (poten tials co ntaining no termsbeyond the quadratic terms proportional to q•q i in Eq. 6.42) it is always possibleto find normal coordinates Q , related to ti by a linear transformation

mo • Q6.46)

which diagonalize both the kinetic and potential energy. In particular, we claimthat these ene rgies become

2T = Qt • Q E 0?

2 V = Q t •A•Qi Q ; 3 ,6.47)

In contrast to expressions (6.44) and (6.42) fo r 2T and 2V in arbitrarygene ralized coordinates , these equat ions conta in no cross terms in 0 i 0ior Q.Q. W e m ay apprec iate the physical significance of such normal coor-dinates by substituting Eqs. 6 .47 into the Lagrangian equations

to yield

d 0 1 _ ,L n

dt 00 i aQ== 1 ,3 N6.48)

c i + 0= 1, ..., 3N6.49)

In con trast to the corresponding coupled e quations i j i + butt = 0 in mass-weighted co ordinates , Eq. 6.49 shows that each normal coordinate Q. oscillates

independently with motion which is uncoupled to that in other normalcoordinates Q i . This separation of motion into noninteract ing normal coor-dinates is possible only if V contains no cubic or higher-order term s in Eq. 6.4.Anharmonici ty will inevitably couple mot ion betwee n differen t vibrational

modes, and then the con cept of normal modes will break down. In the normalmode approximation, no vibrational en ergy redistribution can take place in a nisolated molecule.

T he formal solutions to the secon d-order differential equations (6.49) are

Q i = Q? sin(t .5 )6.50)

so tha t each norm al coordin ate oscillates with frequen cy 1 / 1 = 21 / 2 7 t . Thesefrequencies are ident ical to those found by diagonalizing the secular determi-

nant (6.13) in the mass -weighted coordinate basis. To find the actual form of thenormal coordinates Q, we note that from E q. 6.46

2 V = • II =Q •( M I D B) • Q6 .51)



194 POLYATOMIC VIBRATIONS

Com paring this with Eq. 6 .47 fo r 2 V then implies that

M t° •B•M o = A

In thelight

of Eqs . 6 .28 and 6 . 3 1 ,i t then follows thatM = L

(6.52)

(6.53)

and that

= L Q6 .54)

where L is the matrix of e igenvectors of B. The normal coordinates Q may thenbe obtained via

Q=L 1 q=Ltil6 .55)

6.3 INTERNAL COORDINATES AND THE FG-MATRIX METHOD

W e pointed out in the preceding section that a realistic potential energy functionmay no t be eas ily ex pressible in Cartes ian coordinates , but may be written morenaturally in terms of 3 N gene ralized coordinates related to the mass-weightedcoordina tes by a linear transformation (6.40). In fact, only 3 N — 6 such

coordinates are required to fully specify the potential (3 N — 5 in linearmolecules): 2 V is not sensitive to the center-of-mass position or molecularorientation in space, and a polyatomic molecule exhibits only 3 N — 6 (3 N — 5 )independent bond lengths and bond angles. Such a truncated set of 3 N — 6(3 N — 5 ) general ized coordinates i s called an internal coordinate basis, and i scommo nly denoted S. To i llus t ra te how an internal coordinate basis may beuse d to evaluate normal modes, we conside r the bent H 2 0 molecule in Fig. 6.3.The three internal coordinates are convenien t ly chosen to be

S 1 = r — ro

S 2 = rc — r o6 .56)

S3 = — (P o

(where the subscripts 0 den ote equilibrium values), because to a good firstapproximation the potential en ergy function in water has the form

2 V = k i (Si +k S i6.57)

This potential assumes that independent Hooke's law restoring forces are



INTERNAL COORDINATES AND THE FG-MATRIX METHOD 195

6

3

Figure 6.3 Internal coordinates T B , rc , and 4 ) for the F I 2 0 molecule. Arrows show

orientations of the Cartesian basis vectors throughsed in Eqs. 6.64 and 6.65.

operative in each of the O—H bonds as well as in the bond angle 4 ) . The

3 N — 6 = 3 internal coordinates are displacements in two bond lengths and abond angle, rather than coordinates of individual nuclei as in the mass-weightedbasis. T he matrix formulation of the vibration al problem in Section 6.2 isnon etheles s still applicable, bec ause the S i can be re la ted to th e q i by a lineartransformation. Equation 6 . 57 implicitly defines a force constant matrix F, since

2 V = S t F - S6.58)

with

-- 1 0 F= 0 k 1 06.59)

0 0 k

The force constant matrix need not be diagonal (i.e., more sophisticatedpotentials may be used). For a ben t molecule with three n uclei , an arbitrarilyaccurate force constant matrix will still be a (3N — 6) x (3N — 6) = 3 x 3matrix.

T o obtain a secular determinant e quation analogous to (6.22) in the internalcoordinate basis, both 2T and 2 V must be expressed in terms of S. Since 2 T isreadily given in terms of mass-weighted coordinates ii , we need a transformationof the form

S = D -6 . 6 0 )

Note that becaus e S and II have 3N — 6 and 3N elements respectively, D is arectangular (rather than square ) matrix with 3 N — 6 rows and 3N columns. The



[si

$S2

A S 3

196O LY A T O MI C VIBRATIONS

kinetic energy then becomes

2T if • ii = (D-1 • g)t • (D -1 • g )

= gt • (D • D') -1 • g

— = g t 6 .61 )

According to the rules of matrix multiplication, the product D• D is a(3 N — 6 ) x (3 N — 6 ) square matrix. With 2T and 2 V now consistently expressed

in the S basis, the secular determinant equation IB' — A M " M I = 0 in thegeneralized coordinate basis becomes

IF — AG -1 I =6.62)

in the internal coordinate basis. This is equivalent to

IF • G — AEI = 06.63)

so that finding the normal mode frequencies and coordinates reduces to

evaluating the matrix F • G, and then diagonal izing it [3]. This formalism hasbecome known as the "FG matrix method." Since a physically reasonable forceconstant matrix F is often expressible in internal coordinates, most of the laboris incurred in evaluating the kinetic energy (G) matrix. Wilson, Decius, andCross [3 ] have provided extensive tabulations of G for a number of molecular

geometries.The internal coordinates S 1 = rB — r o and S2 = rc — r o for H 2 0 are related

to the Cartesian displacement coordinateshrough 6 in Figure 6 .3 by

S i = — ( 1 sin 4 ) +2 O S ( 1 ) ) + (5 sin 4 ) + 4 COS 4 )

S2 = sin 4 ) — COS — 4 sin + 4 COS6.64)

To obtain the bending coordinate S3, we recognize that small displacement Ar ofany nucleus in a direction perpendicular to a bond of length ro produces achange d = dr/ro in the bond angle. Consideration of the effects of each of theCartesian displacements on the bond angle then leads to

S3 =— 1

(— COS 4) ± 2sin 4 ) +O S 4 ) ±in 4 ) )6 .65)r o

Equations 6 . 6 4 and 6 .65 now give us D directly. Recalling thatq i = i\bn i and that

D D 16

(6.66)

D3 1 "• D 3 6



The G matrix then becomes

1 CO S 20

G = D• D' —

0

0

MH0cos 2 4 )

m0

1 mo mHo 2v ,, ,

I 011&H

(6.68)

SYMMETRY CLASSIFICATION OF NORMAL MODES 197

the D matrix is by inspection

D=

— sin cos 4 )in 4 ) cos 4 )0

N/rnosin 4 ) cos 4) — sin 4 ) cos ( / )

0 NA%N/moN/mo

— c o s 4)in 4 )os 4 )in 4 ,

r o . .1m H .r0 . 17% r o ...1% r.\/mH

(6.67)

This can be multiplied by the F matrix (Eq. 6.59) and F G can be diagonalized

to find the eigenvalues 2 1 through )13. Such an algebraic procedure can readilybe computerized to vary the force constant matrix in order to optimize the

closeness of fitted vibrational frequencies v = 2 k / 4 7 r 2 to spectroscopicfrequencies.

6.4 SYMMETRY CLASSIFICATION OF NORMAL MODES

When expressed in terms of normal coordinates, the classical kinetic andpotential energies associated with vibration in a polyatomic molecule are both

diagonalized(cf.

Eqs. 6.47).The classical vibrational Hamiltonian becomes

3N- 6

H= T + V= E A i W)6.69)

Under the correspondence principle, the quantum mechanical Hamiltonian for3N — 6 independently oscillating normal modes in then

3N- 6( h2 a2

Rvib = E? + 24202 a

The eigenfunctions of fi v ib are

(6.70)

3N - 6

Illivib(Q 1, • • • 423N - H N„,1 -1 , , ( C i )exp(— a/2)6 .71)




with

(6.72)

and theeigenvalues

are

3N —6

E vib hv i ( v i (6.73)

In Eq. 6 .71 , the functions 1 -1(( 1 ) are Hermite polynomials of order v i in C i , andthe N are normalization constants. The factorization of I l i f y ib > into independentfunctions of the normal coordinates Q . is possible only when the potentialene rgy function is harmonic.

To de rive the selection rules for vibrational transitions, it is ne cess ary to take

the symmetry of the normal modes into account. We assume that th epolyatomic m olecule belongs to some point group G of symm etry operations A .By d efinition, all A in G leave the vibrational H amiltonian invariant, so that [A,f ivid = O . If each normal mode transforms as an irreducible representation (IR)of G , one can s e t up matrices R with eleme nts R u and H with eleme nts Hu in abasis of vibrational states which are eigenvectors of k (i.e., vibrational states thattransform as IR s of G). Since [A, fiv = 0, it follows that [4]

E R E kJ/A = [H ' R ] ik = E H ijRjk

6.74)

However , if the matrix R is evaluated in a basis of eigenvectors of it it follows bydefinition that R u = Ri ik. The n Eq. 6 .74 implies that

RUH ik H ikRkk

and

(R i i Rk )H i k =06 . 7 5 )

Thismean s that if

i k

i.e., vibrational state si and k belong to differen t IRs of

G ), H ik = O. So liv ib has no nonvanishing matrix eleme nts conne ct ing statest ransforming as two different IR s i and k.

Since the molecule is physically unchanged by any symmetry operation A inG , A Q i must be a normal mode having the same frequency as Q i i tself. Hence , if

Q . is a nondegenera te mode, k'Q i must equal +Qi for all operations A in G—so

that Q. must form the basis of a one-dimensional JR in G . If Q. is degenerate ,then

= E Dik0Qk6.76)

where k runs over all modes, including Q i , that are degene rate with Q i . This i s




the case because all of the Q k in this set of modes will os cillate at the samefrequency as Q i .

Since the normal coordinates Q are related by a linear transformation to the

Cartes ian coordinates 4 , the matrices for the transformation properties will havethe same characters (traces) in ei ther coordin ate basis und er all group opera-tions A [5 ] . W e can see this by assuming that 4 = N • Q by hypothes is, and thatthe effect of a group operation A on the Cartesian basis i s

= P(A)6 . 77)

Then the effect of the operat ion A on the normal coordinate s e t Q is

=1 - 4) = N -1 . k4

= N -1 - P (R ) • 4 = [N -1 • P(R)• N] Q

P'(R)- Q6 . 78)

Since the character of a matrix is unaffected by a similarity transformation [2]and P'(R ) = N 1 • P(R) • N, the transformation matrices P ' ( R ) and P(R) in the

normal coordinate and Cartesian bases respectively have the same character for

all group ope rations A in G . This fact cons iderably simplifies the determinat ionof the IR s to which the normal coordinates belong, since the behavior of thenuclea r C ar te s ian d i splaceme n ts unde r the group operations reveals thisinformation even if the form of the normal coordinates i s unknown.

For concreteness , consider the planar, ne arly T-shaped CIF 3 molecule, whichbelongs to the C2, point group (Fig. 6.4) . The group eleme nts a„ and a' , den otethe out-of-plane and in-plane ref lection operations . Any symme try operation on

the Cartesian basis vectors through 1 2 is express ible using a 12 x 12 matrix.U n d e r the a' , Operation, for example, the in-plane vectors c through 8 will beunaffected , whereas 9 through 1 2 will reverse sign. He n ce in the Cartesianbasis

12

1

0

0

0

0

0

0

0

0

0

00

0

1

0

0

0

0

0

0

0

0

00

0

0

1

0

0

0

0

0

0

0

00

0

0

01

0

0

0

0

0

0

00

0

0

0

0

1

0

0

0

0

0

00

0

0

0

0

0

1

0

0

0

0

00

0

0

0

0

0

0

1

0

0

0

00

0

0

0

0

0

0

0

1

0

0

00

0

00

0

0

0

0

0

1

0

00

0

0

0

0

0

0

0

0

0

— 1

00

0

00

0

0

0

0

0

0

0

— 10

0.

00

0

0

0

0

0

0

0

0- 1 12

(6 .79)




Figure 6.4 Cartesian coord inates fo r planar T-shaped molecule of C 2 „ symmetry.

through lie in the a reflection plane, while through 1 2 point normal to this

plane. The a, plane (not shown) is normal to the d, plane; the twofold rotation axis

C 2 lies at the intersection of the a cf, planes.

and the character x(cr'v ) of the matrix a', is 8 — 4 = 4. Under the identity

operation, all of through c 12 are unchanged, so that x(E) = 12. The C2operation is more interesting in that two of the nuclei are displaced:

— 010 —000 —1 0 0100 00 —1 00 1000 0000 0001 0000 —1 0000 —1 0000 —1 0000—1 0000 —1

C2

12 12

(6.80)

For this matrix, the sum of diagonal elements is x(C 2 ) = —2. It can similarly beshown that x (cy„ ) = 2. These characters for the transformation of 4 under the C2,



SYMMETRY CLASSIFICATION OF NORMAL MODES , 201

group operat ions m ay then be summ arized as

EC cr a:,

x = 1 2 —2 2 4 =a i a 3b 0 4b 2

According to the C2, character table, C1 F 3 will have translations transformingas a l , b 1 , b 2 and rotations transforming as a 2 , b 1 , b 2 . Subtracting thes e IR s fromthe above direct sum yields 3a 1 0 b 1 2b. He n ce CIF 3 will have three normalmodes of a l symmetry, one of b 1 symmetry , and two of b 2 symmetry. W e havede termine d this without determ ining what the normal mode coordinates Q 1

through Q6 actually are ; this is possible because the transformation matricesP(R-) and P'(A) have the same character in the 4 and Q bas e s .

Se tt ing up transformation matrices l ike those in Eqs . 6 . 79 and 6.80 is

laborious, and a worthwhile simplification results if one sees [6 ] that only theCartes ian c oordinates a t tached to n ucle i that are undisplaced by a symmetryoperation I co ntr ibute to the character A A ) . In particular, the contributions tothe character are x(a) = + 1 , x(C„n) = 1 + 2 cos (2mir /n) , x(S ,T) = — 1 + 2cos(2mn/n), and x(i) = — 3 for each nucleus that is undisplaced by the symmetryoperat ions a , S , T , and i, respectively. The character x(E) for the identityoperation is always 3 N. These rules are independent of the particular choice oforientation of the Cartes ian a xe s .

The identification of the IR s according to which the normal coordinatest ransform can greatly red uce the computational labor associated with imple-menting the FG matrix method. It is frequently easy to s e t up symmetry

coordinates, as linear combinations of internal coordinates , which trans formaccording to IR s of the point group G . For C1 F 3 , one choice of symmetrycoordinates would be

S i (a ) = (r 1 — r?) + (r2 —

S 2 (a 1 ) = (r 3 — r3)

S3(a1) = (01 — 07) + (02 — 03)

S4 (b 1 ) = 6S 5 (b 2 ) = (r — r?) —(r2 — r 3 )

S 2) =07) — jO (6.81)

where the six indepen den t internal coordinates (bond len gths and bond angles)

are defined in Fig. 6.5. In such a basis, the F•G matrix reduces to block diagonal

form, with subblocks allocated to symm etry coordinates t ransforming asparticular IR s of G (because Rvjb has no matrix eleme nts conne ct ing normalcoordinates belonging to differen t IRs). The block-diagonal form of F-G for

C1 F 3 is shown in Fig. 6 .6 .W e will now cons olidate som e of the ideas int roduced in this chapter byderiving the vibrational freque ncie s of the l inear acetylene molecule C 2 H 2 , in



Figure 6.5 The six internal coordinates for a planar T-shaped molecule. The

coordinate 6 is an out-of-plane bending coordinate; + and — indicate nuclear

motions above and below the plane of the paper.

a l

F • GX 3

b i

I X I

b 2

2 X 2

Figure 6.6 The block diagonal F • G matrix for CIF3.

202




8 13 r

1 2

0 23H Figure 6.7 Five of the seven ind epend ent internal coordinates for D,, acetylene.

The remaining two internal coordinates are bending motions normal to the plane ofthe paper, and are not considered in our treatment.

which th e nuclei are numbered 1 through 4 from le ft to right (Fig. 6 .7). In termsof the bond radii ri; and bending angles 6 ii , we will assume that the potential

en ergy function is given by

2V = k 1 r 3 + k 2 (ri 2 + /14 ) + k b ( c 5 i 3 + Si 4 )6 .82)

(A more detailed poten tial function, incorporating interaction between non -neighboring atoms, fo r ex ample , could be use d to obtain bet ter f it s of derivedvibra t ional f requencies to expe r imenta l f requen cies .) He re k 1 and k2 are

harmonic fo rce cons t an ts fo r C— C and C — H bond s t re t ch ing , respectively, whilek b is a bending force constant. As a first s t ep, we de termine the IRs to which th enormal modes will belon g in linear C 2 H 2 , using th e rules for contributions to the

character by nuclei undisplaced by the symm etry operations in D oh :

2C 4) xav 2S 0 ooC2

X= 1 2 4 + 8 cos 4 0 0 0

Xtrans 3 1 + 2 co s 4 . 1 —3 — 1 + 2 cos 4 ) —1 (=0": CI nu)

X r o t 2 2 cos 4 ) 0 2 —2 cos 4 ) 0 (=n g )

Xvib= 7 3 + 4 cos 4 ) 3 1 1 1o :®2o - g+

nu0n g

Acetylene there fore has three nondegenerate vibrations (one has a : symme -try and tw o have a: sym metry), and two doubly de gene rate vibrations each o fn u and n g sym me try . I t i s now possible to form symm etry coordinates f romlinear combinations of the bond lengths ri; and the bond angles b ii :

Sl(ag ) = r1234

S 2 ( e r g ) = r 2 3

—:) = r1234

S4(irn)— 013 + 024

s 5(n g) =01 — 0 24

(6.83)




These choices of symmetry coordinates are no t all unique; one could takeorthogonal combinat ionsa(r 1 2 + r 3 4 ) + br 2 3 and S '2 a lr 2 • 34.+ bir 2 3 as the tw o a g+ symmetry coordinates instead. The choice S3 =

r12 r3 4 is unique (aside from normalization, which does not concernus here) , because there is on ly one vibration of a u+ symmetry and i t i s non -

degenerate. In our potential energy expression (6.82) we have ignoredbending motions which are analogous to ( 5 1 3 and 6 2 4 , but move in and out of thepaper. These bending motions are simply the degene rate, perpendicular counter-parts of the n u and n g coordinates l is ted in Eq s . 6 .83 . Tre ating them would onlygive us redundant information about the it and 7 1 g modes, and so they areomitted. With these five symme try coordinates, the potential energy expressionbecomes

2V = k lSi'+ l + + (S i +6.84)

and the F matrix is

-k 2 /2 0 0 0 0

O k 1 0 0 0

F = 0 0 k 2 I2 0 0 (6.85)

0 0 0 k 6 /2 0

0 0 0 0 k 6 /2

(Normally the F matrix should be (3N - 5) x (3N - 5) = 7 x 7 in the S basis,but we have left out one of the 7r, and one of the n g coordinates.) To de rive the G

matrix, we begin with the Cartes ian bas is shown in Fig. 6 .8 . (We couldinclude four addi t ional Cartes ian vectors 9 through 1 2 pointing out of thepaper, but these vectors have projections only along the omit ted n u and n gmodes.) In terms of these, the symm etry coordinates are

S 1 =-3r 1 2 and r 3 4 ex pand s imultaneously)

S2 = -

S3r 1 2 and r 3 4 expand out-of-phase)

S4 = -1 - (4 +)rH

25 5 = 1— (6 +7)rHc (6.86)

where rH and rc are the equilibrium C-H and C-C bond distances, respectively.

(The expressions for S4 and S5 arise f rom repeated application of the relation-ship 46 =A O , bearing in mind the effects of increments in the Cartesiancoordinates 4hrough 4on the sign of changes in the angles 6 1 3 and 6 2 4 .) We




e 666-w-e,

e 8

H Figure 6.8 Cartesian displacement coordinates for acetylene. Motions per-

pendicular to the plane of the paper are not considered.

may nowconstruct the matrix D, since

S i

S5

[DIA18

D1558

1 /11

2/8 _(6.87)

Re calling that Il i =nd using the abbreviations C = m c and H = m H ,

D =

i _ 1/,/:i/ \./C —1/1e

01/j 1/.1 -e

—1/1—H1/ie 1j

0 0

0 0

1 /.\/H 0 - 0 0

0 0 0 0

— 1VH 0 0 0

— 1 1 1 — 1

r H O-1i r N /C

H /

H OI

— 1( 2 + 1) ( + 2)

rH \\/CVc rH) . N /CVH rc) ri u N /11

(6.88)

and so

G = DD =

22 HCC

o

0

1 ±2 7 1 ± 2 2 1r4H Cc)0

(6.89)



k 2 (7 +e)

k 2 IC

2k i IC

0 0

k a0

1)1 ±di H

0

0 0k6


The F • G matrix is then

F • G =

(6.90)

Note that this matrix is block-diagonal, with the subblocks a g+ (2 x 2) ,x 1), n u ( 1 x 1 ), and I r g ( 1 x 1 ) . He nce , we could have evaluated the F, G,

and F • G matrices in separate bases of symm etry coordinates t ransforming asone JR at a time. The e igenvalues in the last three modes can now be r eaddirectly from the 1 x 1 subblocks in the F • G matrix,

k2

(

11-H

+ —C

=4 7 r 2v c+

/14b ( 17 r 2 v 4

rH )k2)2-1

6AH C r H rc)

n u

m g

(6.91)

The or eigenvalues 2 1 and / 1 , 2 are the roots of the 2 x 2 secular determinant

— k 2 IC0

— k i ICk 1 IC — A

=

1 (6.92)

yielding

1) 2k1 1 214 2 (1 ±/42 /C 2 = 0 (6.93)2 — A[k 2 e

C




v l J E g+

1 1 3 G.-

V

5

r i g

1 14 n u

Figure 6.9 Cartesian displacements in the normal coordinates of acetylene. Eachof the modes labeled ng

andrri

has adegenerate partner

modeinvolving nuclear

motion in and out of the paper.

The e igenvalues /1 . 1 an d . 1 . 2 then obe y

± A2 = k2 (-1± ) 2k1 = 47 C 2 (Vi 1/ )HCC

214 2 (1 1=/C 2

=1 6

viv,21)12k

C HC

(6.94)g

I t may be se en from the expressions for the symmetry coordinates (Eqs . 6 .86 )that the qualitative normal coordinates in acetylene a re those shown in Fig. 6 .9.



= Hi ( c i )[xp( i;oi3N – 6N – 6

(6.97)


6.5 SELECTION RULES IN VIBRATIONAL TRANSITIONS

Th e s y m m e t r y s e l e c t io n r u l e s fo r El vibrational t r ans i t i ons can be obtained

using th e sy m m etr y p r op er t i e s o f t h e p e r t in e n t v ib r a t i o n a l s t a t e s . A cc o r d in g t o

Eq . 6 .71, th e v ib r a t io n a l g r o u n d s t a t e is

3N-6

'gib> = n NoiH o(cd exp(— cf /2 )

3 N – 6=o) xp( —/2) (6.95)

since th e z e r o t h - o r d e r H ermi t e po lynomi a l i s H0(1) = 1 . Th e e x p o n e n t i a l in th i s

w a ve fu nct ion i s

3 N – 6N – 6—= —E2 7 r v i Q?12h (6.96)

a nd is in va r ia n t u nd e r a l l sy m m et r y o p e r a t io n s o f th e m ol ecul ar p o i n t g r o u p.

H e n c e , t h e n o n d e g e n e r a t e v ib r a t i o n a l g r o u n d - s t a t e w a v e fu n c tio n b e l o n g s t o th e

totally symmetric JR o f t h e p o i n t g r o u p. T h e v ib r a t i o n a l ly exc it ed s t a t e with o ne

q u a n t um in n o r m a l m o d e j a n d z e r o q u a n t a in a l l o t h e r m o d e s is

3N– 6

itkib> = N 1 Hi)exp(— C/2) n NoiHo(i)exP( — a/2)

Since H1 ( ) =n cc Q i , I t / f tl it ,> t h e r e f o r e t r a n s fo r m s a s Q. i t se lf , a nd b e l o n g s t o

th e s a m e JR a s Q.T he symmet r i e s of o v e r t o n e a nd combi na t i on l e ve ls in which n o d e g e n e r a t e

m o d e s a r e multiply excited a r e s t r a i g h t f o r w a r d l y o b t a in e d f r o m direc t produ ct s

of IRs 1 (Q) according to which the normal coordinates Q i t r a n s f o r m . In

p ar t icu lar , t h e symmet ry o f t h e vibra t i ona l s t a t e w ith y 1 quanta in mode 1 , y 2

quanta in mode 2, etc . , i s g iven by

E1(1 2 1 ) Ø IWO 0 • • . 1 ® Er( 22) 0 F(22) 0 • • 0 • •vi factors2 factors

0(I-(23N – 6) 0 r(Q 3N– 6)0 • . 1

V3N _ 6 factors

L e v e ls i n v o l vin g o v e r t o n e s in degenerate modes must be handled with mo re

ca u t i on . C o n s i d e r th e t w o d e g e n e r a t e n u m o d e s Q4 a nd Q4 c o r r e s p o n d i n g t o th e

ben d in g f req u en cy y 4 in acety lene (Section 6.4). T h e s e f o r m a ba s i s for the i r„ JR



SELECTION RULES IN VIBRATIONAL TRANSITIONS09

in D Œ ,h . W e wish to obtain the symmetries of the vibrational states in which y 4

and y'4 quanta are placed in the re s pect ive mode s , under the con dit ion thaty 4 + y4 = 2. Since both modes have nu symm etry, using the above procedurewould imply (incorre ctly) that the res ulting vibrational states should have thesymmetries n u 0 n u = o g+ a- 0 b g . This would mean that four vibrationalstates s upposedly arise from distributing two quanta between the degenerate n umodes (one doubly degen erate (5 g pair of states, and one state each of o - g+ an d ir çsymmetry). However, there are only three distinct (y 4 , y 4 ) combinations possible:(2, 0), (1, 1), and (0, 2). Evaluating the ordinary direct product thereforeovercounts the res ulting vibrationa l leve ls when two quanta are placed in thedegenerate n u vibrations. The symmetries of s ta te s a r is ing f rom multipleexcitation of degenerate modes Q i are ins tead found by evaluating the symmetric

product [F(Q) F(Q)] + [7]. For a doubly exci ted nu mode, the pertinentsym me tric product is (n u 0 nu ) = o - g+ (5grather than a g+ a-6g7]. Iftwo vibrationa l quanta are distributed between nonequivalent degenerate modes( e . g . , a nu and a n g mode , o r two r c u modes oscillating at different frequencies),conventional direct products yield the correct vibrational state symmetries.

The El selection rules for vibrational transitions l i i / v i b > -4 Iti/vib> can beobtained by expanding the dipole moment p in a Taylor series in the n ormalcoordinates,

<11/vibliiltfrvib>

3N6(5p \ 3N-6 /3p

= <Ci131/10 Q i 0 ;)0 QiQ; + • • •Itkvib> (6 .98)

T he leading term is Po<KiblOvib>, which vanishes by or thogonali ty . G rouptheoretically, the terms (a1/0•20 0 Q i , (0 2 0/0Q i 0Q ; ) 0 Q•Qi , etc., all transform asvector components; in effec t, the transformation properties of QiQiQh . in theexpansion (6 .98) are cancelled by those of 0Q i 0Q ; O Q k . . . in the correspondingderivative. Consequently, we obtain the symmetry selection rule thatT(K b ) 0 F(p)0 F(tli i b ) must contain the totally symm etric IR—regardless ofwhich terms do minate in the T aylor series expansion of p . Fo r fundamental

transit ions from the (totally s ymm etric) vibrational groun d state to levels withone quantum in mode Q. and zero quanta on all others,T(C i b ) 0 (p) 0 T(Ik v i b ). r(Q) 0 r(p). Hence the El se lect ion rule fo r funda-mental transit ions is that E(Q) = r(p); that is , Qimus t transform as a vectorcomponent. As two examples of this, we cite BF 3 and acetylene. In BF3 , thenormal mode symmetries are a, a '2', and 2e ' (the prefix 2 den otes that thismolecule has two pairs of degenerate e ' vibrations). In the D3h point group, (x , y)and z transform as e ' and a , respect ively— so that the 2 e ' and a'2 vibrationsex hibit El-allowe d fundam en tals, but the totally sym me tric a '1 breathing modedoes not. In acetylene , we showed that the seve n vibrational mo des are 2o - g+ , a u+ ,

n u , and n g . T he vector componen ts t ransform as a u+ and nu in D c o h , so only thea u+ and 7 E , , vibrations have El-allowed fundamentals in acetylene . These group



210 P O L Y A T O M I C VIBRATIONS

theoretical rules ensure that fundamentals are observed only in normal mode s inwhich the molecule's electric dipole moment oscillates.

The selection rules on tiv can be e xtracted by applying the se cond quantiza-t ion formalism to Eq. 6.98. In particular, the first-orde r term s proportional toQ. permit t rans i t ions with Av i = +1, the second-order terms in Q i Q i are re -

sponsible for the overtone and combinat ion bands with A(v i + v) =0 or ±2,and so on. The sym me try se lection rule must simultaneous ly be satisfied. It iswell known to s tudents in organic chemistry that overtone and combinationbands are frequently prominen t in infrared spectra , and so the s econd- andhigher order term s in Eq. 6.98 are not negligible.

C O 2 presents a good e xample of vibrational selection rules in polyatomics. Itpossesses three normal mode species (Fig. 6.10): a a g+ sym me tric st re tch withfrequency v 1 — 1 3 9 0 cm -1 , a doubly dege ne rate i c „ bend with frequen cyv 2 — 6 6 7 cm', and a a: asymmetric stretch with frequency v 3 — 2 2 80 cm '.T he lowest fe w vibrational levels in C O 2 are shown in Fig. 6 .11 . Each level is

labeled with the number of quanta ( v 1 v 2 v 3 ) in each m ode; the vibrational statesymmetries are also given . T he El allowe d transit ions among these levels areshown by the solid connect ing lines. T hose transit ions originating from the(0 1 0 ) level are hot bands which will have appreciable intens ity in a C O 2 sampleat 3 0 0 K (kT = 20 8 cm -1 ) , because this level lies only 6 6 7 cm' above thevibrationless level (0 0 0 ).

v2 nu

i/3

0 Figure 6.10 Normal modes in D o ,h CO 2 . Note that the relative displacements in the

E: mode are a consequence of the mode symmetry; those in the n u mode may be

found from the requirement that the center of mass is undisplaced in any vibrational

mode.



SELECTION RULES IN VIBRATIONAL TRANSITIONS11

+040g 042 0

04°0

\033 003 1 0

4-0-2 2 0

O i l &

CO2

20°0

10°0

z t;

20°1

12°

04°1z:

1 2 2 0 g= . . . . . .l /0+ 00°2

10°1

02°1

00°0Figure 6.11 Energy level diagram for low-lying vibrational states in CO

2 . Levels

are labeled with the vibrational quantum numbers 1/0/ v 3 . Some of the observed

infrared transitions are indicated by arrows. Adapted from G. Herzberg, Infrared and

Raman Spectra of Polyatomic Molecules, Van Nostrand, Princeton, NJ, 1945.

The in tens e 1 0 . 6 - p m infrared laser transitions in CO 2 are due to the(0 0 1) (0 2 0) and (0 0 1) (1 0 0) transitions. Since these are both o - u+ o:transitions, they are symm etry-allowed (and z-polarized, because z transforms aso u+ ) . It is inte res ting that even though these two El trans i t ion s arise from third-

and s econd-order terms, respectively, in the expansion of p, efficient lasing hasbeen achieved in both of them.Degene rate vibrat ional modes give rise to vibrational angular momentum in

polyatomic molecules. In the case of CO 2 , the degenera te n u normal modes Q2

and V2 shown in Fig. 6 . 1 2 ex hibit nuclear motion in mutually pe rpend icularplanes containing the molecular axis. In cylindrical coordinates, the nuclearpositions during vibration may be specified by the coordinates z, r, and 0 . Sincethe cylindrically sy mme tric vibrational potential ene rgy function is indepen den tof 4 ) , the vibrational wavefunctions must exhibit the 0-dependence exp( + /10)with / an integer. Such a wave function describes vibrational motion in which

the nuclei ex hibit a constant angular mo me ntum lh about the z axis. It may beshown [8] that when y 2 quanta are distributed between the n u modes, theallowed values of 1 are y 2 , y 2 — 2, ... , 0 or 1 . In the (0 1 0 ) level, for example, thevibra tional an gular m om en tum quantum number becomes 1 = y 2 = 1 ; thevibrat ional wave funct ion then bec ome s proportional to exp( + i0). T he vibra-t ional motion in state (0 1 0 ) therefore cannot be confined to one of thecoordinates Q2 or '2 '2 (which exhibit zero angular mome ntum about the z axis).It is class ically des cribed by a linear combination of Q2 and Q ' 2 in which thenuclei follow close d trajectories in a plane n ormal to the z axis (Fig. 6 .13) . In the(y 1 y i2 y 3 ) level notation of Fig. 6 . 1 1 , the 1 quantum num ber is included as a

supe rscript to the number of quanta y 2 in the n u mode . The 1 specification issuperfluous if the vibrational state sym me try is known: levels with Cr, 7E , 6 ,



Q2

0 Q2

Figure 6.12 The doubly degenerate nu bending modes Q, and Q'2 in CO 2 . The

nuclear positions in CO 2 vibrations may be expressed in terms of the cylindrical

coordinates z, r, and 0.

\,, ,

,,47,, ..„...9. \ ,,4.-,, .,/igure 6.13 Nuclear trajectories in a CO 2 vibra-

tional state with nonzero vibrational angular

0 momentum quantum number I.

212



ROTATIONAL FINE STRUCTURE OF VIBRATIONAL BANDS 213

symme try exhibit 1 = 0, 1, 2, ... , respectively. Hence, the symm etric productrepresen tations we des cribed e arlier ( in co nne ct ion with obtaining the symme-tries of states w ith mult iply e xci ted de gene rate modes) embody the rules for

composi tion o f vibration al angular momenta . The foregoing discussion ha stac i tly assum ed that vibration—rotation coupling is negligible; un de r suchcoupling the vibrational angular mo me ntum may no longer be a con stant of themotion. V ibrational angular mom en ta also occ ur in degenerate vibrations ofnonlinear polyatomics, where the nuclear d i splaceme nts m ay trace ellipses,circles, or lines in a plane perpen dicular to the axis of highest symmetry [6 ] .

6.6 ROTATIONAL FINE STRUCTURE OF VIBRATIONAL BANDS

As in diatomics, vibrational transitions in polyatomic molecules are inevitablyaccompan ied by rotat ional fine st ructure . In linear molecules, the vibrationaland rotational selection rules in vibration—rotation spectra are closely analo-gous to the electronic and rotationa l se lection rules , res pectively, in diatomicelectronic band spectra. W hen applied to a p o o h molecule, the general symme tryargumen ts of the previous Section lead to the El se lect ion rules

Al= 0, +1+

4 — X — > a

g 4 — x — > g, u 4 — x — > u

fo r vibrational transitions in linear molecules. He re 1 is the vibrationa l angularmomentum quantum number. The se lection rule on A l i s rem iniscent of thecondition AA = 0 , ±1 fo r El-allowed electronic transitions in diatomics. (Theg/u labels are dropped in the case of C or transi t ions betweentwo a-type vibrational levels (1 = 0 0 ), the vibrational transition momentis polarized along the molecular axis , and the t rans i t ion is called a parallel

t ransi t ion. The rotat ional select ion rule in parallel vibrationa l transit ions isAJ = ± 1 ; that is , only the P and R branches o ccur. For transi t ions in whichA l =+1 rc 4.-.>5 , etc.), the transition mom ent lies perpendicular to themolecular axis, and the trans ition is termed a perpendicular transition. In such atransition, the Q branch also become s al lowed. T he frequencies of the P-, Q-,and R-branch lines in vibration— rotation spectra are given by formulasanalogous to E qs . 4 . 7 6 ; i) 0 r epresen t s the vibrat ional ene rgy change in thetransition. All three rotational branches appear in A l =0 vibrational bands with1 0 (e.g., TC4- r c transitions). These rotational selection rules are all identical tothose that apply to rotationa l fine structure in diatomic electronic transitions, if 1

is replaced by A in the discussion above (Section 4.6).

In symme tr i c top molecules, the ro ta tional se lect ion rules depen d on therelative orien tations of the figure axis (Chapter 5 ) and the vibrational trans ition




moment. When these are parallel, the El rotat ional select ion rules are [8 ]

A K = 0

AJ = 0, ±16.99)

when K 0 . Wh en K is zero, the transi t ion AJ =0 i s forbidden. W hen thet ransi t ion mom en t is perpendicular to the figure axis, on e obtains the contrastingselection rules [8]

AK= +1

AJ = 0, ±16 . 100 )

Analysis of the rotat ional fine structure in vibration—rotation spectra thus offers

potential fo r deducing the d i rec t ion o f the t rans i t ion m om en t (and thus thevibrational symmetry species) of a vibrational band. If the t ransi t ion mo me nthas com ponen ts parallel and normal to the figure axis, then both A K = 0 andA K = + 1 transit ions will be observed.

This variety in rotational select ion rules , coupled with our natural en dow-men t of molecules with diverse rotational cons tants, leads to wide variations inthe rotational fine structure exhibited by symmetric and near-symmetric tops.For definiteness , we consider a prolate symmetric top whose rotational energylevels are given in Eq. 5.26. Ro tational line s will be foun d at the frequencies

=o+ B' (J' + 1) + A ' — B ')1(` 2 — [B"J"(J" + 1) + 1) + A " — B" )K" 2 ](6.101)

where i 3 0 is the frequency of the pure vibrational transition and (A' , B'), (A" , B ")are the rotational cons tants of the upper and lower vibrational states. Accord ingto Eq. 6 . 1 0 1 , the rotational structure can be regarded as a superimposition ofse ts of diatomic P, Q, and R rotat ional branches (corresponding to AJ = —1, 0 ,and + 1 , respe ctively) cen tered a t origins with the K-dependent frequencies

=+A ' — B ')K' 2 — (A " —6.102)

For parallel bands (K' = K" K) the origin po sit ions are

= +(A ' — 1 3 ' ) — (A " — B") ]K 26.103)

whereas fo r perpendicular transitions (K' = K" + 1 K + 1) they become

=+(A ' — B ') — (A " — B" ) ]K 2 + 2 ( A ' — B ') K + A ' — B ' (6.104)

The initial and final rotational leve ls respon sible for a given transit ion may be

specified by writing P , Q , and R as a superscript to denote A K = — 1 , 0 , or + 1 ,and by s upplying the numerical value o f K" as a subscript. H ence the symbol



-15 051 . 1 t l L J 1 l J j ' j t

ROTATIONAL FINE STRUCTURE OF VIBRATIONAL BANDS 215

Figure 6.14 Rotational fine structure in a parallel vibrational band for a prolate

symmetric top in which (A ' -B') - A" - B") is small: A" = 5.28 cm -1 ,

A' = 5.26 cm -1 , and B" = B' = 0.307 cm -1 . The origin positions are closely spaced,and the spectrum resembles the vibration—rotation spectrum of a diatomic

molecule. Horizontal energy scale is in cm -1 .

P Q 2 (3) represents a 0 ,— 1 transition from J" = 3 , K" =2 to J' =3 ,

K' = 1. In a parallel transition, the origin positions (6.103) will frequentlydepend we akly on K, because the rotat ional constants are nearly the same in the

upper and lower vibrational states . In such a case the rotat ional structure willrese mble that in Fig. 6 .14 , which is reminiscent of the HC1 vibration—rotation

spectrum of Fig. 3.3. Since the positions of the Q(J) lines in a paralleltransition vary little with K and J when A' A" and B' B" (Eq. 6 .101) ,considerable intensity is conce ntrated n ear the frequency of the pure vibrationaltransition. Figure 6 .15 illustrates the rotational structure in a parallel transition

in which [ ( A ' — B ') — (A " — B ")] is appreciable; the origin positions are nowwell separated. In man y prolate tops ( e . g . , C H 3 C 1 ) , the rotat ional constant A

about the figure axis is muc h larger than B . The positions of the QK (J) linesthen tend to depen d s trongly on K, but weakly on J (Eq. 6 .101) , so tha t the

spectrum in Fig. 6 .15 i s dominated by a series of bunched Q-branch lines. In

perpendicu lar transitions, the subband origins de pend s trongly on K by virtue ofthe + 2 (A' — B ' )K term in Eq. 6 .104 . The rotational structure then exhibits dense

groups of P Q K an d R Q K l ines , because the position of any "QK (J) lines varies

little with J.



• l i111

216O L Y A T O M I C VIBRATIONS

-15 5Figure 6.15 . Rotational fine structure in a parallel vibrational band for a prolate

symmetric top in which (A' - B') - (A " - B") is substantial: A" = 5.28cm-1 ,

A' = 5.00cm -1 , and B" = B' = 0.307 cm -1 . The origin positions here become well

dispersed. Horizontal energy scale is in cm -1 .

These spectra serve to illustrate the sensi t ivity of rotational fine structure tothe transition moment orientations and rotational constants. In practice,individual rotational lines cannot be resolved in most infrared vibration-

rotation spectra, because the rotational constants are too small. In spectra such

as that in Fig. 6 .14, the bunched groups of Q-branch lines frequently materializeas single intense bands, while the more sparse P and R branches form weak

continua. Rotational structures are frequently analyzed by comparing themwith computer-generated spectra derived from assumed rotational constantsand selection rules. By weighting the rotational line intensities with appropriate

Boltzmann factors (cf. E q. 3.28) and assigning each rotational line a f requencywidth commensurate with the known instrument resolution, realistic simula-

tions of experimental spectra are possible if the rotational constants and

selection rules are properly adjusted.

6.7 BREAKDOWN OF THE NORMAL MODE APPROXIMATION

Our development of the normal mode description of polyatomic vibrations inSections 6 . 1 -6 . 4 r e s t ed on the assumption that the potential energy function

(6.4) is harmonic in the nuclear coordinates. As in diatomics, this assumption



BREAKDOWN OF THE NORMAL MODE APPROXIMATION 217

breaks down fo r sufficiently large vibrat ional ene rgies in real polyatomicmolecules. W hen several quanta of excitation are place d into a normal mode, itbegins to redistribute its energy to other normal modes. Such be havior is

guaranteed by the cubic and higher order term s in the vibrational potential,s ince their prese nce rules out the possibility of finding 3 N — 6 independentlyoscillating normal coordinates that obey the uncoupled differe ntial equations(6.49).

At high vibrational energies, there is co mpelling eviden ce [9 ] that the nuclearmotion canno t be e ven approximately des cribed in terms of normal coordinates .A case in point is the thermal dissociation of benzene, C 6 H 6 A C 6 H 5 + H ,where a hydrogen atom is create d by selective stretching of a single C-H bond.Such motion is incons is tent with the normal mode descriptions of the sixbenzene vibrations composed primarily of C-H stretches (Fig. 6.16), in which

the stretching amplitudes must be identical in at least two of the C-H bonds bysymmetry. The infrared-visible absorption spectrum of benze ne e xhibi ts aprominent series o f overtone bands at frequencies that closely obey the equation[ 1 0 ]

vx i + x 2 )v — x 2 v 2= 1, 2, ... , 86 .105 )

e l u 2 gFigure 6.16 Benzene normal modes dominated by C—H stretching motions. Since

there are six C—H bonds, there are six such modes: one each of a, 9 and 1 3 , .

symmetry, and two degenerate pairs of e l ande 29 symmetry.




with x 1 = 3153 cm'nd x 2 = 58 . 4 cm '. his overtone spectrum is strikinglysimilar to the vibrational s pectrum of the isolated CH radical, and has bee nassigned to an anharmonic local stretching mode confined to a single C— Hbond. The observed frequencies (6.105) are consistent with transitions from

y = 0 to y = 1, 2, ... , 8 in a one -dimen siona l oscillator subject to the quarticpotential

V (q ) = a o + a2 q 2 + a q 3 + a4 q 46 . 106)

If one treats th e anharmonici t ies a 3 q 3 and a 4 q 4 to second and f i rst orde rrespectively in a harmonic osc illator basis. H ere q is the C — H bond displacementcoordinate, and the expansion coefficients in the potential are related to th eovertone spectrum parameters by

X 1 = a 2

X2 = 6a4 — 30a/a 46.107)

Addit ional eviden ce for the validity of the local mode description invibrationally ex c i ted s ta tes i s furnished by over tone s pec tra obta ine d us inghighly sensitive thermal lensing spectroscopy techniques [ 1 1 ] in several otheraromatic hydrocarbons. T he sixth overtone band of the C—H stretching mode i s

foun d at virtually the same frequency in benzene (16 ,480 cm '), naphthalene(16 ,440 cm '), and an thrace n e (1 6 , 4 70 cm"). This same ne ss i s d i f f icult to

rationalize in a normal mode description, in which the nature of the parenthydrocarbon ske le ton i s e xpected to influence the allowed frequencies of thecollective n uclear motions. Furthermore, th e observed width of th e C— Hstretching vibrational band is th e same ( — 3 6 0 cm fo r both th e e lt , funda-mental in benzene (cf. Fig. 6 .16 ) and for the sixth overtone. I f one distributes s ixquanta among th e C—H stretching normal modes in Fig. 6 .16 , one obtains46 2 distinct levels. Using the symmetry classification techniques outlined inSect ion 6.4, i t may be shown that 1 50 of these will ex hibit th e overall a 2 „ ore h, vibrat ional level sy mm etry required in the D6h point group fo r observa-tion of an El overtone transition from the a i g ground s ta te . (T hese are 75

doubly degenerate states of e n , symme t ry . ) H ence o n e would ex pect anoticably broader vibrat iona l band in the s ixth overtone than in the funda-mental, in consequence of the far greater variety of El-accessible statesgenerated using six quanta, if the normal mode approximation were accurate atthese ene rgies . Such a picture is n ot supported by the spectroscopic e vidence[ 11 ] .

In the local mode t reatment [ 1 1 ] of the C— H stretching vibrations inbenzene, the six bonds oscillate independently with energies (cf. Eq. 6 .105 )

Evib(vi) = — 15 62 + 31 53(v + - 12 ) — 58 . 4 (vi + D 26 .108)



BREAKDOWN OF THE NORMAL MODE APPROXIMATION 219

The total vibrational energy residing in the C— H vibrations is

6

Evib = E Evib(vi)

6 .109)

i=1

and the corresponding vibrational states y 1 y 2 • • • y 6 > are products of o n e -dimensional anharmonic oscillator states, which are eigenfunct ions of a Hamil-tonian with the potential function (6.106). For a given total number

(6 .110)

of vibrational quanta, the possible vibrational states may be divided into classesof degenerate states. One such class is the nondegenera te class (1 1 1 1 1 1) , in

which each local mode contains one quantum. An example of a degenerate classis (4, 2), in which one local mode has four quanta and the other two are placedtogether in any of the other six. The degeneracy of this mode is 3 0 [ 1 1 ]. Theclasse s , energ i es , and degene rac i e s of all 462 C—H local mode states in benzeneare shown in Fig. 6 .17. The product vibrational wavefunctions within each class

Class E n e r g y Degeneracy

N o . of E t,

States

( 1 , 1 , 1 , 1 , 1 , 1 )(2,1,1,1,1)

18 .220 I

_

58.103 30

18.0 (2,2,1,1) 17.986 90 14

(2,2,2),(3,1,1j) 3,107.870 20,60

(3 ,2 ,1 ) 207.753 12 0

t o Q

17.5 (3,3), (4,1,1) 17.520 2,105 , 6 0

7(4,2) 17.403 50

> ,h _ (5,1) 57.053 30' 2 17.0

L 1 . 1

16.516.4696) 0 3

Figure 6.17 Classes , en e r g ie s , an d degeneracies of the C—H local mode vibrational

states in benzene. Reproduced with permission from R. L. Swofford, M. E. Long, andA. C. Albrecht, J. Chem. Phys. 6 5 : 187 (1976).




form a basis for irreducible representations of D6h; one may co nstruct linearcombinations of these functions transforming as a g , a 2 g , b 1 „, b 2 „, e 2 g , or e 1 . Inthis way, one f inds that there ex ist 1 50 l inea rly independ ent combinat ions ofanharmonic local mode states with e lu overall symm etry, as shown in Fig. 6.17.

(This counting of s tate symmetries if, of course, independent of whether thenormal or local mode formulation is used.) I t may be s hown [1 1 ] that in thelocal mode formulation, El transitions are possible only when on e of the v ichanges, and the rem aining v i are unaffecte d (i.e. , combination transitionsamong local mode states are forbidden). He nce, for v 6 the only e 1 u stateaccessible from the a 1 g ground state is the one belonging to the ( 6 ) class in whichall six quanta reside in one of the C— H bonds. This is why the absorption bandof the sixth overtone in benzene is no broader than that in the C — H stretching

fundamental. The whole question of whether vibrational motion in polyatomicsis more appropriately des cribed in the normal mode or local mode formulation

has fundamental implications for vibrational spectrosco py, intramolec ularvibrational redistribution (IVR), and dissociation. It is also important inradiationless relaxation processe s such as internal conversion and intersystemcrossing (Chapter 7).

Another manifestation of vibrational a nharmon icity occurs in F er m i re -sonance [8] . W hen two vibra t ional s ta tes of the same overall symm etry areaccidentally degenerate , they can becom e s trongly mixed by the anharmoniccoupling terms betwee n them. T heir ene rgies may be repelled conside rably (inthe language of degenerate perturbation theory), and the intensities of thespectroscopic transitions to thes e levels ma y be red istributed by the mixing.

REFERENCES

1. J. B . Marion, Classical Dynamics of Particles and Systems, Academic, New York ,1965 .

2 . E . D. Nering, Linear Algebra and Matrix Theory, Wiley , New York , 1963 ; F. R.Gantmacher , Th e Theory of Matrices, Chelsea, New York, 1960 .

3 . E . B . Wilson, J. C. Decius , and P. C. Cross , Molecular Vibrations , McGraw-Hill, Ne wYork, 1955.

4. M . Tinkham, Group Theory and Quantum Mechanics, McGraw-Hill, New York ,1964 .

5 . D. S . Schonland, Molecular Symmetry, V an No s t ran d , London, 1965 .

6 . G. W. King, Spectroscopy and Molecular Structure, Holt , Rinehart, & Winston, Ne wYork, 1964 .

7. C. D. H. Chisholm, Group Theoretical Techn iques i n Quan tum Chemistry, Academic ,London, 1976 .

8 . G. Herzberg, Molecular Spectra an d Molecular Structure, II. Infrared and RamanSpectra of Polyatomic Molecules, V an Nos t r and , Princeton, NJ, 1945 .

9. P. Avouris, W . M . Gelbart , and M . A. El-Sayed, Chem. Rev. 77: 793 (1977 ).



PROBLEMS21

1 0 . J. W. Ellis, Phys . Rev . 32: 906 (1928); 33 : 27 (1929) ; Trans. Faraday Soc. 25: 888 (1924).

11. R . L . Swofford, M . E . Long, an d A. C. Albrecht, J. Chem. Phys. 65: 179 (1976). Se ealso W . Siebrand, J . C h e m . Phys . 44: 4055 (1966 ); W . Siebrand and D. F. Williams, J.

C h em . Phys . 49: 1860 (1968) ; B. R . Henry and W . Siebrand, J. Chem. Phys. 49: 5 3 6 9(1968); R . Wallace, C h em . Phys. 1 1: 189 (1975) .

PROBLEMS

1. The force constants of the H-C and C-N bonds in l inear HCN are5 . 8 x 1 0 5 and 1 7 . 9 x 1 0 5 dyne/cm, respectively. Use the t reatment of the linearAB C molecule in Section 6 . 1 to predict the HCN stretching frequencies in cm -1 .Compare these with th e actual stretching frequencies, 2 0 6 2 and 3312 cm 1 , andcomment on the

validityof the

harmonicapproximation

tothe vibrational

potential.

2. The frequencies of the stretching fundamentals of linear CS 2 are 6 5 7 and1523 cm -1 . Calculate th e C-S bond force constant in two differen t ways. Are theresulting values consistent? W hy or why not? Which of these fundamentals is Elinfrared-active?

3 . Determine th e symme try species of the normal modes in SF 6 (O h ) , P4 (TAand C 5 H 5- (D5 h ) . Which of these normal modes have El-allowed fundamentals?

4 . Cons ider a planar, T -shaped mo lecule of C2, symmetry (C1F 3 has approx-imately this geome try). The pertinent coordinate systems are shown below.

(a) H ow many vibrational modes will involve on ly nuclear displaceme nts in

th e molecular plane, and according to what irreducible representat ionsmust they transform?



222 P O L Y A T O M I C VIBRATIONS

(b) W e may form the symm etry coordinates

S i = (r i + r 2 )/2

S2= r 3

S3 =

S4 = (r 1 — r2)/2

S5 =

from the internal coordinates r i , r 2 , r 3 , (/), and ( 5 . According to whichi r reducible representat ion does each of these transform? Assuming th epotential energy function

2V = 1 ( 1 (ii + ri) + k r i + k3 4) 2 + k46 2

determine the F matrix.(c) Obtain the G matrix for the in-plane vibrations, form the matrix F G, and

determine the value(s) of any vibrational frequen cies that can be obtainedwithout solving quadratic or higher order equations for A .

5. Consider a hypothetical square-planar A4 molecule of D 4 h symmetry.

(a ) H o w m a n y in-plane vibrational modes does A4 have, and what are theirsymme try species?

(b) As symm etry coordinates for the in-plane vibrations , we m ay take

S i = (r i + r 2 + r 3 + r4 )/4

S2 = (r i — r 2 + r 3 — r 4 )/4

S3 = (r 1 —

S4 = (r 2 --r4)/2

S5 =



PROBLEMS 223

where th e internal displaceme nt coo rdinates r 1 , r 2 , r 3 , r 4 , 4) are defined inthe accompanying figure ( 4 ) is ac tually the displacement of the bond anglefrom its equilibrium value of 90 0 ) . W hat i rreducible represe ntat ion of D4h

does each of these belong to ? Assuming th e potential

2V = k i (r? + r +r + ri) + k 2 4 ) 2

se t up the F matrix for the in-plane vibrations.( c ) Obtain the G and F • G matrices for the in-plane vibrations, and determine

th e A4 vibrational frequencie s in terms of the force constants, the equilib-rium bond length and the nuclear m a s s .



7ELECTRONIC

SPECTROSC OPY OFPOLYATOMIC MOLECULES

M any of the ideas that a re es se nt ial to understand ing polyatomic electronicspectra have already bee n deve loped in the three preceding chapters. As indiatomics, the Born-Oppenheimer separat ion between electronic and nuclearmotions is a use ful organizing principle for treating electronic transitions inpolyatomics. Vibrational band intensities in polyatomic electronic spectra arefrequently (but not always) governe d by Franck-Con don factors in the vibra-tional modes. The rotational fine structure in gas-phase electronic transitionsparallels that in polyatomic vibration—rotation spectra (Section 6.6), except thatthe rotational selection rules in symmetric and asymmetric tops now depend onthe relative orientations of the electronic transition moment and the principalaxes. Analyses of rotational contours in polyatomic band spec tra thus providevaluable clues about the symmetry and assignment of the electronic states

involved.Polyatomic band spectra s till abound in features that have no antecedents indiatomic spect ra . Polyatomic spectra are often far more conges ted (in the sensethat they e xhibit man y more vibrational bands per frequency interval), becauseth e number of vibrational modes scales with molecular s ize as 3N — 6 . Athermal gas sample of naphthalene (C 1 0 H 8 ) cannot be s electively pumped into asingle vibrational leve l in its lowes t excited singlet S l s tate at 30 0 K, because th erotational fine structure at this temperature me rges the closely space d vibra-tional bands into a barely resolved continuum. A qualitatively new phenomen-on ar i ses from the presence of nontotally symmetric modes in polyatomics. Such

modes can cause vibronic coupling betwee n e lec t ronic s ta tes belonging todif fe rent sym me try species , a l lowing e lec tronic transitions which wouldotherwise be E l-forbidde n to gain appreciable E l intensi ty. This coupling is

225



226LECTRONIC SPECTROSCOPY OF POLYATOMIC MOLECULES

respons ible for the "first allowed" electronic transition in SO 2 , the well-known26 00- A S i 4 - - S o band system in benzene, and the rich S 1 state photochemistry o fthe carbony l group in aldehydes and ketone s. Predictions of vibrational bandintensities in such transitions require quantitative theories o f vibronic coupling.In contrast, many other electronic band spectra arise from intr insically El-allowed transitions in which vibrational band intensities are straightforwardlygiven by products of Franck-Condon factors and squared e lectronic transitionmome nt s (c f. Eq. 4.51). Examples of these are the "second allowed" transition inSO 2 and the S i 4 — So spectrum of aniline, C 6 H 5 N H 2 .

In isolated polyatomic molecules of sufficient s ize, electronically excitedsta tes de cay nonradiatively and i rrevers ibly into states with lower e lectronicenergy. (Since such a process is nece ssarily i soenerget ic in an isolated molecule,th e electronic energy difference is converted into excess vibrational energy.) Suchspontaneous radiationless relaxation processes , unknown in collisionles s dia-

tomics, pervade th e photophysics of molecules with Z , 4 atoms. Their discoveryprompted fundame ntal questions about the nature of quantum mechanicalstat ionary states in molecules with d e n s e vibrational level structure, and theirinvestigation became one of the most active research areas in chemical physicsduring the 1 9 6 0 s and 1970s .

Discussions of polyatomic band spectra in a text o f this scope can cover onlya small fraction of the molecular types that have been explored in this vast field.W e begin by treating electronic transitions in triatomic molecules, which a re o finteres t to en vironm en tal scien tis ts (viz. NO2 , 03 ) and astrophysicists. T heelectronic band spectrum of SO 2 i s considered in detail and presents us with a

prototype example of vibronic coupling. We then de al with several aromatichydrocarbons: ani l ine , naphthalene, and benzene. These chemically similarmolec ules ex hibit sharply con trasting S i 4 - - S o spectra arising from transitionsf rom the ir ground s ta te s to the ir lowes t exci ted s ingle t s ta te s , and serve toilluminate th e sensitivity of band spect ra to sym metry , vibronic coupling, an dgeometry changes accompanying transitions. This chapter con cludes by de vel-oping quantitative theories for vibronic coupling and radiat ionless relaxation inpolyatomics.

7.1 TRIATOMIC MOLECULES

As in diatomics (Sect ion 4.3), th e molec ular orbitals in polyatomic moleculesmay be expressed as linear combinations of atomic orbitals (A0 s ) centered onthe nuclei. A minimal basis set of AO s contains all of the AO s that are occupiedin the separated a toms [1 , 2] . In the bent o z o n e molecule, fo r example, theseparated 0 atoms have the ground s tate configuration (1s) 2 (2s) 2 (2p)4 . T heminimal basis set for ground-state 0 3 therefore consis ts of the is, 2s , and three2p orbitals centered on each of the oxygen n uclei (Fig. 7.1). The orientationsselected for the 2p AO s in the basis set are of course arbitrary (aside from th e

con straint that basis AO s centered on any n ucleus must be l inearly independ-



2pi

2 P 1 1

2s

scy'ccoIs

A

Figure 7.1 Minimal basis set of atomic orbitals (A0s) in O

227



228 ELECTRONIC SPECTROSCOPY OF POLYATOMIC MOLECULES

ent); the particular orientations shown in Fig. 7.1 are simply one choice whichfacilitates the construction of MOs appropriate to the molecular symmetry.

The electronic states must transform as one of the irreducible represe ntationsF of the molecular point group, and so linear combinations of the AO s in Fig.

7.1 must be found which transform as thes e re presen tations. Such symmetry-adapted linear combinations (SALCs) may be ob tained us ing the projectionoperator technique [3]. Application of the projection operator

(7.1)

to any basis AO yields a linear combination of AO s transforming as the JR F,provided such a linear combination exists which contains the original AO. Herez r(R ) is the character in represen ta tion F of the class to which the group

operation A belongs, and the summation is carr ied out over all operations A inthe point group. B ent AB 2 molecules belong to the C2 v point group, whichcontains the operations E , C2, o in a plane perpend icular to th emolecular plane), and a', (reflection in the molecular plane). The projectionoperators for the a l and b 2 represen ta tions of C2 v are then

P(al) =2 +

13(b 2 ) = Ê — 0 2 — +

By applying these projection operators tothe is

AOs I ls A> and Ilk> in Fig. 7.1,

we obtain the unnormal ized SALCs

Aai)I lsA> = 2(11 sA> + l is s> )

Aai)lisc> 4110

P (b 2 ) 1 1 s A > = 2(11sA> — I lsB>)7.3)

Neglecting overlap between the is A0s, we then have the normalized SALCs

j o - i (a i p= (11s A > + I lsB > )/ N /2

1 0 - 2(ai)>sC >7.4)

1 0 - 02» = (i s A > — 1 sB>)/ N/

Application of the projection operators fi(a 2 ) or /3(b 1 ) to any of the is AO s yieldsa null res ult, and the l inear combinations P (a i )Ils B > , fl(b 2 ) 1 1 s 8 > , and 15 (b 2 )Ils c >all either van ish or reproduce the unnormalized SALCs (7 .3). Hence, only threel inearly independe nt SALCs are generated from the three is basis A0s, asexpected. In an L C A O — M O — S C F calculation, the two SALCs of a l symmetrywill mix to y ield the lowest two MOs of a l symmetry, I la i > and I2 a 1 > (Fig. 7.2).

•••

aV

(7.2)



TRIATOMIC MOLECULES 229

/)\

230a1

o - 1 (b2 )0/

I b 2

I a t

SALCsO s

Figure-7.2 Symmetry-adapted linear combinations (SALCs) and molecular or-

bitals (MOs) generated from the i s AOs in a bent AB2 molecule.

The extent of this mixing will be sm all in triatomics composed of atoms with

valence electrons in their 2 s or higher energy A0s, since the overlap betwee n theinne r-shell is AOs in these molecules will be insignificant . The rest of theminimal-basis SALCs can similarly be generated using the projection operatortechnique on the 2s and 2p A0s. Mixing between SALCs having similar energyand belonging to the same symmetry species then yields valence MOs with the

qualitative nodal patterns shown on the right side of Fig. 7.3.

Wh en the bond angle (/) in a bent AB 2 molecule is increas ed toward 1 80 0 ,

these nodal patterns must be preserved. As the molecule approaches l ine argeometry, the resulting orbital symmetries in D o, may be found by noting thatthe symme try operations C2, a e , and o- ', in C2,, correspond to the classes C'2 , h ,

and a„, respectively, in the linear point group. The correlat ions between AB 2orbital symmetries in C2 v and Do p h may then be worked out by requiring that thecharacters of the corresponding classes of opera t ions be iden t ica l in bothirreducible represe ntat ions . (Som e of the characters for a h , which are n otordinarily listed in D o, character tables, are + 1 (o - g± and o- g- ), — 2 (m g ) , — 1 (oand o ,;), and +2 (m e)) This yields the correlations

C 2 , E C 2 (C2) v (ah) av) pooh

a l

b 2

a l + 1 3 1

+ b 2

1

1

22

1

— 100

1

— 12

— 2

1

100

a +g+a.

itu

ag



C 0 2O 2

50-g

21 I u

3 cru0=0

4 0 -9)-0

2c •u 0

3 c r g 0-0-0

b 2

3a 1

9

230 ELECTRONIC SPECTROSCOPY O F P O L Y A T O M I C MOLECULES

Figure 7.3 Nodal patterns and qualitative energies of MOs in linear and bent

triatomic molecules according to Walsh [4]. The lowest three MOs, composed of 1sAOs on the constituent atoms (Fig. 7.2), participate little in the chemical bonding

and are excluded. Horizontal coordinate is the bending angle O . Orbital occupancies

are shown for linear CO2 =80 0 ) and for bent NO 2 = 134°). Irreducible

representations are given for the MOs in D c o t , and C2 , point groups at left and right,respectively.

and these are reflected in the orbital correlation diagram in Fig. 7.3. Note thatwhen a linear molecule becomes bent, its doubly degenerate nu orbitals in D c „ h

plit into pairs of nonde generate a l and b 1 orbitals in C2,.

Orbital corre la t ions a lone ca nno t ra t ional ize the electronic structure intriatomic molecules; one needs to know how the M O energies are ordered an dhow they are influenced by the bond angle 0 . In a remarkably prescient se ries o f

papers published in the early 195 0 s (long before accurate wave functions becameavailable for polyatomic molecules), W alsh [4] developed sem iempirical rules



TRIATOMIC MOLECULES31

fo r predic t ing geom etry e f fec t s on orbital ene rg i e s in smal l polyatomics. Walsh'srules fo r tr iatom ic AB 2 are incorporated in the correlation diagram in Fig. 7.3,where the orbital energies and bond angle are qualitatively plotted along the

vertical and horizontal axes, respectively. Most of the orbital ene rgi e s a r e s e e n toincrease slightly when linear AB 2 becomes bent. The 2 7 r „ orbitals in linear AB 2present a key anomaly: one of them correlates with the 6 a 1 orbital, whose ene rgyfalls violently as the bond angle is reduced.

The electron configurations in AB 2 may now be constructed using th eAufbau prescription of placing electrons in successive M O s according to th ePauli principle. The first s ix electrons go into the nonvalence orbitals 1 a 1 , 2 a 1 ,and 1 b 2 (Fig. 7.2). The remaining electrons are placed in the valence orbitalswhich are shown in the correlation diagram (Fig. 7.3). The C O 2 molecule, whichhas 16 valence electrons, exhibits the ground-s tate configuration

(3o-g ) 2 (20-„) 2 (4o- g ) 2 (3c7J 2 (1704 (17cd 4E g+

Since th e majority of the occupied orbital ene rg i e s a r e minimized at 4 ) = 1 8 0 0 ,C O 2 has linear geometry. In N O 2 , which has 1 7 valence electrons, the bentgeometry becomes favored because the "extra" electron goes into the 6 a 1 orbitalwhose energy drops sharply when the bond angle is decreased. The electronconfiguration of NO2 then becomes

( 3 a1 ) 2 ( 2 b2) 2 ( 4 a1 ) 2 ( 3 b2) 2 ( 1 b1 ) 2 ( 5 a1 ) 2 ( 1 a2 ) 2 ( 4 b2) 2 ( 6 a1 ) 1 A

In accordance with these predictions, ground-s tate t r iatomics with 1 6 or fewervalence electrons (C O 2 , CS 2 , OC S, N 2 0) are experimentally found to be linear,while those with more than 1 6 valence electrons (NO2, 03, SO 2 ) are bent.

The low-resolution absorption spectrum of SO 2 is shown in Fig. 7.4. Theintense band system lying between 1 9 0 0 and 2 3 0 0 A is called the "secondallowed" band. It exhibits a decadic molar absorption coeff ic ien t E ,̂ 3 0 0 0 Lmol cm' at maximum, which is characteristic of a strongly El-allowedelectronic transition in tr iatomics. (Absorption coefficients are de f ined in

Appendix D.) The less prominent "first allowed" band system between 2400 and3 4 0 0 A shows s values on the order of 3 0 0 L mol' cm'. In the "forbidden"band be tween 3 4 0 0 and 4 0 0 0 A, the absorption coefficients of

0 .1 L mo1 -1 cm -1 are typical of those found in spin-forbidden transitions.The valence structure in SO 2 is isoelectronic with that in 03 , and we will use theorbital nomenclature in Fig. 7.3 to discuss electronic transitions in SO 2 . In s odoing, we should bear in mind that the "6 a 1 " orbital in Fig. 7.3, for ex ample , isnot actually the sixth lowest-energy a l MO in SO 2 : This molecule (unlike 03 )

has additional inner-shell a l MOs arising from SALCs of 2s and 2 p A O scentered on the sulfur atom.

The ground state in SO 2 has the closed-shel l configuration

( 1 a2 ) 2(4 b 2 ) 2 (6 a 1 ) 2i




4000

3000

2000

to

1000

0

E X 5000

A k _4000

ANGSTROMS

Figure 7.4 Absorption spectrum of SO 2 gas from 1900 to 4000A. The strong

"second allowed" band system appears between 1900 and 2300A ; the "first

allowed" band occurs between 2400 and 3400A; and the very weak "forbidden

band" lies between 3400 and 4000A . Reproduced by permission from S. J. Strickler

and D. B . Howell, J. Chem. Phys. 4 9: 1948 (1968).

and is a totally symm etric singlet state. According to W alsh's rules (Fig. 7.3),some of the lowest-lying exc ited states in SO 2 should be

. ( 1 a2 ) 2 (4b2 ) 2 (6a1) 1 (2 1 31 ) 11 B i, 3/3 1

. ( 1 a2 ) 2 (4b2 ) 1 (6a1) 2 (2 1 31 ) 1 1A2, 3 A2

( 1 a2 ) 1 (4b2 ) 2 (6 a1 ) 2 (2 1 3 1 ) 1 1B 2, 313 2

Each of these open-shell configurations gives rise to a singlet and a triplet state;the triple t state in each configuration exhibits the lower energy due to Hund'srule. T he overall electronic state sym metries are given by the direct products ofirreducible representations for singly occupied MOs (e.g., b 2 b = A ). Thesecond allowed band between 1900 and 2300 A is due to the 'B 2 4 — 'A Itransition, which promote s an electron from the 1 a 2 orbital to the 2b 1 orbital.This transition is group-theoretically El-allowed (A l 0 B2 = B2) and y -polarized. This i s consis tent with an analysis of rotational fine s tructure in this

band system [5], which indicates that the electronic transit ion is polarized in themolecular plane. According to Fig. 7.3, this transit ion removes an electron froman ess en tially non bonding it-type orbital (1a 2 ) and places it into an antibondingir* orbital (2b 1 ). This should weake n the S-0 bond in the 1 B 2 state relative tothe 1 A 1 ground state , and thus endow the 1 B 2 state with longer bonds. T his is infact what happens : T he S-0 bond lengths in the 'A l and 'B 2 states are 1 . 4 3 2and 1.560 A , respectively. Furthermore, the la2 orbital ene rgetically favorslarger bond angles (its correlation curve minimizes at (/) = 1 8 0 ° ) , whereas the 2b 1orbital energy varies more weak ly with 4). Hence the 13 2 < — 1 A 1 transit ionshould produce an excited s tate with a sm aller equilibrium bond angle. This i s

borne out by the experimental bond an gles in the 'A l an d 1 B 2 s tates (119.5° an d



TRIATOMIC MOLECULES 233

104 . 3 ° , respectively). Ben t SO2 exhibits three vibrational modes: an a l s y m -metric stretch (y 1 ), an a l bend ing mode (y 2 ) and a b 2 asymmetric st re tch (y 3 ) .

Since the equilibrium bond angles are so different in the 'A l and 1 B 2 s tates , the

resulting displacement along normal coordinate Q2 between the minima in therespective potent ia l energy surfaces causes the Franck-Condon factors

1 <ei v'2 v' 1v7*;>1 2 to assume appreciable magnitudes for many values ofAv 2 = v'2 - v'2'. This is re sponsible for the progression of ne arly equally spacedvibrational bands which is o bserved on the long-wavelength side of the secondallowed system in Fig. 7.4. These arise principally from transitions from v'2 = 0in the 'A l state to v'2 = 0, 1, 2, 3, ... in the 'B 2 s ta te . Such progressions aregene rally asso ciated with significant changes in equilibrium geometry betweenthe upper and lower electronic states.

The ass ignment of the first allowed band system was controversial for many

years [5 ]. It is n ow kno wn that the low-energy portion of this s ystem (where thespectrum m aintains a nearly regular band spacing in a progression between

2800 and 3400 A in Fig. 7.4) is due to the 'A 2 +- 'A l transition. Since novector component t ransforms as A2 in C2 v , this transition is El symmetry-forbidden . It is obse rved an yway (though with lower intens ity than the secondallowed '1 3 2 4 - 'A l transition), due to a breakdown in the Born-Oppenheimer

approximation. We recall in Section 3.1 that motion in two electronic states Ilk>and lo k ) in a diatomic molecule may be coupled by a nuclear kine t ic e ne rgyterm propo rtion al to 0/410/0/0/4,>. Generalizing this t reatment to triatomicSO 2 , one finds that nuclear kine tic ene rgy coupling is possible between the I A 2

state and som e other ex cited s inglet s tate (which we provis ion ally call 1 B ),provided that

oB la/aQkliA2> o7.5)

Here Q k is one of the SO 2 vibrational mode coordinates . If Q k were one of the a l

normal modes, the '13 state wo uld have to be another 1 A 2 s ta te to r end er theintegrand in Eq. 7 .5 totally symm etric as required for a nonvanishing matrixelement. If mode Q k is the b 2 a symmet r ic stretching mode, however, the

integrand will trans form as F('B) 0 b2 0 A2. This

become s totally symme tric ifthe electronic state 'B ha s 'B 1 symmetry. As it happens, we have already shownthat Wa lsh's rules predict the existence of a low-lying 'B 1 exci ted state . Theclose ene rgy se paration between this 'B 1 state and the 'A 2 sta te then make s forsubstant ia l kine t ic ene rgy coupling be tween thes e s ta te s by the nontotallysymmetric b 2 vibrational mode. As a result, the 'A 2 state no longer has purely1 A 2 character , but contains an admixture of 'B, character as well. Since a

1 • 1H electronic transition is El sym me try-allowed (and polarized per-

pen dicular to the molecular plane), this 'B, admixture to the 1 A 2 state rendersthe otherwise forbidden 'A 2 4 - 'A l transition partly allowed. This coupling of

different electronic s tates by n uclear motion in nontotally symmetric modes isan example of vibronic coupling, a phenom enon widely observed in polyatomicband spectra.




In the W alsh picture, the 'A2 4-'A I t ransi t ion e xci tes an electron from the4 b 2 orbital to the 2 b 1 orbital (Fig. 7.3). Such a transition from a non bonding toan antibonding n* orbital should produce an increased bond length, as i s

observed (1.53 versus 1 .432 A ) . In analogy to the 'B 2 4 — 'A I t ransi t ion, it also

decreases the bond angle as predicted (from 119.5° to 99°). The question arises asto where the 'B, s ta te , f rom which the 'A l 4 — 'A i transition presumablyborrows i ts inten si ty via vibronic coupling, can be found in the absorptionspectrum. This 1 1 3 1 state i s not conspicuous in the SO 2 spectrum in Fig. 7.4; itmay co ntribute a weak continuum to the high-energy side ( — 2 4 0 0 -2 8 0 0 A ) ofthe first allowed sys tem.

T he forbidden band system ( 3 4 0 0 - 4 0 0 0 A) arises from a 3 1 3 1 4 — 1 A 1 spin-forbidden transit ion. (This 3 1 3 1 state is the tr iplet coun terpart to the afore-mentioned "B 1 state.) In this transition, an electron jumps from the 6 a 1 orbitalto the 2 b 1 orbital, orbitals that favor sm aller and larger bond angles, re-

spectively (Fig. 7.3). T hus W alsh's rules predict that SO 2 in the 3 B 1 state willexhibit a larger bond an gle than in the ground state, as indeed i t does (126.1° ascompared to 119.5 ' ) .

Investigators have searched in vain [5 ] for absorption bands due to the tworemaining low-lying triplet states ( 3 A 2 an d 3 1 3 2 ) whose existence is predicted byWalsh's ordering of orbital ene rgies . E lectronic structure calculations havepredicted that the 3 1 1 2 state should absorb in the region around 6 0 0 0 A , and thatthe 3 A2 state should produce a band sys tem at 3 4 0 0 - 3 9 0 0 A . Perturbations inthe 3 131 4-'A l spectrum have bee n at tr ibuted to the presence of a nearby 3 B 2state.

7.2 AROMATIC HYDROCARBONS

The lowest energy electronic transitions in homocyclic aromatic hydrocarbonsoccur at ne ar-ultraviolet , visible, and infrared wavelengths from 2 5 0 0 A out tobeyond 7 0 0 0 A . They involve excitat ions of electrons in delocalized 7r-type MOs,which are composed principally of carbon 2 p orbitals oriente d perpen dicular tothe aromatic plane . T he rem aining minimal-basis carbon valence orbitals (the 2sorbitals and the 2 p orbitals oriente d in the molecular plane) are utilized to form

in-plane ci -type MOs directed along the chemical bonds. E xcitations of e lectronsin c i-type M O s to unoccupied MOs require far higher photon energies (in thevacuum ultraviolet), and are not considered in this Section.

A well-studied aroma tic molecule is aniline, C 6 H 5 NH 2 , which differs frombenzene in that one of the six hydrogens has been replaced by the amino group—NH 2 . T o a first approximation , the itMOs may be forme d f rom l inea rcombinations of the six out-of-plane carbon 2 p A0s. In benzene, these M O swould have to t ransform as irreducible representations of the D 61 , point group(see below). It is reason able to ex pect that amino group substitution will perturbthe 7r-electron system in aniline, however, reducing its effective point groupsymm etry to C2,. (In fact, the plane containing the three atoms in the amino



AROMATIC HYDROCARBONS 235

4b,

202

2b 1

3b 1

I a 2

I bl

Figure 7.5 Nodal patterns and irreducible representations in C 2 , of orbitals in

aniline, C 6 1 - 1 5 N H 2 .

group is inclined from the aromatic plane by 39 0 in ground-s tate ani l ine , so thatth e environment experienced by th e 7r-electron system does not even show C2,

symmet ry . This point will be ignored in our treatment.) The nodal patterns forthe six lowest-energy n M O s are shown in Fig. 7.5. This diagram also gives thealignments of the x, y, and z axe s which are standard in discussions of exci ted-state symmetries in aniline. Since each carbon atom contributes one electron tothe n s ys t em, ground-state aniline will have the closed-shel l n-electron

configuration

( 1 b1 ) 2 (2b1) 2 (1a2) 2A 1




while the lowest four excited s tates should have the configurations

(1 b 1 ) 2 (2 b 1 ) 2 (1a2) 1 ( 2 a2 ) 1A1 , 3 A1

(1 b 1 ) 2 (2 b 1 ) 2 (1a2) ' ( 3 b1 ) 111 3 2, 3132

(1 b 1 ) 2 ( 2 b 1 ) 1 (1a2) 2 (2a2) 1 'fl1 2 , 3B 2

( 1 1 3 1 ) 2 ( 2 1 3 1 ) 1 ( 1 a2) 2 (3b1) 1 lA1, 3A 1

The s inglet groun d s tate S o is therefore totally sym me tric, while the lowestexc ited singlet S I is ei ther a 1 A 1 state or a 1 1 3 2 sta te. B oth types of excited statesare El-accessible from the 1 A 1 ground state: the 1 A 1 <– 1 A 1 and 1 B 2 1 1

transitions should be polarized along the z and y axes, respectively. Thissituation poses an intere sting contrast to that in benzene , which is considered atthe end of this section. U n d e r the D6h symmetry of benzene , some of theanalogous transitions (including the S I 4– S o transition)

prove to be El-forbidden.The S 1 4– S o fluorescence excitation spectrum (total fluorescence intensity as

a function of excitation wavelength) is s hown in Fig. 7.6 for aniline vapor atroom temperature. Also shown in Fig. 7.6 are several of the aniline vibrationalmodes (labeled in accordance wi th the Varsanyi [6] normal mode nom-enclature). The most prominen t bands in the spectrum are assigned using anotation that concise ly specifies the aniline vibrational le vels in the upper an dlower states. The 1 4 , band, for example, corresponds to a transition from an S o

- s t a te m o le c u l eero quanta of vibrational energy in the a l mode number 1 2 ,

to an S I -state molecule with one quantum in mode 1 2. The origin band (arisingfrom a transition between vibrationless S o and S I s tates) is de noted the 0 2 b and .The very fact that Fig. 7.6 shows a s trong 0 2 band— implying that the S I 4 - - S otransition occurs without succor from vibronic coupling through vibrationalexcitation of either the S I or S o state—means that the S l 4 — So electronictransition is intrinsically strongly El-allowed. H en ce the original D6h symmetryof the 7r-electron sys tem in benzene is significantly distorted by the presence ofamino group. In contrast to the allowed band systems in SO 2 (Fig. 7.4), noregular progressions of ne ar ly e qually spaced bands occur in the anilineexcitation spectrum. For that matter , the great m ajority of the aniline bands

arise from transitions in which no vibrational quantum number changes b ymore than 1 unit. Unlike the allowed transitions in S O 2 , then, the S < – S oaniline transition is not accompanied by large geometry changes along its totallysymmetric modes.

To de termine the symmetry of the S 1 state, Christofferson e t al. [7] analyzedthe rotational structure of several of the bands. The appearance of the 0 2

absorption band under high resolution is s hown in the top portion of Fig. 7.7. I n

a molecule as large as aniline, such a contour comprises some 3 0 , 0 0 0 rotationaltransitions, and so there is little hope for resolving individual line s. Ins tead, thecontour is compared with a computer simulation that calculates the asymmetricrotor ene rgy level differences , weights the intensities of allowe d rotational



i1 6 a ,

1 6 a 1,T ,I 1 1,6a°11t I

Ii6 a , T ,

1

2975

:IG

1 6 a,2T 2

2u , I6O:T 1 16 0 1 T i .

28258508759009259509754A

o o°

6aio

o

29004

a l

ØØ6o

02

-

0

•*)

I6o0 •

•

C )

-_

z

„ „

c

0

00la

1 ;

,.c .: :; . :

c 1 6 0

0 tD

1)

-

0

c

Z

0

0

c

e l l

b 2•

Figure 7.6 Fluorescence excitation spectra of aniline vapor: (a) 0.3-torr pressure,

0.32 A resolution; (b) 0.24-torr pressure, 0.08-A resolution. Also shown are severalnormal vibrations that are active in the fluorescence excitation spectra. Used withpermission from D. Chernoff and S. A. Rice, J. Chem. Phys. 70: 2521 (1979).

23 7



238 ELECTRONIC SPECTROSCOPY OF POLYATOMIC M O L E C U L E S

lal

1

lb )

, , ,, ,. . " ,„\A„

t

.C I

_

_

• . , Id

_

+10 1020

wavenumber (cm 1

Figure 7.7 Rotational fine structure of the o g absorption band in aniline vapor. The

experimental rotational contour is shown at top; the theoretical simulations labeled

(b), (c), and (d) were generated using identical rotational constants, but assumed< — S o transition mom ents polarized along the h, a, an d c principal rotational axes,

respectively. This work established that the S, <— S o transition in aniline is polarizedalong the b axis (Fig. 7.8), or equivalently, y-polarized (cf. Fig. 7.5). Reproduced bypermission from J. Christofferen, J. M. Ho Ilas, an d G. H. Kirby, Mol. Phys. 16, 4 4 1(1969).

transitions with the degeneracies and Boltzmann factors of the lower levels, and

superimposes the l ines (broadened to replicate the known e xperimental re-solution) to gene rate a theoretical spectrum. T he rotational constants A", B " , C "

of S o state aniline were already kno wn from m icrowave spectroscopy. T he S 1

s tate rotat ional constants A', B', C ' may be varied to optimize the theoreticalspectrum to fit to the experimental con tour. A crucial point here is that t h econtour shape is extremely sensitive to the rotational se lection rules, which i nturn hinge on the re lative orien tations of the S 1 4-- S o transition mom ent and theprincipal rotat ional axes a, b, and c (Fig. 7.8). W e have already see n in Se ction6 .6 ho w the rotat ional structure in vibration— rotation spectra is influenced bythe alignment of the vibrational t rans i t ion mom ent with respect to t h e

rotat ional axes . The three theoretical contours in Fig. 7 .8 were generated using

75

50

0

50

0

50

50



1B 3 u


a

Figure 7.8 Orientations of principal rotational axes in

aniline. The c axis is perpendicular to the paper.

ident ical upper state rotational constants, but assuming different directions forthe S 1 4 — So transition moments. It is clear that the observed contour is only

consistent with an electronic transition polarized along the b principal rota-tional axis , which is equivalent to th e y axis in Fig. 7.5. He n ce the S 1 s tate is a1 B 2 rather than 'A l state. This ex ample illustrates the power of rotat ionalcontour analysis fo r assigning electronic transitions.

Naphthalene, C 1 0 H 8 , is a planar m olecule belonging to the D2h point group.T en i t-type SALCs may be derived by application of the D2h projectionoperators to the out-of-plane carbon 2p orbitals; the qualitative nodal patternsof the resulting it MOs are shown in Fig. 7.9. W e use the Panser coordinatesystem, in which th e in-plane x and y axes are aligne d with the long and shortaxes in naphthalene; the z axi s i s perpe ndicular to th e molecular plane . Since

naphthalen e has ten e lectrons (one donated by e ach carbon) in it s it orbitals, itsground state S o will have the closed-shell configuration

(1 b 1„)1 b 2 g ) 2 (1 b 3 g ) 2 (la„) 2 (2 b 1 „) 2A g

The lowest few ex cited singlet states are then expected to be

. . . (1 a,,) 2 (2b .) 1 (2b 2 d i

• • • (1a„) 2 (2b 1 „) 1 (2b 3g) 1

• • (la)'(2biu) 2 (2 b2)1

The S l 4 — So transition to the lowest ex cited singlet state in napthalene is the'B 3 „ <— 'A g transition. Since the 'A g state is totally sym me tric and the vector xtransforms as B 3 „ in D 2 1„ this transition should presuma bly be El symmetry-allowed and polarized along the long axis. The ne xt higher spin-allowe dtransition (S 2 4 — S o ) i s a symmetry-allowed B- 2u 4-- ' A g transition. The vector ytransforms as B2 u , so this transition should be polarized along the short axis.The S2 4 — So transition is in fact respon sible for an intense electronic band

system in the near ultraviolet.In Fig. 7 . 1 0 , S I < — S o fluorescence excitation spectra are contrasted fo r room-temperature collisionless naphthalen e (at a pressure of — 5 x 1 0 - torr) and for



2blu

I b3g

I b2 g

I blu

3 b3 g

3 bi u

2 a u

2 b 39

2 b2g

Figure 7.9 Nodal patterns and irreducible representations in D 2h of n orbitals in

naphthalene, C 1 0 H 8 .

FLUORCENCNTENSITY

a o ) 1 4

\

ANgeta4

o d i

320002003600300

E X C I T A T I O N FR E QUE N C Y (cm')

Figure 7.10 Fluorescence excitation spectra of naphthalene: (a) in room-

temperature vapor at 5 x 10-5 torr; (b) in supersonic jet, 0.07-torr hydrocarbon in

4 atm helium. The 8(b 19 ), peak in the jet spectrum extends considerably offscale.

Used with permission from S. Behlen, D. McDonald, V . Sethuraman, and S. A. Rice,

J. Chem. Phys. 75 : 5685 (1981).

240




naphthalene in a supersonic jet. The "bulb" spectrum is broadened by rotationalprofiles at room tem perature. The supersonic jet spectrum differs radically fromthe bulb spectrum because the effect ive rotational and vibrationa l tempe raturesin the jet are less than — 1 0 K. The rotational cooling sharpen s the vibrationalbands, while the vibrational cooling rem oves hot bands caused by transitionsfrom vibrationally excited S o molecules. The jet spectrum is clearly more usefulfor making spectral assignments and for s tudying the S i state vibrationalstructure.

It can be shown using the group theo ret ical methods in Chapter 6 thatnaphthdlene possesses nine a g , eight b i g , three b 2 g , four b 3 g , four a n , four b i „,eight b 2 „, and eight b 3 „ normal modes. Each normal mode i s deno ted with anum erical prefix ind icating i ts rank in frequency among modes of the samesymmetry. For example, mode 8(b i g ) is the b i g mode having the eighth highest

( i . e . , the lowest) frequency am ong b i g modes. This notation is employed in thevibrational band assignments in Fig. 7.10 : The band 7(a g ) arises from a

transition between an S o molecule with zero vibrational quanta in mode 7(a g )and an S i molecule with one quantum in mode 7(a g ) . A distinguishing feature of

the naphthalene S i 4 — So spectrum is the weakness of the 02 transition betweenvibrationless S i and S o states (this transition is labeled "origin" in Fig. 7.10).Even though the S i 4 — S o transition is El symmetry-allowed in naphthalen e, i tso happens that the pertinent electronic transition moment M e (Section 4.4) isnume rically much smaller than expe cted for a st rong El transition. (In theoscillator s trength language which will be d eveloped in Chapter 8 , the S i < — S o

transition exhibits f 0 . 0 0 1 ; the strongly allowed S2 4— So transition thatoccu rs at higher en ergies e xhibits f — 1.0.) The presence of b i g vibrationalmodes allows vibronic coupling betwee n the 1 B 3 „ S i state and the nearby 1 B 2 uS2 state (since B 3 „ 0 b i g 0 B2u = A g ). The 1 B3 u SI state then gains some of the'B 2 „ character of the S2 state, and consequently borrows some of the intensity ofthe s trong S 2 < — So transition. This vibronic coupling ac counts for the 8(b 1 g ) , , ,

7(b i g ), and to some exten t the 8(b i g ) 1 , 8 ( a g ) (1) bands, all of which are an order ofmagnitude more intense than the origin band at 32,018.5 cm -1 . Since most ofthe intensity in these bands is due to the 1 B 2 n admixture into the 'B 3 n S i state ,these bands are polarized a long the short axis (rather than along the long axis as

would be e xpected for an intrins ically stron g 1 1 3 3 n 4- 1 A g 5 1 +-S 0 transition). Itwill be shown in the following section that the vibrational se lection rule in thesevibronically i nduced transitions is dv = ±1, ±3, ... in the normal moderesponsible for the vibronic coupling. This is why one observes transitions suchas 8 (b 1 g ), but not 8(b i g ) 6 .

The bands designated 9(a g ) 1) , 8(a g ) , !) , 7(a g ) 1) , and 3(a g ) 1 ; are also vibronicallyinduced bands, occasioned by vibronic coupling of the 1 1 3 3 n S i state and somehigher energy 1 B 3 „ S n state by the respective totally sy mm etric modes(B 3 „ 0 ag 0 B3u = A g ) . W hile this mixing o f S n into S i does n ot modify the B3 u

character in S i levels having excitation in a g modes, it will enlarge the S i < — S otransition moment if the S „ 4 — S o transition is more intense than the intrinsicS i 4— S0 transition in the absence of vibronic coupling. These bands in the



e 2 u

el g


SI 4- S o spectrum should exhibit the long-axis polarization of the intrinsicS n 4 - - - S o ('B 3 u 'A) transition. All of these band polarizations have beenconfirmed by obtaining absorption spectra of napthalene doped as an impurityinto host crystals, where the napthalene orientations relative to the crystal axesare independently known.

Another notable feature in the napthalene S I 4 - - S o spectrum is the lack ofprogression formation—the absence of intense bands like 8(b ig) O - (a g ) 4 withn> 1 . This indicates that large geometry changes do no t occur along totallysymmetry modes in the S l 4 - - S o transitions. In this respect, napthalene is s imilarto aniline and contrasts with SO 2 .

The benzene molecule C 6 H 6 exhibits D6h symmetry. The nodal patterns ofthe six it-type SA LC s formed from l ine ar combinat ions of out-of-plane carbon2 p AOs are shown in Fig. 7.11 . With s ix i t e lectrons, benze ne has the totally

a2u

Figure 7.11 Nodal patterns and irreducible representations in D et, of n orbitals inbenzene, C61-16.



BENZENE IN ;H i,

WAVELENGTH,

—

0


symmetric A ig ground-state configuration (a 2 u ) 2 (e lg ) 4 . Its lowe st lying ex citedstates arise from the configuration ( a 2 u ) 2 ( e i g ) 3 ( e 2 u ) 1 ; they have the overallelectronic symme tries e lg 0 e 2 „ = B 1 . C- B2u 10 E h , . T he intrinsic ( i .e . , no tvibronically induced) electronic transitions B

— 1. 4-- l

A lg and1 1 3 2 u 4 - -

iA lg aresymme try-forbidden. The E2u 4-

lA lg t ransi t ion i s El-allowed, and has beenassigned to an in tense (f —1 . 0 ) band sys tem at 1 8 5 0 A. T he lowest spin-allowedtransition S 1 4 — S o in benzene is associated with a much weake r sys t em(f —0 . 0 0 1 ) at 2 6 0 0 A ; its spectrum is shown in Fig. 7 . 12 . This must be either the

ABSORPTION FLUORESCENCE

1.0 101 - •

BENZENE VAPORENZENE VAPOR

0.5 - -5

o o1.0

II 11. 10 10

OPICAL DSTY

0.6

BENZENE IN C6 F,4 BENZENE IN C,F.

WAVELENGTH. ma

INTENT Arir Us

Figure 7.12 S i < — S o absorption spectra of benzene between 2300 an d 2700A in

the vapor, in C 6 F 14 , and in C 6 1 - 1 14 . Used with permission from C. W. Lawson, F.

Hirayama, and S. Lipsky, J. Chem. Phys. 51: 1595 (1961).




' 1 3 1 u 4 - - l A i g or the 1B2uA i g transition, with El intensity gained throughvibronic coupling.

T he 30 normal vibrations available in benzen e include two a i g , o n e a 2 g , twob 2 g , on e e i g , four e 2 g , one a 2 „, tw o b iu , two b 2 u , three e iu , and two e 2 „ modes .For a vibronically induced El transition from S o to S i , the latter state must be

coupled to either an E li, state (which transforms as x, y) or an A2u state (whichtransforms as z ) . In the latter case, the S i S absorption spectrum wouldexhibit parallel bands polarized along th e z axis. For S i states having B iu and

B2u symm etry, respectively, b 2 g and b i g modes would be required to effe ctcoupling to an A 2 „ state. Be nzene has two b 2 g modes, but no b i g m o d e s . Since noparallel bands have ever turned up in the S i 4 - - S o spectrum, th e S i state is a B2u

rather than a 1 B iu state.Perpendicular bands (polarized in the molecular plane) may appear d ue to

vibronic coupling of the 1 1 3 2 u state with the higher en ergy E iu s tate throughvibrations of e 2 g symmetry. The lowest frequency (605 cm') e 2 g mode, calledmode 6 in curren t spectrosco pic li terature , yields th e largest B2u 4—° 1 E1 u

vibronic coupling in benzene. (In the H erzberg-Te ller theory of vibroniccoupling d eveloped in the following sect ion , lower-frequency modes are predic-ted to yie ld stronger coupling.) Hen ce, the vibronically induced bands 6, 6 7 , 6,

_

I

60

61 1 1

00

6'1 200

•• •' in0 0

o6

1

01

6 1 1 0

026

110

036

11 0

0 n— • 6 1 1

0

261 612

S I

S O

1 2 6 1

1 2

1 1 6 16

2

61

0

62

6,

00

Figure 7.13 Energy level diagram for vibronically induced S, 4-- S o transitions in

benzene.



QUANTITATIVE THEORIES OF VIBRONIC COUPLING45

6 1 are prominent in the S 1 4 — S o spectrum; the last three are hot bands whichoccur in bulb spectra. The distinctively regular progres s ion in the benzenespectrum of Fig. 7.12 appears because the geometry change associated with the

S 1 4 — So transition ha s a large component a long the totally symmetric "brea-thing" mode (which is called mode 1 ) . Fo r this reaso n, one observes , in addit ionto the transitions 6 1 ) , 67 , 6 ? , 6 1 , the transitions 6 1 , 1 14 , 674, 61 4, 614 with n 1

(Fig. 7.13).

7.3 QUANTITATIVE THEORIES OF VIBRONIC COUPLING

In our qualitative discus sions of vibronic coupling in Sections 7.1 and 7.2, we didnot develop any method fo r pred icting vibrationa l band intensities or

vibrational selection rules in electronic band spectra in which vibronic couplingis important. Vibronic coupling ge ne rally arises from interactions betweennuclear and electronic motions [8]. The Herzberg-Teller (HT) theory of vibroniccoupling de als with the ramifications o f the vibrational coordinate depen den ceof the electronic t ransi tion mom ent; it thus examines one a spec t o f couplingbetween nuclear and electronic mot ions . T he B orn-Oppenheimer (BO) vibroniccoupling theory is con cern ed with interactions arising from the nuclear kineticene rgy operator T(Q) = — (h 2 /2 ) a2/aQ7 ; these interactions are analogous tothose described fo r diatomic molecules in Section 3.1. Earlier in this chapter, werationalized vibronic coupling in the 1 A 2 s tate of S O 2 in terms of such nuclear

kinet ic en ergy coupling. T he HT theory has accounted fo r vibronic bandintensity distributions to a first approximation in many organic molecules ; theB O theory has bee n invoked to improve on the quan titative predictions of H Ttheory.

Let the initial and final states in an El optical transition be represented by theBorn-Oppenheimer states

nv(q, ) X 7 ( 0 2)>7.6)

and

W..(q, I t f r n ( q , Q)X(Q)>7.7)

where q and Q represent the electronic and normal coordinates , 1 1 / / . > and I t / i n >

are the ini t ial and final electronic s tates , and le, > and bc> are the respe ctivevibrational states . In analogy to Eq. 4.49 fo r diatomics , the transi t ion mome ntintegral is given by

p..,„< znivr - l e y >

7.8)

where Mr" is the electronic t rans i tion m omen t

AK"n =7.9)




In the H erzberg-Te ller theory, Mrn (whose notat ion we n ow truncate to M „ , „ )

i s expanded in a T aylor ser ies in some normal coordinate Qk about itsequilibrium value Qk = 0 ,

(O Mm n

\Mnin(Q) = Mnin(0) + k

" This renders the E l t rans i tion mom en t equal to

/O M \Mmn0XXIvnbar>n841X7 1 42kba> ± • • •

0

(7.10)

(7.11)

M„,„(0) i s nonzero on ly if the electronic transition is intrinsica lly El-allowe d(as in the an iline S 1 < — S o sys tem or in the seco nd allowed s ystem of S O 2 ) . It issmall for the naphthalene S 1 4 — So system, and i t vanishes for the S 1 < — S o

t ransi t ion in an isolated benzene molecule. In the lat ter two molecules, intensevibrational bands can thus arise only from the first-order term (or in principlefrom higher-order te rms) in Eq. 7 .11 . H T theory reta ins on ly the first-order termin Eq. 7 .11 . IfQ k is a nontotally symmetric mode having the same frequency an dequilibrium value in both e lec tronic s ta te s m and n, the Franck-Condonamplitude < X 1 , m 1 X n . > vanishes by sy mme try unless the number of quanta in modeQk changes by A v = 0 , ±2, + 4, .... However , (Z n i IQkley> vanishes unlessA y = + 1, + 3, ... If Qk is totally symmetric, however, both terms in Eq. 7.11 mays imultaneously be n onzero, allowing interference s to occur between the zeroth-order (intrinsic) and first-order (vibronically induced) components of p .„ ,„„ , .

T he vibronically induce d compo ne nt, which is proportional to (aMinnial2k)o,controls th e vibrational band intensity according to HT theory when t h e

intrinsic electronic transition is forbidden . Since the electron ic dipole momentoperator x i depends on ly on the electronic coordinates (Eq. 4.47), we have

(7-n) 0 = okinipei K Z : Peilikn>

-P -lkm1PollfriXtkii aNf r ik) + (aa l f rQm > <O illieliOnd (7.12)

This leads to an explicit expression for the vibronically induced component o f

the t rans i t ion mome nt according to H erzberg-Te ller theory,

a 4n' v. =O f i l> — < 0 .1 i>>i<x ivnIQkIK>

( 7 . 1 3 )

T he first s e t o f terms on the right side of this expression may be physically

interpreted as follows. T he upper electronic state I//> in the trans ition n w < — m v



QUANTITATIVE THEORIES OF VIBRONIC COUPLING47

can be mixed by vibration in normal coordinate Q k with som e other electronicstate IC> to an extent that is proportional to <tk i 1 0 / 0 4 2 k I l k „ > . T he nw m y

transition moment, which would be zero in the absence of vibronic coupling if

the intrinsic transition is forbidden (M m „ = 0 ), can then borrow in tensi ty from thei m electronic transition provided that <0.ipe l i t i i i > i s nonze ro. Since aiaQ k

transforms as Q k unde r molecular point group operations, th e matrix elemen t< i i i i 10/0Q k l 0„ ) required fo r intensity-borrowing is nonvanishing only i f the d irectproduct F(C) C ) F(Q k ) /(0„) contains the totally sym me tric irreducible rep-resentation. This symm etry requirem en t i s ident ical to the one we used in ourmore qualitative discussions of vibronic coupling in S O 2 , naphthalene andbenzene. W e may interpret Eq. 7.13 less formally by s aying that when state 1 0 „ >

is vibronically coupled to so me other s ta te I vi i > , i t becom es re placed by th emixed state It/i n > + O k i > , where a is a small number proportional to

<C10/0Q k i t / i n > . The transition moment Prnn =< mX IP eiI'n X v> is then supersededin the H erzberg-Te ller picture by

=<0„,x'vnlpeilOixn

which says that if the intrinsic nw m y transition is forbidden

( < One,Weilika nw> = 0 ), it can still occur if the i < — m transition is allowe d(<IP.X71Poil 1 ' i X nw > 0 0) and if states I'l',> and IC > are vibronically coupled (a 0 0 ).

The second set of terms in the summation of Eq. 7.13 represents vibroniccoupling of the initial state I0 , n > with other states IC > through vibration in mode

Q k . The transition nw my then borrows intens i ty from the electronic trans-itions i < — n. In most situations of intere st, Ilk.> and I> are ground and excitedstates, respectively, in an absorptive transition. Other e lectronic s tates tend

to lie closer in ene rgy to I I I i n > than to 'O m > , and so workers have frequentlyassume d that the "intensity-lending" states I t k i > a re much more strongly coupledto the exci ted sta te Il/',> than to th e ground s ta te 10 , ,> . This approximation

(which ignores the second set of terms) has been challenged, however [9]. It maybe shown that

=tkilallolaQk10.>EaQ k 0i — Enili> (7.15)

where 1 1 0 is th e electronic H amiltonian and the Ei , E n are electronic-stateenergies. For a nontotally symm etric, harmonic mode with identical frequenciesin both electron ic states, we have the selection rule dy + 1 in the vibrationalintegral <Iry 4 2 k 1 X nw > . Use of second quantization then shows that

h )1/23N-6< X 7 1Q / c1K> (e,/[ 14, Wk] E oe7(wix(2;)>7.16)

*k

where y k and co are the reduced mass and frequency in mode k, and [v k , w k ] isthe larger of the two quantum n umbers o f vibrational states y and w in normal




coordinate Qk. Combining Eqs. 7 .1 5 and 7.16 yields the conventional expression

• HTP•mn,vw

i>la H 01 aQkil n >=

i*.r

n. —E

(2,

n

-04)1/2N — 6

X/[14, wit] H <x7 (2i)Ix(Q i ) >7 . 1 7 )j* k

for the vibronically induced transition moment in the H erzberg-Te ller theory.(The last factor in Eqs. 7.16 and 7.17 is simply a product of Franck-Condonamplitudes in all modes other than mode Q k . ) H erzberg-Te ller coupling appearsto acco unt fair ly we ll for the relative intensities of the vibronically inducedbands in the naphthalene S 1 < — S o spectrum. Equation 7.17 predicts thatvibronically induced band intens ities will vary as lko k with the frequency of thenontotally symmetric mode. This i s in accord with the observation that lowe r-frequency modes of appropriate symmetry tend to be more active in vibronicallyinduced spectra.

Vibronic coupling through the nuclear kinet ic e nergy operator t(Q) ratherthan through Q - d e p e n d e n c e in the electronic transition moment M. can betreated in a man ne r that paralle ls o ur discussion of the Born-Oppenheimerapproximation in diatomics (Section 3.1). The Born-Oppenheimer theory ofvibronic coupling pred icts that the induced transition moment will be

h c o k p,1 (n) w E

E i —

ko k + E EmLTIQkiK>7.18)

i —vk

If vibronic coupling of the lower electron ic state O .> to higher s tates 1 0 > i s

ignored (by se t ting < 0 . 1 a 1 a 4110i> = 0 ), comparison of Eqs. 7.13 and 7.18immediately shows that

pnmO vw

pH nT v w

= hcok /( E i — (7.19)

if the coupling is dom inated by one of the higher states I t k i > . The vibrationalspacing h c o k in mode Q k is frequently small compared to the energy gap (E i — E ,)between co upled electronic states. For example, I t i / i > and O n > can represent th eS 1 ( 1 1 3 3 .) and S 2 ( 1 B 2 ) states that are separated by — 38 0 0 cm -1 in naphthalene;they are vibronically coupled by b lg modes with hco k = 512 and 94 4 cm in thejet spectrum shown in Fig. 7 .10 . Hence , BO coupling has frequen tly bee nass umed to be insignificant relative to H T coupling, a presumption that hasbeen quest ioned by Orlandi and Siebrand [9]. I t is ne cess ary to invoke BO as

well as H T co upling to reproduce the details of the naphthalene S 1 4— So band



RADIATIONLESS RELAXATION IN ISOLATED POLYATOMICS 249

intensity distribution, because the intensity borrowing and lending sta tes (S 2

and S 1 ) are unusually close together in this molecule.W e finally comment on the vibrational selec t ion rules for vibronically

induced transitions. The intensityof a

vibrationalband

which occurs throughvibronic coupling in normal coordinate Q k is proport ional to I < Z 7 1 Q k i e w > 1 2 inboth the Herzberg-Teller and Born-Oppenheimer theories . W hen mode k i s

nontotally symmetric, the sym me try se lect ion rule in that mode will bedy + 1, ±3, ... if the equilibrium position is undisplaced along Q k in the

electronic transition. If mode k is also harmonic, with similar frequen cies in bothelectronic states , the more restrictive selection rule Av = + 1 applies. In such acase, the band intens i ty becom es propor tional to h[v k , wk ]/2,u k c o k times aproduct of Franck-Condon factors over all vibrational modes other than m odek according to Eq. 7 .1 6 . For this reason , one observes the vibronically induced

transitions N , 67 , 6? , 61 in the S I S spectrum of benzene , but not 61 , 6 6 , or 6.One similarly obtains the vibronically induced bands 8(b ig ) and 7(b 1 g ) (1 . , in the

naphthalene S 1 4-S o spectrum, but no t bands like 8(b 1 g ) 6 . In the first allowe dband progression of SO 2 (in which the upper state 1 A 2 is vibronically coupled toa higher '1 3 1 electronic state by asymm etric stretching mode 2, which has b 2

symmetry), the selection rules permit the transitions 1 7) 2 ( ) and 126 with variousn, but not 1 ' 4 2 S or l '426.

7.4 RADIATION LESS RELAXATION IN ISOLATED

POLYATOMICS

Wh en an isolated (collision-free) molecule is prepared in an excited vibronicleve l, its probable subsequent fa te depen ds fundame ntally on whether the

molecule is "small" or "large" (the criterion for "largeness" will be developed in

this section). If one excites the 1 3 1 1 1 „ state in y = 5 of Na 2 at sufficiently gas lowpressures, that pumped level will decay almost exclusively by emitting a

fluorescence photon—and one can be confident that essentially all of thefluorescence in a system of Na 2 molecules so excited will be e mitted specificallyby y = 5 in the 1 3 1 Il u state . However, a pumped vibrational level (say 6') in

collision-free S I benzen e (which is assuredly a "large" molecule in the context ofradiationless relaxation theory) has acces s to many decay routes that do notinvolve e mission of a photon. These radiationless decay routes include conver-sion of a 6' S 1 molecule into a vibrationally excited S o -state isoenergetic withthe 6 1 S state (internal conversion), and conversion into som e vibrational levelof T 1 (lowest triplet state) benze ne with a total ene rgy that closely matches thatof 6 1 S I benzen e ( intersys tem cross ing). Internal conversion and intersystemcrossing are generic terms for spin-allowed and spin-forbidden nonradiativeelectronic-state changes, respectively.

Since the molecule is isolated, these radiationless processes must be ene rgy-conse rving. Since they are energy-conserving, it might seem a priori that theyshould be re versible. For example, T 1 benzen e formed by intersystem crossing



250 ELECTRONIC SPECTROSCOPY O F POLYATOMIC MOLECULES

(ISC) from 6 1 S benze ne s hould be capable of reverting back to the initial state.The processes are in fact irreversible, even in isolated large molecules. (Innondilute gases an d in solution, these radia tionless de cay proces se s wouldappear irreve rsible in any case , because collisions would rapidly remove th eex cess vibrational ene rgy from the vibrationally hot S o or T 1 molecules formedin IC or ISC, rend ering them en erget ically incapable of recreating a 6 1 S 1molecule. The point we are making here i s tha t the energy-conserving non-radia t ive de cay processe s themse lves are irreversible in collision-free largemolecules.) Unlike diatomics, large molecules can thus spontaneously decaynon radia tively into e lec t ronic s ta tes other than the pumped s ta te , and emitluminescence from those states. 6 1 S 1 benzene can emit a fluorescence photon, ori t can und ergo ISC to som e vibrational level within th e T 1 manifold of levels(which may then phosphoresce), or i t may un de rgo IC to the S o state (Fig. 7 . 1 4 ) .

The peculiarities of large-molec ule photophysics and their contrasts to s mall-molecule behavior occupied the attention of a number of foremos t theore ticiansduring the 1 96 0 s and 1 970 s [ 1 0 ].

T he general problem of isolated-molecule nonradiative relaxation may bes tated as follows. A Born-Oppenheimer molecular state itp s > in electronic statemanifold A i s prepared in a molecule by photon absorption f rom som e lower

Fluorescence

Phosphorescence

Figure 7.14 Possible radiationless processes following creation of 6 1 So benzene,one of the vibronic levels which is El-accessible from ground-state benzene (cf. Fig.7.13). This level may undergo internal conversion (IC) to an isoenergetic, vibration-ally hot S o molecule, or it may undergo intersystem crossing (ISC) to an isoenergeticlevel in triplet state T.,. The T 1 S phosphorescence transition can be monitoredfor experimental evidence of ISC. Time-dependent S 1 -4 S o fluorescence decayfurnishes a probe for depopulation of S, through radiative (fluorescence) and

nonradiative (IC, I SC) decay.



RADIATIONLESS RELAXATION IN ISOLATED POLYATOMICS51

Born-Oppenheimer stateltk g > as shown in Fig. 7.15. Electronic state B possessesa manifold of Born-Oppenheimer states IC> with energies E. which span theene rgy region of the sta te IC> having energy E. One wishes to calculate the rate

of decay of the initially excited level lc> into the IC ) manifold due toperturbations like nonadiabatic and spin— orbit coupling, and also to determinethe conditions under which such decay will be irreversible. The Born-Oppenheimer s ta te s and Hamiltonian are analogous to those we used fordiatomics at the beginning of Chapter 3 . The total Hamiltonian

R = (q) + D(Q) + U(q, Q) + V(Q) + fl 07.20)

contains the electronic kinetic ene rgy, nuclear kinetic ene rgy, electronic potent-ial en ergy, nuclear repulsion, and spin—orbit operators, respectively, which are

functions of the electronic and/or nuclear coordinates q and Q. The reducedSchrödinger equation for electronic motion is

Ci"(q) + U(q, Q)]Itk i (q, Q)> = E 1 (Q)10 1 (q , Q)>7.21)

where the II', > are fixed-nuclei electronic wave functions and the E 1 (Q) arepotential energy surfaces for nuclear motion in the electronic s tates > . The

total wave function is the superimposition of Born-Oppenheimer states

=) > I X 1 ( 0 >7.22)

E

Figure 7.15 General problem for nonradiative decay of an excited Born-

Oppenheimer state with energy Es in electronic state A, prepared by photonexcitation of a level with energy Eg in the electronic ground state. The prepared state

s > is connected by perturbations (spin—orbit coupling, nonradiative coupling, etc.)to a set of Born—Oppenheimer states 1 0 > with energies E. in electronic state B. The

states IC> are not accessible by El transitions from the ground state.




and satisfies the total Schrbdihger equation

ROW, 42» = wItlf(q, $ 2 ) >

7.23)

This requires that

+ D (Q) + U(q , Q) + V (Q) + H . — W])x l (Q)> = 0

(7.24)

In analogy to what was don e in Section 3.1, we m ultiply this equa tion by <0.(q ,and use the facts that

<0 i l t (q ) + U (q , E n Ad

<0.1 1 1 (0101> = 1 7 ( Q ) (5 .17.25)

<0.1W101 > = W6 .1

to obtain the coupled equations

CD(Q) + E m (Q) + V ( 12) + <Cn I t (Q)0 .> + <0 . 1 1 s010.> — W ilX.(Q)>

--E < 0 . 1 1 V)1 0 > I x i ( Q

h 2 l b >2 —tik <0 .1 a Q k I t f r i > a Q k

_ — O f r . 1 h 1 s o l t f r i > l x i (Q ) > where U k is the reduced mass in normal coordinate Q 1 , and l x „ , > and Ix> are thevibrational wave fun ctions in Born-Oppenheimer states h l / m > and I I l ' , > . W h e nthe right sid e of this e quation vanishe s (i .e . , nonadiabatic and spin—orbitcoupling are negligible), the motion is confined to a single Born-Oppenheimerstate ( 1 0 „ , > in this ex ample). All of this parallels what we de mo ns trated fo rdiatomics in Section 3.1.

Returning to the general problem, the pumped BO s tate IC> may be coupledby terms like those on the right s ide of Eq. 7.26 with varying s trengths to a large

number of levels in the > manifold with irregularly spaced ene rgies E i . Sincethis gene ral problem is not tractable to analytic solution, Bixon and Jo r tn e r [ 1 1 ]studied an idealized s ystem in which the energies of the IC > manifold are equallyspaced,

E . = E, — a + is= 0, +1, +2, ...7.27)

+ E (7 .26)

where Es is the energy of BO state It/i s > and a, E are real constants. They also



RADIATIONLESS RELAXATION IN ISOLATED POLYATOMICS 2 53

made the simplifying ass umption that the coupling matrix eleme nt

<C1 1 : 1 1 0 i> = y7.28)

ha s the same value regardless of the final level i , so that all B O levels IC> haveidentical nonadiabatic couplings to II'>. The density of final states pi is 1/s , aconstant independent of E i , since the latter energies are equally spaced (Fig.7.16) . I t was also ass ume d that

o f r s f fli > =

= o

Mc> = E.f r g i f i lo7.29)

<tk ilill > = o

so that nonadiabatic coupling occurs only between i t k s > and the states in themanifold. In the presence of the perturbations ( i . e . , nonadiabatic and s p i n —

orbit coupling) which are in 'cluded in the total Hamiltonian fi, the B O statesIC > and the IC> will becom e m odified into the mixed s tates

I W .> = a n I t k s > +i>7.30)

which satisfy

fli t> = E. w n >

7.31)

It is important to differen t iate here betwee n the energies E s , E. (which are

e igenvalues of the B O Hamiltonian t(q)+ U(q,Q)) and the energies E „ (which

E .

E s

Figure 7.16 Simplified level scheme studied by Bixon

and Jortner. The levels E , in electronic state B are

equally separated with spacing c; the level E 0 is offset

from the photon-excited level Es by an arbitrary energya The coupling matrix element 0/ / 5 1 / 4 1 0 1 > = y is as-

sumed to be the same for all states IC>.



254 ELECTRONIC SPECTROSCOPY O F POLYATOMIC MOLECULES

are eigenvalues of the complete H amiltonianncluding nonadiabatic andspin-orbit co upling). The expansion coef f ic ien t s in Eq. 7.30 must satisfy

-

a

-

=E (7.32)b 7

H

Using the matrix eleme nts of nassigned by Bixon and Jor tner in Eqs. 7.28 and7.29, this is equivalent to sayin g that

a n a„

0

b 7=E

b 7(7.33)

where the only nonzero off-diagonal elements in the H amiltonian m atrix arethe Hs i . Expanding the matrix e quation (7.33) leads to

(E s - E n )a n b7 = 07.34)

(E i - E n ) b 7 + va n = 07.35)

From the second of these equat ions, we have

so that

-va nva n=E i - E„

=+ iE - E„(7.36)

co2 an( E , - E „ ) a „ = E7.37)

i= _ 0 3 Es - a + ic - E„

and

00

E s - E„ = -v2 E (E n - E s + a - is) -1 (7.38)

This is an equat ion tha t can be solved for the perturbed-state en ergies E n interms of E s , the coupling y, and the constants a and e . It can be shown [ 1 1 ] thatEq. 7.38 is ma thematically e quivalen t to the stateme nt that

E s - E„ = —

1 1 1 2Cot

[(

-r)E „ E s + o c ) ]E (7.39)



RADIATIONLESS RELAXATION IN ISOLATED POLYATOMICS55

which may be solved graphically for the perturbed ene rgies E n (Fig. 7.17). It isclear from inspection of the graph that there will be one new eigenvalue E n of the

perturbed H amiltonian between each pair of B O eigenvalues E. (Fig. 7.18). For

large I i i (E, « or »E s ) En will diffe r little fro m E i , so that the E „ are little shiftedfrom the E. for sta tes W O which are far off-resonance. Normalization of the

perturbed states 1 1 11 „> in Eq. 7 .30 requires that

CO

or

oo a„= a n2 ± v 2 7ao (Es — a + is — E „) 2(7 .40)

1a n2 = (1 + v 2 E

_ (E s — + is — E „ ) 2 ) - 1

V2

(7 .41)

(E. — E 5 ) 21,2 (nv2y

—(2c + a)(c + a)a —a€— a€ —

E„-E,

Figure 7.17 Graphical solution of the equationE - E n ) = ( r 1 1 , 2 I e )x cot[(rr e)(E - E5 + a)]. The eigenvalues are given by intersections of the straightline with th e periodic cotangent function.




E i

E 3

E l

Es_

E 0

E-2

Figure 7.18 Eigenvalues Es andE . , +1, ±2,...) of the Born-Oppenheimer

Hamiltonian, solid lines; eigenvalues E (n = 0, ±1, ±2,...) of the perturbed

Hamiltonian, dashed lines. The latter eigenvalues are obtained from the graphical

solutions in Fig. 7.17. Note the general property that there is one perturbed

eigenvalue between every pair of Born-Oppenheimer levels.

This expression gives the admixtures a n2 of the original BO state lc> in thevarious mixe d states io n > which have energies E. Because of the (E„ — E s)2term

in the denominator of Eq. 7 . 4 1 , these ad mixtures will only be appreciable ne arresonance.W e are now prepared to examine what happens whe n a molecule is e xcited

with a short light pulse. W e assume that the transition from t k g > to i t i / s > i s El-allowed, but that transitions from 1 1 / / g > to the states jt/i> are forbidden, i.e. ,

< 0 l o b P g > o

i l p 1 0 > =

7 . 4 2 )

This will be the case, for exam ple , when 1 0 s > is an ex cited s inglet s tatevibrationa l leve l that is E l-con ne cted to the ground s tate , and when the i t f r i > are



RADIATIONLESS R E L A X A T I O N IN ISOLATED POLYATOMICS57

vibrational levels be longing to one of the t r iple t electronic states. Excitation by aphoton of appropriate en ergy will then mo me ntarily produce the pure BO state

> , which may be ex panded in terms of the mixed s tates as

ItGs> = E ion>< onios> Eanlon>7.43)

at the time t = 0 when excitation occurs. At later t ime s this prepared state willevolve as

1 0 ( 0 > = E a n e - i E . t r n i o n >

= E a n e —t/h ( an i t ps\ Eb)

n (7.44)

The probability of finding the molecule in BO state l iP s > after excitation will be

Kti/s10(01 2s(t)2

E oksiane -œ.trnianiks + E b 7 t P i >

E a n2e -E„t/h2

V2

e - E„t/h2

(7.45)(E n — E ) 2 ± V 2 +

(v2)2

This is difficult to e valuate a priori, because the mixed s tate ene rgies ,E n are n otgiven by an alytic expressions (Fig. 7.17). W e therefore coarsen our approxi-mation by ass uming that they are given by

Es ± nc= 0 , + 1 , ±2, ...7.46)

(This is no t totally unreaso nable, because there will be an E „ level between e verypair of E . levels, will and so th e two se t s of levels have similar average spacings inFig. 7.18). Letting

=

we have the t ime-de pend en t probability

Ps (t) =

(v2 )2

v 2 +

2

(7.47)

(7.48)2e

i n E t t h

Zr

22A 2n nE




that th e molecule will be found in state I t f r s > after excitation. In the limits y » E

(o r vp i » 1 ) and t « hie , this sum may be replaced by th e integral [ 1 0 ]

P (t) =

fc°

_ 0 9 dn2n 2 e 22 C O s(entl h)

k= e

-2n2t ich e

2

( 7 . 4 9 )

Hence when y » e and t « hle, th e exci t ed B O state IC> undergoes irreversiblefirst-order decay with a rate constant

kNR = 2ny 2 /e h ( 7 . 5 0 )

which is proportional to the square of the nonadiabatic coupling matrix elementand to the density of f inal states pi =f, however , we drop the assumptions

» e and t « hle, th e sum fo r P5 (t) in E q. 7.48 must be evaluated explicitly. Thiswas done by Gelbart e t al. for several model systems [10] , and we show theirresults fo r y = 0 .5e in Fig. 7.19. P 5 (t) initially decays exponentially, becoming

I.0

08

0.6

0.4

0.02 —

0.2r e c

00104680TIME (nonosec.)

Figure 7.19 The time-dependent decay function Ps (t) evaluated using Eq. 7,48

with y = 0.5e . Ps (t) exhibits exponential decay (manifested by a straight line in this

semilog plot) at early times; a partial recurrence occurs at t 2rrhle T r ej,The

energy spacing e was chosen to render the exponential lifetime r = 1/kNri between 1

and 2 ns for realism. Reproduced by permission from P. Avouris, W. M. Gelbart , and

M. A. El -Sayed, Chem. Rev. 77 : 793 (1977).



RADIATIONLESS RELAXATION IN ISOLATED POLYATOMICS 259

very small for t »11kN R . However, it then begins to build up again to a large

value at the recurrence time T r e e 2nhle, s ince Eq. 7 .48 contains a superimposi-

tion of periodic terms proportional to exp( ine t1h) . In real molecules, the

population of pumped state IC> is depleted by spontaneous emission (flu-ores cence , phosphorescence) long before this recurrence time is reached. Forexample, we can consider the radiat ionless decay of vibrationless S 1 benzeneinto T 1 benzene (ISC). The density of T 1 vibrational states i soenerget ic with the

0 0 S 1 state is p — 3 x 10 5 cm (i.e., 3 x 10 5 levels per cm -1 ) in benzene, and thisnumber can be identified with 1 / c in our discussion. The correspondingrecurrence time is then Tree = 27thle — 1 0 -5 s, which should be compared withthe — 1 0 _ 8 s fluorescence lifetime in benzene . The radiative d e c a y t imescale in

benzene preempts that of recurrence by several orders of magnitude.One of the nominal criteria for the validity of the integral approximation in

E q. 7.49 was y » e . The foregoing discussion shows that this is too strict, s incethe calculations of Gelbart et al. prove that irreversible decay can be obtainedfor y0 .5c . A more realistic criterion is v ie 1 , or vp 1 in molecules with anenergy-dependent density of final s ta tes .

The problem of estimating densities of vibrational states p is a large one thatwe will only touch on here . For single harmonic oscillator with uniform energyspacing hv, p is of course l/hv. The number of ways of placing n vibrational

quanta in q identical oscillators (total ene rgy E = nhv) is [12]

W ( E ) = ( n + q — 1 ) !1 ( q — 1 ) ! n !7.51)

The total number of states in such a system with vibrational energies between 0and E is

n— 1 ) ! (n + q )!E

i=c) i4q-1)!!q !(7.52)

and the density of states p ( E ) at vibrational energy E is obtained from

p(E) = dG(E)1dE7.53)

For a s ys t em of three such oscillators, the following table desc r ibe s the behaviorof G ( E ) with increasing n = Elhv:

G ( E )

0 1

1 4

2 1 03 204 35




G(E) and p(E) can becom e incredibly large at mode rate vibrational ene rgies (e.g.,5 0 0 0 cm -1 ) in molecules of respectable size (benze ne has 30 vibrational modes) .Approximations are then required to evaluate them, particularly when thenormal modes have a range of frequencies [13] .

The criterion that vp I now implies that a critical numbe r of vibrations i s

required to make an irreversible radiat ionless relaxation process possible. W emay cons ide r a hypothetical case in which th e ene rgy gap be tween thevibrationless electronic states is 1 e V (8066 cm -1 ) ; that is , state l i k s > decaysinto a se t of f inal states I il'> which have — 1 e V of exce ss vibrational energy. ForIC and ISC, typical values of the coupling v may be take n to be — 1 0 -1 and10 -4 cm -1 , respectively [11 ] . A table of products vp calculated by Bixon an dJortner fo r nonlinear molecules with N atoms in which all (3N — 6 ) vibrationalm o d e s oscillate with frequenc y 1 0 0 0 cm -1 is shown below; th e densities of statesp were evaluated according to the method of H aarhoff [1 3] :

N p (E = 1 eV ) vp (IC) v p (SC)

3 0 . 0 6 cm 6 x 10 -3 6 x 1 0 -64 4 0 .4 4 x 10 -45 50 5 5 x 10'

1 0 4 x 10 5 4 x 104 40

He nce , in te rnal conversion is typically expec ted to o ccur in molecules with 4

atoms, and intersystem crossing set s in when N 1 0 . These are rough guidelines

for the "large-molecule" regime in which nonradiative relaxation is prevalent inisolated molecules. It includes all aromatic molecules (the smallest common oneof which is ben zene ); formaldehyde and larger molecules with the carbonylchromophore; and all laser dyes such as rhodamines , oxazines and commarins(Chapter 9 ) .

REFERENCES

1 . H . F. Schaefe r III , The Electronic Structure of Atoms and Molecules: A Survey of

Rigorous Quantum Mechanical Results, Addison-Wesley, Reading, MA, 1972.2 . W . H . Flygare, Molecular Structure and Dynamics, Prentice-Hall, Englewoo d Cl iff s ,

NJ, 1978.

3 . D. S. Schonland , Molecular Symmetry, Van Nos t r and , London, 1965 .4 . A. D. Walsh, J. Chem. Soc., 2260 , 2266 , 2288 , 2296 , 2301 , 2306 , 2318 , 2321 , 2325 , 2330

(1953).

5 . J. Heicklen , N. Kelly, and K. Partymiller, Rev. Chem. I n t e r m e d . 3 : 315 (1980) .6 . G . Varsanyi , Assignments for Vibrational Spectra of Seven Hundred Benzene

Derivatives, Wiley , New York, 1974.7 . J. Chris toffersen, J. M. bilas, and G. H. Kirby, Mol. Phys. 16, 441 (196 9).8 . G . Fischer, Vibronic Coupling, Academic, London, 1984.



1 4 1 za4 31 4303

006

A

004

00

330 340 350 390

H E COVO

0.10

244 4 1 2 54 /44 1

244 3

24 e2 343

234344 3343

23 5 1 +2g sT+ 2 , e+1 1 2 2 4 32 4. I 1 4

2%

0 2 . 14 3

\A A2T09090001020

X (nm)

PROBLEMS , 261

9 . G. Or l and i and W. Siebrand, J. Chem. Phys. 58 : 4513 (1973) .10. P. Avouris, W. M. Gelbart , and M. A. El-Sayed, Chem. Rev. 77 : 793 (1977 ).11. M. Bixon and J. Jor tner , J. Chem. Phys . 4 8 : 715 (196 8).

12. W. Forst, Chem. Rev. 71 : 339 (19 71) .13. P. C. Haarhoff , Mol. Phys. 7: 10 1 (1963).

PROBLEMS

1 . T he S 1 4 — S o band system of formaldehyde (CH 2 0) has bee n thoroughlystudied, and it has bee n e stablished that the vibrationless S I and S o states have'A 2 and 1 A 1 symmetry, respectively, in C2 v . T he S 4-- S o absorption spectrumand des cr ipt ions of no rmal vibra tions in the SFigure P7 .1 and in the list below.

an d S o s ta te s are shown in

Normal mode Symmetry l' A A l A 2

v 1 C—H symme tric s tretch a l 2 7 6 6 . 4 2 8 4 7v 2 C=0 stretch a l 1 7 4 6 . 1 1 1 7 3v 3 H—C—H bend a l 1 5 0 0 . 6 1 2 9 0

v 4 O ut-of-plane wag 1 3 1 1 1 6 7 . 3 1 2 4 . 6V 5 C—H asymme tric s tretch b 2 2 8 4 3 . 4 2 9 6 8v 6 In-plane wag b 2 1 2 5 1 . 2 9 0 4

(a ) Is the intrinsic S l 4-- S o t ransi t ion El-allowed in formaldehyde?(b) Assuming that the exhibited bands gain in tens i ty by H erzberg-T el ler

coupling between S I an d higher ex cited singlets S n , what are the symmetriesof the electronic states S n ? W hat polarizat ions do these bands e xhibit?

(c ) W hat information about the re lative geometries of the S I an d S o s tates canbe inferred from the progressions in this spe ctrum?

Figure P7.1 Reproduced with permission from E . K. C. Lee, Adv. Photochem. 12:

18 (1980).



0-0

262 ELECTRONIC SPECTROSCOPY OF P O L Y A T O M I C MOLECULES

2 . In s-triazine (a D3h molecule), the three lowest ex ci ted s inglet states arepredicted to be closely spaced in ene rgy with symme tries 'A' 1 " , and E". Ans-triazine crystal absorption spe ctrum (taken with unpolarized light) is shown inFigure P7.2. The weak 0-0 band is El symmetry-forbidden, and appearsbecause the D3h symmetry of s-triazine is slightly dis torted by the crystalenvironment. From analysis of the polarized crystal absorption spectra, thefollowing fundam en tals are found in the Si 4— S o spectrum:

Normal mode Symmetry Polarization

6 e ' 'Iz)4 a 2 (x, y)5 a' 1_ (x, y)

1 0 e" 1 (x , y)1 6 e" 1 (x , y)

(a ) Assuming that these fundamen tals gain intensity through Herzberg-Tellercoupling, deduce the symmetry of the lowest excited s inglet state. Show thatthis choice is consistent with all pertinent data given in this problem.

(b) Mode 1 2 in s-triazine ha symme try. By what mechanism (other thanen vironme nt sym me try-breaking) can the 1 2 (1 ) band appear in this crystalabsorption spectrum?

(c ) W hat symmetries of electronically ex cited s tates are vibronically coupled tothe S l state in the 6 1 , 4 1 , 5 1 , 10% and 1 6 1 vibrational levels? Consideringthis, why doe s the 6 1 , band ex hibit such large intensity?

F

300140500

1000

(cm - ')

Figure P7.2 Reproduced with permission from N . J. Kruse and G. J. Small, J .

C he m . Phys. 56 : 2987 (1972).



0

b -POLARIZATION

r-• z t .r-

o eoN-o )ltO ) . -oo W C VP 'O\ /O0re )o0 6 7 )

O )0

10

C'- POLARIZATION

nrs .

1jJ

TRANSMIT

NTENSIT

l o o

10 0

O

PROBLEMS 263

3. Polarized S 1 < — S o absorption spectra of tetracene in a transparent orderedhost crystal at 4 . 2 K are shown in Figure P7 .3 . The upper spec trum was obtainedusing light polarized along the tetracene y axis, and the lower s pectrum was

obtaine d using z-polarized light. The y and z axes lie in the molecular planealong the short and long axes of tetracene, respectively. T he tetracene S o s tateha s l A g symmetry in D2 h .

(a ) Most of the assigned bands in the y-polarized spectrum ar i se fromfundamentals and combinations in a g modes: the 308-cm" band is 12 ( a g ) ( 1j,the 308 + 609-cm ' band is 12 ( a g )4 1 1 ( a g ) 1 ) , etc. W hat is the symmetry ofthe electronic state S 1 ?

(b ) The 479-, 1166-, and 1 5 0 6 - c m -1 bands in the z-polarized spectrum arefundame ntals arising from vibronic coupling betwee n S 1 and a higher lying

1 1 3 1 u state. W hat is th e symmetry of the normal modes responsible fo r thisvibronic coupling?(c ) Use H erzberg-Te ller theory to develop an algebraic expression for the ratio

of intensities for the 4 7 9 - and 1166-cm" bands in the z-polarized spectrum.What structural information about tetracene would be ne eded to test thetheory against these data?

2050010001500

v (c m -I)

Figure P7.3 Reproduced wi th permission from G. Fischer and G. J. Small, J.

C h e m . Phys. 56: 5937 (1972).




4. One way to evaluate energy-dependent vibrational densities of states p(E) i s

to invert the classical vibrational partition function

Q =(E )e - E i k T(E ) -E s dE = Q (s)

The desired density of states is the inverse Laplace transform of Q(s),

P(E ) =1 [Q(s)]

For a harmonic oscillator, the classical partition function is

Q(s) = lim [1/(1 — exp( — hylk T))] = kT Ihv = 1 1 shyT— s co

and for y independent oscillators with frequencies y i it is

Q(s) = (kT)v I1 1 h y i =111 i

(a) Show that the classical vibrational density of states in a molecule with y

normal modes is

p ( E) =

EV-

( y — 1 ) !v(b ) Obtain numerical values of p(E) at E =8000 cm' for CO2 (whosevibrational frequencies are given in Section 6 .5 ) and for e t hylene , whos e 1 2nondegenerate vibrational frequencies are 8 2 5 , 9 4 3 , 9 5 0 , 9 9 5 , 1 0 5 0 , 1 3 4 2 ,1 4 4 3 , 1 6 2 3 , 2 9 9 0 , 3 0 1 9 , 3 1 0 6 , and 3 2 7 2 cm -1 . What circumstances lead tolarge dens i t i e s of states for given E ?

(c ) The classical density of states is accurate only when E is large compared tothe zero-point vibrational energy E 0 = - 1 - h E y i . A more accurate expression

for p(E) at lower energies is the semiclassical expression

p(E) =(E + Eo) v

(y — 1)! nhv 1

which results from replacing E in the classical density of states with(E + E0 ). Reevaluate p(E) for CO 2 and ethylene at 8000 cm -1 using thesemiclassical density of s ta tes . At what energies do the tw o approximations

agree to within 10% in these two molecules?



P R O B L E M S 265

5 . In the isolated benze ne mo lecule, the 3 B 1 u T 1 s tate lies 9 2 0 0 cm -1 below the

1 B2 u S1 state. Assume that T 1 benzene has the following vibrational frequenciesin cm:

3 0 6 0 a ig 85 0 eig

990 a ig 3 0 0 0 elu1 2 0 0 a 2 g 1 5 0 0 elu

6 5 0 a 2 u 1 0 5 0 elu3 0 5 0 I)" 3 0 0 0 e2g

1 0 0 0 b li , 1 5 0 0 e2 g

1 5 0 0 b 2 g 1 2 0 0 e2 g

550 b 2 g 6 0 0 e2 g

1 9 0 0 b 2 g 1 2 0 0 e2u

1 1 4 5 b 2 u 4 0 0 e2u

(a ) Calculate the semiclassical density of states p(E) in T 1 benzene a t the energyof the vibrationless S i state. (Be sure to cons ider the vibrational modedegeneracies.)

(b ) If the value of the spin— orbit coupling matrix eleme nt v between S i and T 1is on the order of 1 0 -4 cm -1 , is irrevers ible T i 4— S i intersystem crossingl ikely to o ccur in benzene?

6 . The lowest excited triplet state ( 3 A 2 ) in formaldehyde lies some 3 5 0 0 cm -1below the S i origin. Assuming that the vibrational freque ncie s in T 1 are thesame as in S i (Problem 7.1), evaluate the sem iclassical dens ity of states in T 1 atthe ene rgy of the vibrationless S 1 state. If the relevant spin— orbit couplingmatrix elemen t v is 1 0 -4 cm', what will be the first-order rate constant k N R ins -1 for T 1 4— S i intersystem crossing?



SPECTRAL LINESHAPES AND

OS CILLATOR STRENGTHS

W e have tacitly assumed in the preceding seven chapters that spectroscopictransitions occur at sharp, well-defined frequencies, leading to absorption l ineswith zero frequency width. In reality, numerous mechanisms (including lifetime

broadening, Doppler broadening, and collisional broadening) endow experi-

mental absorption l ine s with f ini te widths and characteristic lineshape funct ions .We develop a framework in the first part of this chapter fo r predicting these

lineshape functions in gases, where they assume a particularly simple form. In

condensed media, the absorption lineshapes (e.g., in solid naphthalene orNd' glass) are controlled by electron–phonon interactions, thermal broaden-

ing, and broadening that ar i ses from the fact that not all molecules experienceidentical environments within the medium (inhomogeneous broadening). Suchcondensed-phase broadening mechanisms lie beyond th e scope of this chapter.

Our discussion of Einstein coeff ic ients an d oscillator strengths in Sections 8 .3

and 8 .4 yields fundamental relationships among absorption coefficients, lumin-e scen ce li fe t imes , and probabilities fo r stimulated emission. The latter process i sresponsible fo r light amplification by stimulated emission (laser action), and

these relationships figure prominently in the derivation of lasing criteria inChapter 9 .

8 .1 ELECTRIC DIPOLE CORRELATION FUNCTIONS

According to th e formalism we developed in Chapter 1 , the probability

amplitude for an El one-photon transition from state 1 k > to state 1 m > is

1 it4 1 ) ( t ) = —'rnk t i K n i llk>dt i

ih o267



268 SPECTRAL LIN ESHAPES AND OSCILLATOR STRENGTHS

comk Oyler - AA>h

for a molecule that is expose d to a vector potential A(r, t) = A o ex p[i(k • r — cot)]

turned on at time t = O. Since the time integral in this e quation is

oe i ( 'mk -"tidt - =

i(conik — co)(8.2)

the probability that the transition will occur betwe en time s 0 and t becomes

ei( ""lk -w)t — 1

Zlac o

( t) 2 =h 2

m0 1 k > 1 2— cos[(co m k — co)t]

(w.k - 02

dt i8.1)

=1 < m l o A o l k > I 2coLn 2 sin 2 [(w, n k — w)t12]

( c o m k

) ) 2

(8.3)

In the limit of long times t» con-2, the co-dependent factor in Eq. 8 .3 approachesa constan t times the Dirac de lta function,

— > n t 6 (w „ , k — c o )((ontk —

(The proportion ality factor nt is required here to ens ure that the delta function is

normal ized to uni ty.) T he external electric field E = E0 ex p[i(k • r — wt)] i s

related to the vector poten tial in the Co ulomb gauge by E = — OAP. Notingthat the delta function (8.4) will con strain c o to equal c o n ik in Eq. 8.3, we may write

Ic(t)1 2: ti <OP' Eoik>1 26 (comk — (0 )8.5)

T he t rans i t ion probabili ty is cle arly proportional to the t ime duration t f o r

which the external field i s applied. I t i s therefore m ean ingful to de f ine t h etransition probability pe r unit t ime,

Pk =

MIP E0101 2 4 5 ( a) m k a ) )

m h

2 1 r8.6)

This express ion coincides with the well-known G o lde n R u l e formulation [1] o fthe molecular transition probability under the external perturbationW= —p• Eo .

W e now wish to generalize this expression to a system of molecules at thermalequilibrium. Let P k an d p m be the probabilities that a molec ule will be found i n

state k and in state m, respectively. The r a t e o f en ergy loss from the radiation

2 5i11 2 [((O„, k — co)t/2](8.4)



ELECTRIC DIPOLE CORRELATION FUNCTIONS69

field due to El one-photon absorption/emission proces ses will be [3]

E ha m k Pk-m

k m

2 7 r= (0.k(pk — P.)IonlE0 -pi1> 1 2 6(0 ).k —

r& km

2 7 r= wm kPk ( 1 e -h w m k

/kT)Kin iE 0 ' P l k>1 26 ( c o m k

k m

— co)(8.7)

The average ene rgy dens ity stored in the electromagnetic field is E = 80 E0 /2 [2].Since the delta function in Eq. 8.7 requires c o to equal co n i k in each term of thesummation (8.7), the optical absorption coe fficien t will be proportional to

—É 4nc ( 0 ) == — ( 1 — e -hwIkT ) E pki<m lf.pik>1 2 6 ( 0 ) m k — (0 )coE eohm (8.8)

where Ê is a unit vector directed along the electric field. Since we are nowconcentrating on the shape (rather than inten si ty) of the absorption lines, wedefine the lineshape function

3 e o he(co)1(o) ) =h I(T -3E pki<miÊ.pik>1 2 6 ( c o n i k — ( 0 )4 7 4 1 — e w-- )m

In view of the integral represe ntation (1 .11 3) of the Dirac delta function this isequivalent to

/(w) =2 7 r

E pk<kltinXplmf plk>k mte i"E k ) h — w ] t

3

3 r= —t e P k < kl plm > < ml eiE„,t/hoe -iE k t m ik>

2 n _m(8.10)

In the Schrödinger representat ion of the latter matrix elemen t in (8.10), th emolecular states are regarded as time-dependent basis functions exp( — iEk t /h)lk>and e x p ( — iE „ , t / h ) lm> , and the operator .8•11 is considered to be time-independent. For present purposes, it is more illumin ating to use the Heisenberg

representation, in which th e molecular states are the t ime-independ ent basisfunctions ik> , Im> and the operator is viewed as t ime-dependent . Since

> = Ei li> for each of the molecular s tates I i> , we have

1 (w) = 2 r° dte -iwtY p k <kit - plm><mlet/hf.pe-ifilk>27r i - x

m

(8.9)

CO

- 00

(8.11)



270 SPECTRAL LINESHAPES AND OSCILLATOR STRENGTHS

and s o th e time-dependent electric dipole moment operator in the Heisenbergrepresentation is

A t). ifit/hfle -ifit/h

8.12)

In terms of this, the lineshape function becomes

31(o)) = -z 7 r fte"k <klf p(COE'Atlik>8 . 1 3 )

Performing the summation in (8.13) with the Boltzmann weighting factors P kamounts to evaluating an ensemble average. Using the subscript ze ro to denotethe ensemble average of a matrix element, we then have

1(w) = —3ir it e 1 "<kl • p(0)f p(t)11c>8 . 1 4 )

Since the orientation of the unit vector Ê is arbitrary in an isotropic sample, ands ince

<kitt .(0)1i,c(Olk> 0 = <klily ( 0 ) 1 4 1 , ( t) lic> 0 = <ki 1 .(0), z(t)lk>

= i<kili(0 ) • 1 1 (t)ik> o

our lineshape function assumes the final form [3 ]

1(w) = — J

,E

dt i "<p(0) • P( t)> o27r _

( 8 . 1 5 )

( 8 . 1 6 )

1(w) is then given by the Fourier transform of the electric dipole correlationfunction <p(0)p(t)> . For a single molecule , the quantity p(0) • p( t ) gives theprojection of its dipole moment at time t along its initial direction at time t =O .In a collection of molexules , the total dipole moment must be used in p(t) , so thatthe correlation function will generally contain cross terms between dipolemoment operators belonging to different molecules. At high concentrations ( e . g . ,

in pure polar liquids), where orientational motion between neighboringmolecules may be highly correlated, < 0 ( 0 ) • go o therefore cannot be interpretedin terms of reorientation of a s in gle molecule. Such c ros s t e rms are unimportant

in gas es and in dilute solutions of polar molecules in nonpolar solvents (wherethe motions of neighboring polar molecules are essentially uncoupled), and inthese systems the dipole correlation function gives a quantitative measure ofhow a single molecule loses its orientational memory as a result of collis ions o rother perturbations. The correlation function typically approaches zero at longt imes in liquids in gases , as the molecular orientations become randomizedthrough stochastic processes (e.g., angular momentum changes in bimolecular

collis ions).



LIFETIME BROADENING71

8.2 LIFETIME BROADENING

W e nex t evaluate the lineshape function (8 .16) fo r two concrete si tuations in

gases. (The complexity of molecular motions in liquids precludes com putation oftheir dipole c orrelation functions in a text of this sco pe.) In the first situation, weimagine that we are examining lineshapes in the far-infrared spectrum of acollision-free, rotating polar molec ule. Its dipole moment p o is assume d to rotateclassically without interruption with angular frequency co o about an axis n ormalto p o . In a dilute gas, we would then have

= pi, cos ( D o t

2Po •=(e + e i œ a)

fo r — o o < t <+ co The relevant lineshape function is then

2 1o

/((.0)te i m t )_

2

= —2

[ 6 ( c o — c o o ) + (5 ( 0) + C o o ) ]

(8.17)

(8.18)

These two delta functions correspond to absorption an d emission of radiation atfrequency (D o , res pectively, with spectral lineshapes exhibiting zero full width athalf max imum (fwhm). Such uninterrupted molec ular rotation, in which thedipole correlation function (8.17) maintains perfect s inusoidal coherence for anindefinite period of time, produces no broadening in the lineshape function 1(w).

Suppose now that we have the m ore realist ic co rrelation function

< M O ) • p(t)> 0 = 0 <0

2 — yt/2= [ t o eos c o o t> 0 (8.19)

This would represen t a molecule rotating classically with frequency co o , with at ransi tion mome nt that dec ays e xpone ntially with 1 /e lifetime t =2 /y followingexcitation at t ime t = O. Such decay may occur via radiationless transit ion,spontaneous emiss ion (Section 8.4), or collisional deactivation in the excitedstate. T he corresponding (real) lineshape function is

2oACO) =

— P o Ret 1 2 ( e i w 0 t e - imt)dt4 n

2 [/2/24 n ( c o — c o 0)2+ y 2 /4 ( c o + w 0)2+ y2/4]8.20)




Like Eq. 8.18, this expression exhibits terms corresponding to absorption andemiss ion, respect ively. The normalized absorption lineshape function

sa t i s f ies the condition

1

PL(w) 2Y 7 r (a ) — 0) 0 ) 2 + y 2 /4

P L ( w) c i co = 1

(8.21)

(8.22)

PL (co) is called the Lorentzian lineshape f unc t ion . Its fwhm is equal to y, and i s

inversely proportional to the lifetime r = 2/y. It approaches zero as c o -4 + cc ,and maximizes at c o = co o (Fig. 8.1). Physically, y itself will have several

components in any real absorption line, arising from spontaneous emission(f luorescence or phosphorescence), nonradiative exci ted-s ta te de cay ( intersys temcrossing, internal conversion, photochemistry), collisional deactivation, etc.:

= Yradnonradcoll ± • • •8.24

Figure 8 .1 Lorentzian and Gaussian profiles P L (w) and PG(w) (solid and dashedcurves, respectively) with the same peak height and the same fwhm. The two profiles

are very similar for l c u l fwhm/2, but the Lorentizian profile falls off much more

slowly than the Gaussian profile at large I L O .



DOPPLER BROADENING AND VOIGT PROFILES73

or

1 /T = 1/T rad + 1 trnonrad + 1 P r e o n +8.24)

Lifetime broadening is ubiquitous, because all excited states decay. T heobserved absorption lineshape i s s e ldom Lorentzian, however , s ince o thermechanisms than lifetime broadening usually dominate the actual l ineshape .

8.3 DOPPLER BROADENING AND VOIGT PROFILES

As a result of the Doppler e ffect, a molecule traveling with velocity component v z

along the propagation axis o f an inciden t light beam will ex perien ce a light

frequency that is s hif ted from that experienced by a stationary molecule by

= w (D o = — wo(v./c)8.25)

Here w o is the frequency experienced by the stationary molecule. In a thermalgas sam ple at tempe rature T , the one-dimensional Boltzmann velocity distribu-tion in v z will be

P(v)0e—rrivR21cT

8.26)

where m is the molecular mass. (This equation should not be con fuse d with thethree-dimensional velocity distribution, which is proportional tov 2 exp( — m v 2 12kT) . ) Using v z = c(co — w)/w 0 , we obtain

PG (w ) = P o exp[ — m(co — w 0 ) 2 c 2 /2co6k 718.27)

This i s the Gaussian lineshape function, which arises from Doppler broadening.Like the Lorentzian function PL (w), i t maximizes at w = w o , and approacheszero as w + oo. T he fwhm of PG (w ) is

fwhm (2wy2kTln 2y 1c A m)

2

(8.28)

so that the Gaussian lineshape broadens with temperature as T1 1 2 . T heLorentzian and Gaussian lineshapes are contrasted in Fig. 8.1.

T he Lorentzian and Gaussian lineshapes physically differ in one importantrespect. All molecules in a homogeneous gas (excepting specialized situationswhere the gas is s ubjected to a nonuniform external field) in a given excited statehave an identical probability per uni t time of decaying e i ther radiatively ornonradiatively. All molecules in such a sam ple thus contribute equally to PL (w )

at all frequencies co. This is an example of homogeneous broadening. In Dopplerbroade ning, i t is clear from Eqs. 8.26 and 8.27 that slowly moving m olecules




contribute more to th e center of the Gaussian profile PG ( c o ) , while fastermolecules account for the wings where co » C O o or w « c o o . Doppler broadening

is thus an example of i nhomogeneous broadening.Both lifetime and Doppler broadening a re always present in gas sample s . If

these two are the only important broadening mechanisms, the resulting profile is

given by the convolution

P(w) =G ( c o —co')PL (a)dol8 . 2 9 )

of the Gaussian and Lorentzian profiles (Fig. 8 . 2 ) . This function, called the Voigtlineshape function, is nonanalytic (i .e . , inex press ible in c losed form), and it has

been extensively tabulated fo r analysis of gas sample absorption lineshapes.With our current access to interactive computers, such tabulations are rapidlybecoming unnecessary.

Doppler broadening in gases can readily be eliminated by manipulation ofexperimental conditions. One way to achieve this is to do spectroscopy onsupersonic gas jets, in which the translational velocity distribution can be madeto resemble a delta function along the jet direction (i.e., the velocity distribution

Lk )

Figure 8.2 The Voigt profile, formed by the convolution (8.29) of the Gaussian

profile with the Lorentzian profile, effectively sums Lorentian profiles (solid curves)

centered at all frequencies af, weighted by the value of the Gaussian profile (dashed

curve) at those frequencies.



EINSTEIN COEFFICIENTS75

is highly non-Boltzmann), while the velocity distribution transverse to the jetdirection is characterized by low temperatures T 20 K. At such low temper-atures, Eq. 8.28 yields an absorption line fwhm that is far narrower than thatobserved in room-temperature gases. Supersonic jet spectroscopy has thereforeafforded experimentalists an opportunity to obtain highly detailed, well-resolved vibrational structure in large-molecule electronic band spec tra (cf. Fig.7.10). An alternative way of obtaining Doppler-free spectra (applicable tot h e rma l gases) i s to do two-photon spectroscopy with counterpropagating laserbeams; this will be discussed in the section on two-photon absorption inChapter 1 0 .

Experimentalists must always contend with ins t rumen ta l broadening aswell, since spectrometers and excitation lasers never operate with ze ro bandpassor output bandwidth. The resolution of a grating spectrometer (defined as

R = I A A , with 2 and Al equal respectively to the operating wavelength and thewavelength bandwidth passed by the instrument) is ideally inversely pro-portional to the width of the instrument's ex i t slit. Any real instrument suffersfrom aberrations that lower R from its ideal value: coma, spherical aberrations,and chromatic aberrations are s o m e of the artifacts that can contribute toinstrumental broadening in the less well-designed instruments. Additionalproblems arise in grating spectrometers with mechanically ruled gratings,because the groove spacing in such gratings cannot be made absolutely uniform;these gratings inevitably diffract more than one wavelength at a time into anygiven direction. This drawback has been greatly reduced by the introduction of

holographic gratings, in which the diffraction grooves are automatically formedwith uniform spacing upon exposure to an i n t e r f e r ence pattern of two coherentlaser beams with well-defined wavelength.

8.4 EINSTEIN COEFFICIENTS

The Einstein coefficients prove to be useful fo r understanding th e relationshipsamong the probabilities fo r spontaneous emission, stimulated emission, andabsorption. They are thus valuable fo r understanding the criteria for achieving

laser action, where the competition between spontaneous and stimulatedemission in the laser medium is crucial. The Einstein coefficients also lead toimportant insights into the relationships between the absorption and flu-orescence properties o f molecules, relationships that are often taken fo r grantedin the chemical physics literature.

We begin our discussion with an ensemble of identical two-level systems inwhich the upper and lower state populations are N2 and N 1 , respect ively. T heene rgy levels are spaced by A E = h v , and the sys t ems are at thermal equilibriumwith a radiation energy distribution over light frequencies y given by p(v). It isassumed that only three mechanisms exist fo r transferring systems between

levels 1 and 2 : one-photon absorption, spontaneous emission (radiation of asingle photon), and stimulated emission (Fig. 8.3). In the latter process, a photon



276PECTRAL LINESHAPES AND OSCILLATOR STRENGTHS

N 9A

B1 2

NIp(v)

2IN

2

2IN

2p(v)

NI

Figure 8.3 Optical processes in a two-level system with level populations N, and

N2 . The rates of photon absorption, spontaneous emission, and stimulated emission

are B 2 N,p(v), A 2 1 N2 , and B21 N2 p(v), respectively.

with frequenc y y (which matches the ene rgy level differen ce A E ) and givenpolarization is incident upon à system in upper leve l 2. It causes the system torelax to level I, giving u p a photon of ident ical ene rgy, polarizat ion, and

propagation. (Since two identical, coherent photons emerge from one incidentphoton, s t imulated e mission forms the basis for laser amplification in Chapter9.) T he number of photons absorbed per unit t ime will be proportional to bothN 1 an d p(v), and we define the Einstein absorption coefficient B12 by se tting thisphoton absorption rate equal to .1 3 1 2 N p(v ) . T he number of photons emitted perunit time w ill be e qual to N2 A 2 1 + N 2 B 2 i p(v), where we have now implicitlydefined the Eins te in coefficients B21 fo r st imulated e mission and A 21 forspontaneous e miss ion (whose rate i s necess ar ily independen t of p(v)). Theabsorption an d emission rates m ust balance at thermal equilibrium, s o that

N2(B2 1P (v) + A 2 1 ) = N1B 12P (y)

8.30)

W e may s olve this for the radiation energy distribution

N 2 A 2 1p(v ) =N 113 1 2 — N 2 B 2 1

(8.31)

However , p(v) also be given by the Planck blackbody distribution at thermalequilibrium [4 ]

8 7 r h y 3 p(v ) =c3e

hv/kT W e also know that N2 and N 1 are related at equilibrium by [4 ]

N 2 /N 1 = ( g 2 1 9 1 ) e-hvIkT

(8.32)

(8.33)

where g i an d g 2 are the statistical weights of levels 1 and 2. It is easy to show thatEqs. 8 .31 through 8.33 are mutually consis tent only if

B21 - (g1ig2)B12

8.34)



OSCILLATOR STRENGTHS 277

and

A21 =

87rhy 3B21

C 3

(8.35)

Hence , the Einstein coefficients for absorption, spontaneous emiss ion , andst imulated e mission are all sim ply related. The y 3 factor that en ters in thespontaneous emission coeff ic ien t A21 (Eq. 8.35) has had historical importance inthe development of lasers , since i t implies that spontaneo us em ission compe tesmore effectively with stimulated e mission at higher frequen cies. H igh-frequen cylasers have therefore bee n more difficult to construct. This i s one of the reasonswhy X-ray lasers have only rece ntly bee n built , and why the first laser wa s anammonia maser operating on a microwave umbrella-inversion vibration rather

than a visible laser.I t may be shown [5 ] that in the El approximation the Einstein absorption

coefficient for the two-level system is given by

8 7 E 3B 1 2 =

3 h2l < 1 1 1 1 1 2 > I 2 (8.36)

This leads imm ediately to ex plicit expressions for the tw o Einstein emissioncoefficients,

and

8 7 r 3 (g 1B21= -

3h'—g 2

) Kliiii 2 > 1 2

64ev 3(A

2'=

3hc3

g,

)

(8.37)

(8.38)

8.5 OSCILLATOR STRENGTHS

We now exploit the re la tionships amon g Einstein coeff ic ients to obta inrelationships between the electronic absorption and emission propert ies inpolyatomics. For simplicity, we assume that an absorptive transition originatesfrom the ground vibrational state 0 in electron ic state 1 , and terminates withvibrational state n in electronic state u. (These vibrational level de signations areshorthand labels for collective vibration al states in volving all of the normalmodes.) The Einstein coefficient for this absorption process is

8 m 3B 1 0 _, =

3 h2K 1 0 1 1 / 1 1 4 1 0 1 2 (8.39)




which in the B orn-Oppen heime r approxim at ion (not good fo r vibronicallyinduced transitions!) is

870=< 1 1 P 0 1 0 1 2 1 0 1 n > 1 2

8 n 3=imi--012K01023 h 2

(8.40)

Here K 0 1 0 1 2 is the Franck-Condon factor for the 0 -÷ n vibrational bandintensity in the electronic spectrum, and ME!'" is the electronic transitionmome nt . I f one sums Eq. 8.40 over all of the upper s tate vibrational leve ls nreached from the vibrationless lower state, one obtains

E B 4 O , 87E3 IMuI 2 <010<n10>

3 h 2

8 7 r 3 iw, 1 2

3 h 2(8.41)

which depen ds only on the electronic t ransi t ion moment and is independent ofthe upper s tate vibrat iona l structure . This sum is therefore a measure of theallowedness of the electronic transit ion. To describe the latter, it is conventionalto use the dimensionless oscillator strength [6]

8n 2 mu =3he2e1.11‘4!""12 (8.42)

where m e is the electron mass, e is the electron charge, ands the meanfrequency of the electronic transit ion. Defined in this way, the sum of oscillatorstrengths > ,

fo r electronic transit ions from electronic state 1 to all otherelectronic states is supposed to equal unity, but in practice this depen ds on ho wT ) 1 u is specified. As a rule,fi _ , u on the order of unity is associate d with a stronglyallowe d El transit ion in aromatic hydrocarbons, while El-forbidde n transitionscarry f 2 .

Ne xt we exam ine the fluorescen ce propert ies em bodied in the spontaneousemission coefficient. In a two-leve l system with low radiation den sity (i.e.,ne gligible s timulated e miss ion and ne gligible pumping of level 2 by absorptionfrom level 1 ) , we have

dN 2N2 A 2 1dt

This has the time-dependent solution

(8.43)

N 2 (t)= N 2 (0)e -A21tN2(0)e - o t r a d8.44)



OSCILLATOR STRENGTHS 279

meaning the upper level decays by spontane ous e mission (fluorescen ce in thecase of a spin-allowed transition) with an exponential l ifetime rrad = 1 /A21-W hen we cons ide r the analogous case of a polyatomic fluores cing from its

vibrationless electron ic state u down to a manifold of vibrational levels m inelectronic state 1 , the radiative lifetime is given ins tead by

1 /rad E Auo-oim8 .45)

In the Born-Oppenheimer approximation, this become s

647t 4lltr"=

3 hc 3 (g1/92)KUllie1l 1 > 1 2 E qo--.1.1<01 m > 1 28 . 4 6)

which simplifies into

647T4ittra =9 )1A,1- s /412 , ,3

1 vu- s 1d 3hC

(8.47)

if the value of v 3 i s as sume d constant over the fluorescence spectrum. At thislevel of approxim ation, i t is clear that r rad - -S independent of the upper statevibrational level. (To see this, assume that upper vibrational leve l / rather thanlevel 0 emits, replacing the right sid e of Eq. 8 .45 with E. A a j,i ,a . The 1 -

dependence of 1/Trad then drops out by the time the analog of Eq. 8 .47 i sreached.) As we pointed out at the end of Section 8.2, T ra d i s re la ted to theobserved lifetime T by 1/-c = 1/-r lhrad • -f nonrad + 1 /"Ccoll + • • .9 so that the ob-served lifetime e quals T rad only if the other ex cited-state dea ctivation pathwayshave negligible rates. The fluorescence quantum yield of the excited s tate u i sdefined as

QF = YradAYrad + Ynonradcoll + • • *)

8 .48)

and approaches un ity - v, rad » Ynonrad + ycoll + • . Since y r a d is nearly indepen d-ent of the fluores cing vibrational level 1 in the Born-Oppenheimer approxi-mation, a st rong /-dependence in QF implies that there are vibrational level-depen dent intersystem crossing, internal conversion, photochemical, or colli-sional deactivation processe s presen t if the Born-Oppenheimer approximationholds in Eq. 8 . 4 6 .

W e n e xt combine Eqs. 8.41 and 8.47 to find a relationship between the

absorption spectrum and the fluorescence lifetime. In particular, the equation

8nhv 3 ., 1 /Trad3g1/g2) L , B10- un (8.49)

implies that the fluorescence lifetime for the u 1 fluorescence transition may bepredicted by s umming the Einstein absorption coefficients for all of the 0 - -+ n




vibrational bands in the 1 -* u electronic absorption spectrum. In manypractical situations (i.e., outside o f supersonic jets) the measured absorption

spectrum is n ot a series of discrete vibrational lines, but a continuous spectrum(inhomogeneously broadened in a solution, for example). A working formulathat has bee n evolved fo r singlet-singlet transitions in such case s is [6 ]

8 7 c • 230314 litrad =<V3

f >v8.50)Nc 2 n a

where n a and n are the medium (solvent) refractive i ndex at the absorption andf luorescence wavelengths, y is frequency in s -1 , T r a ci is in s , and e ( y ) is the molarabsorption coefficient in L/mol cm. Equation (8.50) has been verified for anumber of rigid molecules by Strickler and coworkers [7]. In nonrigid moleculeshaving differe nt equilibrium geo me tries in electronic s tates 1 and u,(which depen ds on the electronic wave functions) will not have the same value in

both Eqs. 8.46 and 8.41 , so that Eq. 8.50 will no t be valid.The strongest commonly observed absorption bands in organic molecules are

ex hibited by laser dyes such as rhodamine 6 G, a xanthene dye with a rigidchromophore which exhibits g m a x-0 5 L/mol cm a t 5 3 0 0 A (Chapter 9 ) . Itsfluoresce nce lifetime r is in the neighborhood of 5 ns (depending o n solvent), andi s domina ted by t r a d because Q F i s near ly uni ty unde r conditions in whichs t imulated e mission is suppresse d. At the other e xtreme, e m .„ fo r 1 2 in an inertsolven t l ike cyclohexane is about 7 x 10 2 L/mol cm for the spin-forbiddenV E g+ B3ni , transition. In accordance with Eq. 8.50, T r a ci for this transition i sin excess of 10j. Little 1 2 B

3 Hou VEg+emis s ion can be s ee n in solution,

however (Q F < 10), because collisions of B 3 H0u 1 2 with th e solvent inducesrapid predissociat ion into I atoms on a time scale of - 15 Ps.

REFERENCES

1 . K. Gottfried, Quantum Mechanics, Vol . 1 , W. A. Benjamin, New York, 1 966 .

2 . M . H. N ay fe h and M. K. Brussel , Electricity and Magnetism, Wiley, New York, 1985.

3 . R. G. Gordon, Adv. Magn. Reson. 3: 1 (1968) .

4 . F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill , New York,

1965 .5 . E . U . Co n do n and G. H . Shortley, The Theory of Atomic Spectra, Cambridge Univ.

Press, Cambridge, England , 1935 .6 . J. B. Birks , Photophysics of Aromatic Molecules, Wiley-Interscience, London, 1970 .

7 . S. J. Strickler and R. A. Berg, J. Chem. Phys. 37: 814 (1962 ).

PROBLEMS

1 . The El transition moment <2pl 4 ti ls> for a ls -4 2p transition in H can be

evaluated an alytically using the hydrogenic wave functions.



P R O B L E M S8 1

(a ) Calculate the Einstein coefficients B 12 , A21 and B21 for this transition.

Include units.(b ) For what radiation ene rgy density p(v) at the transition frequency will

stimulated e mission from a 2 p level becom e co mpetitive with spontaneousemission?(c ) Determine the radiative lifetime of a 2p state in hydrogen.

2. The absorption and fluorescence spectra of the organic dye rhodamine 6 Gin ethanol are shown in Figure P8.2. Use reasonable approximations todetermine ;a d and the oscillator strength to within 20 % for the S 1 4- S o

ransition. (Assume that the S 1 4- S o system is confined between 4 0 0and 6 0 0 n m; the absorption bands at wavelen gths s horter than 4 0 0 n m arisefrom transitions to higher s inglet s tates.)

3 . Calculate the expected fwhm of the R(0) line in the H C I vibration-rotationspectrum of Fig. 3.3 if the l inewidth i s domina ted by Dopper broadening.Assume that the gas temperature is 3 0 0 K. Is the observed broaden ing primarilyDoppler broadening?

4. An absorption spectrophotomete r is operated with a resolution of 1 cm -1 inthe near ultraviolet. W hat excited-state lifetimes will yield Loren tzian l ine shapefunctions with fwhm larger than this?

5. Figure P8.5 shows the normalized dipole correlation functions< A O ) 72( t )>/<02 (0)> obtaine d by taking the inverse Fourier transform of near-

10

74,

05X

200000000

00

Wavelength (nrn)

Figure P8.2 Reproduced with permission from K. H . Drexhage, in Topics in

Applied Physics, Vol. 1, Dye Lasers, F. P. Schafer (Ed.), Springer-Verlag, New York,1973, p.168.




1.0

0.9 -

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

\.\ \

"\ \ 1 t-- 0.1.2.3.4.5TIME (10 -1 2 sec) --o

-0.1

-0.2

-0.30 0 6

1

0.7

1

0.8 0.9

Figure P8.5 Reproduced with permission from R. G. Gordon, Adv. Magn. Reson.

3; 1(1968).

infrared absorption l ines of CO in several contrasting environments [3].Account for the qualitative d i ff e r ences be t ween the various correlation functions

(assume that the solutions are dilute enough to render cross t e rms in <(0) • p(t)i>insignificant). Why is the correlation function for "free" CO (i.e., isolatedmolecules) not a simple sinusoidal function of time, like that of a s e t of classicaldipoles rotating at a uniform frequency c o o ? How would the correlation functionfor free HC1 differ from that of free CO?



LASERS

Since their introduction to spectroscopy laboratories in the 1 9 6 0 s , lasers have

revolutionized the f ield to the point where most current spectroscopy would beunrecognizable to observers from th e pre-laser era. This stems partly from th efact that tunable lasers can be made to operate with intense output which is

highly monochromatic (output bandwidths AT ) 10 -3 cm are now availableusing specialized ring lasers), so that high-resolution spectroscopy which wasonce barely feasible can now be done routinely. The fact that lasers can easilygenerate picosecond and subpicosecond light pulses with selectable wavelength

and repetition rate has brought time-resolved spectroscopy securely into thet ime scale of vibrational motions and fast photochemistry. The truly novel laser-

based spectroscopies, however , have evolved primarily from the ability of lasers

to generate such extraordinarily high power densities that the nonlinear terms inthe Dyson expansion of c m ( t ) in Eq. 1 .96 , which are barely noticeable withclassical light source s , become prominent in laser-excited molecules. Processes

like two-photon absorption, second- and third-harmonic generation, four-wavemixing, stimulated Raman scattering, self-focusing and self-phase modulationwere all discovered in laser laboratories. Ordinary Raman scattering was well

established in classical spectroscopy before the 1950s , but required cumber-

somely large excitation lamps; laser excitation has enormously expanded itsresolution and productivity. We discuss the characteristics of lasers in thischapter, and we deal with several topics in nonlinear optics in the last two

chapters of this book.

283



284 ' L A S E R S

9.1 POPULATION INVERSIONS AND LASING CRITERIA

To establish the conditions required for laser action, we cons ide r a slab of a lasermedium containing a large number of two-level species . A light beam initially

with intensity / 0 will have the intensity

I = I e9 .1)

after passing a distance 1 through the medium if the latter obeys Beer's law.According to Eq. 9.1, the absorption coeff ic ien t a is

dl9.2)

so that the change in beam intensity after passing through a distance d l is

—dI = aI dl9.3)

We now rephrase Eq. 9.3 in terms of the microscopic dynamics of the two-levels ys t em. If W1,2 and 14/2„1 are the probabilities fo r upward and downwardtransitions per unit t ime in a two-level sys tem , the change in beam intensity willbe

—dl = hv(W1 _ . 2 N — W2,N 2 )d t9.4)

where dt = ndllc is th e time required for a photon to traverse distance d lthrough a medium with refractive index n. Using the Einstein coefficients andtwo-level notation introduced in Chapter 8, this becomes

—dl = hv(B 1 2 N — B 2 1 N 2 )I • n9.5)

From Eq. 9.2, we may now obtain an expression for the absorption coeff ic ien t o fthe laser medium,

dI nhvŒ = — — -=1 3 1 2 N — B 2 1 N )I dl

(9.6)

Note that we include no term in A21 (for spontaneous emission) in Eqs. 9.4through 9.6: Fluorescence is so widely dispersed over all propagation directionsthat its contribution to the coherent beam directed along the propagation axis ofthe incident light can be ignored. Using the relationships among the threeEinstein coeff ic ien ts , the absorption coefficient of the laser medium is finally [1]

Œ =C2A21 (g2X

N1 — f?.!i N 2 )8 7 c v 2 n 22 (9.7)



POPULATION INVERSIONS AND LASING CRITERIA85

This absorption coefficient can have either sign a priori, with the consequences

tabulated below:

Sign of a

esult

N11

N20mplification, / > Iog2

N1> —l

N20bsorption, I < Ig2

N1=—1 N20hreshold, / = /0

92

At thermal equilibrium, N2 = ( g 2 /g 1 )N 1 exp(— hylk T), me aning that one inevit-

ably has

Œ = c2A21 (2 1[1 — exp(— hv f icT)] > 08nv 2 n 2 g

(9.8)

for a collection of two-level systems in a Boltzmann distribution with finite,

positive temperature. Hence, no system at thermal equilibrium can l ase ; to get

the required "negative absorption coefficient," N2 must be artificially increased

by external pumping of the upper level through optical or other means.

To s e t up a prototype laser cavity (Fig. 9.1), we place a laser medium of lengthL between parallel mirrors with light intensity reflectivities r 1 1nd r 2 1 t

the optical frequency v = A E l h corresponding to the transition between the

upper and lower levels of the two-level system. Upon traveling twice through the

gain medium with absorption coefficient a in a single round-trip pass through

the cavity, an incident light beam with intensity / 0 will emerge with intensity

exp(— 2aL). At the s a m e time, it will be reduced by the factor r 1 r 2 exp( — 2y)as the beam strikes each reflector once in a round-trip pass; th is de f ines a cavityloss coefficient y = —iln(r I r 2 ) . For lasing to occur, the round-trip gains must

LASER

MEDIUMr-

M(r ) M(r2 )

.11

Figure 9.1 A laser cavity. The laser gain m ed ium has physical length L ; the en d

reflecting mirrors M have light intensity ref lectivities ri and r2 •



286 LASERS

e x c e e d the cavi ty losses , or

This requiresv satisfy

In view of E q.

or

Io e - 2 2 1 ' e - 2 ) ) > /0

that the negative absorption coeff ic ien t a(v)

— a(v) > y/L

9.7, this implies the condition

—2 A211 -1 2 1(g2 ) (N I —2 ) >

(9.9)

at the las ing f requen cy

(9.10)

) ±9.11)

y(9.12)

L

87rv 2 n 2

1)

9 2

9 2,

8n (gi)n2v2

N21 c A21 2As a f inal detail, we allow fo r broadening (e.g., Doppler broadening, whichdetermines the lasing lineshape i n H e / N e laser s) of the two-level transition.Assume that the spectra l lineshape function is g(v) , normalized so that

g(v)dv = 1 . Then the negative absorption coefficient in Eq. 9.7 at the l ineshapecenter frequency v o should be multiplied by g (v 0 ), which will be inversely

proportional to the l ine shape fwhm Av s ince g(v ) is normalized; the productg (v o )A v will be about uni ty. This implies that the lasing criterion in E q. 9.12should be replaced by [1 ]

() N2 —N1

87r2

n 2 v 2 (g )y dv

92 A2i g 2 9.13)

The factors favoring lasing are now e as i ly ident i f ied . T he right side of inequality(9.13) can be minimized by selecting a strongly allowed El transition (large A 2 1 ) ,

by using a large laser me dium length L (in principle), by using sharp transitionbandwidths dv, and by minimizing the cavi ty loss coe f f icient y. (In practice, y hascontributions from diffraction, scattering, and reflection losses at the mediumboundaries as well as from the cavity mirror reflection losses mentioned earlier.)T he upper state population N2 must be kept large enough to maintain theinequality.

Not all of these requirements are symbiot ic . In an argon ion (Ar k ) laser, theupper lasing level population N2 is created in a plasma in which Ar atoms areionized and excited by bombardment with ho t electrons. However, the resultinghigh temperatures markedly increase the transit ion bandwidth A v through

Doppler broadening, raising the lasing threshold. As another example, a laser at



TH E He/Ne AND DY E LASERS 287

the Lawrence Livermore Laboratory amplifies 1.06-pm pulses through one ofth e world's longest Nd' :YAG gain media—but this fact can incur problems,because the intense laser pulses themselves can cause the gain medium to

become lenslike (with nonuniform refractive index) , leading to pulse self-focus ing and gain medium damage over such di s t ance s .In the next sect ion , we describe how these lasing criteria are met and

maintained in two real laser types, the He/Ne laser and the dye laser .

9.2 THE He/Ne AND DYE LASERS

Discovered by A. Javan and coworkers in 1 9 6 0 at Bel l Laboratories, the He/Nelaser is the most commonly used laser for alignment and holographic purposes.

The upper and lower laser levels in this system are pairs of highly excited( 150,000 cm -1 ) electronic states in the Ne atom. Lasing has been achievedbetween many such pairs: The 6328 A line is the one most commonly used, butthe system can lase at 3.39, 1.12, 1.21, 1.16, and 1.19 pm, and many other linesstemming from th e complicated multiplet structure of rare gas atom exciteds ta tes .

When a mixture of He and Ne is subjected to a suitably energetic electricdischarge , the following energy transfer processes take place:

1. He + ee* + e

H e * + N e — > H e + N e * (N 2 )

hv(laser)

Ne* (N 1 )

2. Ne +ee* (N 2 ) + e

ihv (laser)

Ne* (N 1 )

(The asterisks denote electronic excitation.) In the first mechanism, electroni-ca lly exc i t ed Ne * produced in the discharge transfers its excitation collisionallyto Ne, which subsequently lases down to a lower excited level. In the secondmechanism, excited N e * is produced directly by the elec t r ic di scharge . The lattermechanism can cause las ing a t se veral lines in pure Ne; however , the addition ofHe greatly increases the las ing ef f ic ien cy for the following reasons. He ga ssubjected to an electric discharge is initially pumped to a wide range of exci tedstates, which can then rapidly relax radiatively to a succession of lower lying

levels by El transitions (Fig. 9.2). The latter process, known as cascading ,

proceeds according to the selection rules A l = + 1 , A j = 0, + 1 for the exc i t edelec t ron. A large fraction of the plasma-excited He atoms accumulate this way inthe 2 1 S and 2 3 S states (both with the (1s) 1 (2s) 1 configuration) of He. Once



cm- 1

0

20,000

40,000

60,000

80,000

180,000

200,000

2

0

288 - L A S E R S

Figure 9.2 Energy level diagram for He. Since the 2 1 S an d 2 3 S levels are El -

metastable, populations accumulate by cascading toward these two levels. Repro-

duced by permission from B . Lengyel, Lasers, 2d ed., Wiley-Interscience, New York,

1971.

populated, these levels are El-metastable because neither one is El-connected to

any lower level (only the (is) 2 PS ground state lies below the 2 3 S s ta te) . The selevels do not lase in He , because (as ide f rom the lack of any El-connected lower-

lying level) the 1 S ground state invariably has a larger population N 1 than

either 2'S or 2 3 S. H e n c e , 2'S and 2 3 S H e atoms typically retain their excitation(at suitably adjusted H e and Ne pre s sure s ) until they collide with a Ne a tom,creating Ne *. The efficiency of this collisional excitation transfer is abetted by

the near-resonance of the 2'S and 2 3 S energy levels with several of the• • • (2p) 5 (5s) and • (2p) 5 (4s) ex ci ted levels in Ne , respectively (Fig. 9.3).

The multiplet structure in Ne* arises from a pair-coupling (not Russell-

Saunders) angular momentum coupling scheme, which is obeyed in rare-gasatom excited states. T he ground and excited states of Ne exhibit the electron



170

4 }3P

3s8(• 2p

THE He/Ne AND DYE LASERS 28 9

Figure 9.3 Energy level diagram for Ne excited states in the energy region of the

metastable He 2'S and 2 3 S levels, which are shown at extreme left. The strongest

laser transitions are indicated with their wavelengths. Adapted from B . Lengyel,

Lasers, 2d ed., Wiley-Interscience, New York, 1971; used with permission.

configurations • • (2 p ) 6 and • • • (2p) 5 (n1) 1 , respectively. In the latter configura-tions, the angular mom enta of the 2 p "core" electrons form the resultants L„ S,of their orbital and spin angular momenta , which couple to form the coreangular mom entum J. Clearly L e = 1 and S, = 4 (since these are the quantumnumbers of the missing electron in the otherwise-filled (2p) 5 core shell), and sothe pos sible J, values are 4 and 4. The excited "valence" electron (ni)', whoseangular momenta are compara t ively we akly coupled to the core angularmomenta because i ts orbital is cons iderably more voluminous than those of thecore e lectrons, exhibits 4, = I and S, =4 . T he angular mome nta J, and L, thencouple to form a resultant

K = J, + L,9 .14)



290 L A S E R S

and the total Ne* angular mom entum is given by

J = K + S,9.15)

The Racah notation for rare gas exci ted s ta tes n eat ly summ arizes all of thesepertinent angular mom en ta in the term symbol nl[K],. As two examples o f N e *multiplets, we consider the 3s[K]. , and the np[K], states. T he former are thelowest-lying • • • (2 p ) 5 (3s) 1 excited s tates with L i , = 0 and S , = 4 . Since on e alwaysha s L, = 1, S, = 4, and J, =4 or 4 in the rare-gas atoms, the possible K valuesare from Eq. 9.14

K = J, + L„, . . •Lyi = =9.16)

since L, = 0 in the 3s[K] JMultiplet. Then the possible J values are [Eq. 9.15]

J = K + S „ , . . • , IK — S y l

when K =

when K =(9.17)

The 3s[K], multiplet therefore has four sublevels: 3s[4 ] 0 , 3s[1 ] 1 , 3 s [4] 1 , and3s[4] 2 ; the higher ns[K]J multiplets ex hibit similar sets of four sublevels. In asimilar vein, the np[K]j multiplets each e xhibit 1 0 components , because L e = 1can com bine with J, = to form K = I I;

W ith these multiplet splittings, a large number of N e * laser transitions i spossible. The common 6328-A line ar i ses from a 5 s[ K] — > 3 p [ K] JEl transition,the 3.39-pm infrared line ar i ses from a 5s[K]j 4p[K] t ransi t ion, and theseries of infrared l ines betwee n 1 .12 and 1 .19 pm ar i ses from transitions fromvarious 4s[K]j levels to lower-energy 3p[K] jlevels. Most laser applications inspectroscopy require m onochromatic output, and the undesired laser lines areeasily suppresse d in favor of the selected one. One can make the cavity losscoefficient y = — - 1 1 n ( r 1 / - 2 ) considerably larger for the infrared wavelen gths thanfor the 6328-A wavelength, fo r exa mple , by using laser cavity mirrors withcoatings exhibit ing larger r 1 r 2 at 6328 A than in the infrared. One may

alternatively insert a prism or other dispers ive elemen t into th e cavity s o thatonly the de sired wavelen gth will strike both cavity m irrors at normal incidencein a self-consistent, recycling trajectory (Fig. 9.4).

One important property of the above N e * ex cited states which favors theirinvolvement in lasing is their radiative lifetimes. T o maintain favorable N2 an dN 1 in Eq. 9.13 and achieve continuous laser action, the upper state populationN2 should have a high probability fo r stimulated e miss ion prior to deexcitat ionby other pa thways , and the lower s tate population N 1 should be deple tedrapidly afte r the laser transition to preven t i ts buildup. T hus, the upper andlower levels ide ally s hould have relatively long and short radiative lifetimes,respectively. This condition is met in each of the He/Ne transitions we have



THE He/Ne AND DYE LASERS91

'eor = I2<

Figure 9.4 Laser cavity with intracavity tuning prism. Laser oscillation at the

selected frequency y0 is refracted by the prism in a direction normal to the reflector

with r 1 = 1, and retraces its trajectory through the gain medium. Oscillation at other

frequencies y strikes the reflector at non normal incidence, and leaves the cavity.

described, since the ns[K]J and np[K]j levels have radiative lifetimes o f 1 0 -7and — 10 8 s, respectively. This condition is not met in the well known N2 laser ,which osc illates with high efficie ncy in a series of Cll. - ÷ 1 3 3 11 g transitionsbetween 3370 .44 and 3371 .44 A . Since the upper s tate lifetime is s horter than th eB 3 1 1 g state lifetime , the laser action terminates in about 20 ns , and N2 laseroperat ion is re stricted to the pulsed m o d e .

Prior to th e discovery of organic dye lasers in 1 9 6 6 , laser action had already

been demonst ra ted in several hundred gases (e.g. , H e / N e , CO, N 2 ) and solids(e.g., ruby, Nd' glass, Nd' YAG). The truly n ovel property of dye lasers wastheir tunabili ty—the fact that their output wavelengths could be varied over abroad range (tens of nm) by adjus tmen t of the dye concentration and/orresonator conditions. W hile dyes may lase in vapors and solids as well as inliquid solutions, the latter rapidly becam e the media of choice owing to theireconomy and ease of handling. Population inversions in early dye lasers werefrequently achieved by broadband excitation with xeno n gas-filled flashlamps.Dye lasers are now more commo nly mon ochromatically pumped using anotherlaser (typically a 5145-A argon ion laser or a 5320-A beam from a frequency-

doubled Nd' YAG laser).The processes that are critical to lasing in an organic dye [2 ] are summarized

in Fig. 9.5. Absorption of a visible photon creates a vibrationally excited S 1

molecule. In aqueous or alcoholic solution, the exc ess vibrational ene rgy is lostwithin seve ra l picosecon ds to the medium, leaving an S 1 molecule whosevibrational en ergy distribution is therm ally equilibrated. Laser action may thenoccur , terminating in S o molecules with varying degrees of vibrational e xcita-tion. To o btain S i 4°o en ergy gaps appropriate for visible laser transitions, onemust reso rt to molecules with ex tende d n electron s ystems (alternat ing singleand double bonds)

whichare larger than those in

anyof the

ultraviolet-absorbing molecules we con sidered in Chapter 7 . A typical size is e xhibited by- the rhodamine 6 G cation (Problem 8.2), which has 6 4 atoms and therefore 1 86



292 LASERS

Tn

Triplet — Tripletabsorption

TI

Figure 9.5 Energy levels for an organic dye laser. IC and ISC denote internalconversion and intersystem crossing, respectively (Chapter 7). Wavy arrows indicatevibrational relaxation. S o and S., are singlet electronic states, while T., and T

triplet states.

vibrational modes. In view of the resulting potential for spectral congestion, it isno surprise that this d ye 's S 1 < — S o absorption and f luorescen ce spectra are

vibrationally unresolved continua. It is this feature that en dows a dye laser withi ts m ost conspicuous ass et , cont inuous tunability.T wo of the processe s which detract from lasing, S i — > S o internal conversion

(IC) and S 1 — > T 1 intersystem crossing (ISC), are shown in Fig. 9.5. IC de pletesthe S 1 population, reducing the laser gain. The consequences of ISC are oftenmore serious. Since the T 1 s ta te i s m etas table (the T 1 — * S o phosphorescencetransition is spin-forbidden ), molecules accum ulate in T 1 if the ISC quantumyield is substantial. M any dyes exhibi t large T 1 T„ bsorption coefficients atthe wavelengths of the S 1 — > S o laser transition, due to t ransi t ions from T 1 tosome higher triplet state T.Such triplet—triplet absorption can d ramatically

reduce the laser gain. Since rapid IC and/or ISC occur in the vast m ajority o forganic com pounds (Section 7.4), a survey of some one thousand dyes that werecomm ercially available in 1 9 6 9 gleane d only four that proved useful in dye lasers[2]. Most of the dyes tha t are curren tly use d fall into three structural classes:xanthenes , ox azines , and coumarins.

It has been em pirically e stablished [2 ] that the presence of torsional modes(intramolecular rotations about bonds) accelerates S i — * S o IC in dyes. Forexample, the dye phenolphthalein has es se ntially zero fluorescence yield, whilefluoresce in has 90 % yield (Fig. 9.6): The torsional modes in phenolphthalein arefrozen out by the presence of the oxygen bridge in fluorescein. In the absence of

such torsional modes, most of the S 1 < — S o electronic energy differen ce in IC is



Phenolphthalein Fluorescein

Figure 9.6.

absorbed as S o state vibrational energy in high-frequency C—H stretchingmodes. (It may be shown that the IC probability is larger if thi s ene rgy di ffe rence

is deposited into a smaller number of vibrational quanta in the lower electronicstate [3]. For a f ixed ene rgy gap, this implies that higher frequency "accepting"modes are more effective in promoting IC.) For this reason, the pre sence of moreH atoms attached to the chromophore will reduce the fluorescence yield (andalso the lasing efficiency). Dyes freque ntly show larger fluorescence yields indeuterdted solvents , because proton exchange with the solvent replaces the C— Hmodes with lower frequency C—D modes via the isotope effect.

Intersystem crossing will obviously be accelerated by heavy-atom sub-stitutients, and no efficient laser dye contains them. The four bromine atoms ineosin (Fig. 9.7) increase its S 1 — > T 1 ISC yield to 76%, as compared to the 3%

value observed in fluorescein. Solvents containing heavy atoms (e.g., CBr 4 ) alsocontribute to T 1 buildup in laser d y e s , and are avoided.

Drexhage [2] has formulated a remarkable rule that relates the ISC rates indyes to the topologies of their 7r-electron structures. If the ring atoms thatcontribute to the 7r-electron structure form a closed loop of adjacent sites, themolecule exhibits a higher ISC yield than if the loop is broken by the pre sence of

a ring atom uninvolved the 7c-electron sys tem . This rule can be rationalizedsemiclassically by noting that the orbital angular momentum in dyes with suchclosed loops will produce larger spin—orbit coupling and enhanced S 1 — > T 1

y i e ls an example, acridine dyes (Fig. 9.8) have not materialized as a class of

laser dyes . The two resonance structures shown in this figure both exhibit the

Eosin

Figure 9.7.



294 LASERS

Figure 9.8 Resonance structures contributing to n electron densities in acridine

dyes (X = N) and rhodamine dyes (X = 0). The resonance structures containing

=X+ < are dominant in acridines, but are less favorable in rhodamines.

ammo nium group =N+ <. Both structures have comparable weighting in the i t -

electron s tructure of acr idines , so that considerable i t-electron d ens i ty re sideson the bridging N atom. In the important rhodamine laser dyes, the bridginggroup < in the se cond resonance structure is replaced by the en ergeticallyless favored oxonium group =O— . Appreciably less i t-electron de nsi ty is thenfound on the bridging atom in rhodamines than in acr idines , with the re sult thatthe latter show far higher S i T yields. Likewise, carbazine dyes (in which thebridging atom is a tetrahedral C atom, Fig. 9.9) ex hibit very low triplet yields.

M any laser dyes (xanthenes and oxaz i nes ) are employed as cations in polarsolvents, and the S i state in such dyes m ay be quenched by electron t ransferfrom the negatively charged counterion. The quenching rate decreases with theidentity of the counterion as I - > B r - > Cl > C1C4. Perchlorate is thereforethe counterion of choice, part icularly at higher concentrations in nonpolarsolvents, where rapid diffusion occurs between the dye and counterion.

In the xanthene dyes (Fig. 9.9), the absorption a nd lasing wavelengths aresensitive to the substituents on the xanthene chromophore, with the result that acontinuous range of las ing wavelen gths is accessible between 5 40 and

6 5 0 n m through appropriate choice of dye . (The carboxypheny l • group

comm on to all rhodamine dyes i s not part of the rigid xanthene chromophore,and it has little effe ct on its lasing properties.) Longer lasing wavelengths (630-

7 5 0 n m ) are available us ing oxaz ine dyes, in which the =C— bridge group inxanthenes is supplanted by =N—. Since the S 1 4 - S o ene rgy gap is smaller in theoxaz ine s than in the xanthenes, IC is more problematic in the oxaz ine s , and theuse of deuterated solvents improves their lasing efficienc y. In both the xantheneand oxaz ine dyes, the S i < - S o t rans i t ion moment is polarized a long the longmolecular axis. By s ymme t ry , ne i the r the S i nor S o state in these dy echromophores exhibits a permanent dipole moment.

In co ntrast , the coumarin dyes , while relatively nonpolar in the ground state,



H2

N NH 2

Rhodamine 110 (560)

Et2

N NEt2

Oxazine 1

Rhodamine 101 (640)

C H3

NEt2

Oxazine 4

Figure 9.9 Typical xanthene and oxazine dyes.

So

I

Et2

N Et N

Coumarin 1

oumarin 6

Figure 9.10 Dominant resonance structures for S o and S, electronic states in

coumarin dyes, and two typical coumarins.

are highly polar in the S 1 state (Fig. 9.10). Excitation of a coumarin S 1 state in apolar solvent (e.g., methanol) then produces a sudden increase in the molecule'spermanent dipole moment, followed by rapid solvent dipole reorganization [2].This lowers the S 1 state energy relative to that in nonpolar solvents , andd e c r e a s e s the S 1 < — S o energy separation; the result is a large wavelength shift(Stokes shi f t ) be tween the absorption and fluorescence maxima. For example ,

the absorption and las ing maxima of coumarin 1 in methanol are at 373 and460 nm. Owing to their smaller chromophore s ize, the coumarins lase at shorterwavelengths (440 to 540 nm) than the xanthenes and oxazines.



296 LASERS

Lyot plate

r v iDye laser

Pump

laser

Figure 9.11 Folded-cavity configuration for dye laser pumped by an argon ion or

frequency-doubled Nd 3 :YAG laser. The optics denoted Mare >99.9% reflecting at

the incident wavelength. The dye jet is flowed along an axis perpendicular to the

plane of the paper. The Lyot plate provides wavelength tunability (see text).

A common ly employed configuration for a cw (continuous-wave) dye laserexternally pumped by an argon ion or frequency-doubled Nd' : YAG laser i s

shown in Fig. 9.11 . A rhodamine dye solution in ethylene glycol (a viscoussolvent) is flowed in a je t stream 0.25 nm thick at — 7 m/s. The use of such a jetobviates the thermal inhomogenei t ies , window damage, and extensive dyephotochem ical de gradation that would attend exposure of a stationary dyesolution to an intense laser beam in a cell. To attain the large populationinversions requisite fo r lasing, th e pump laser beam is t ightly focuse d using aspherical concave reflector to -.0 .0 1 m m diameter at th e center of the jet. Thedye laser beam inside the cavity is similarly focused , using two confocal sphericalreflectors (i.e., reflectors whose focal points coincide at the center of the jet) to

provide goo d spatial overlap between the amplified dy e laser beam and thesm all pumped volume of the jet. All of these s pherical surfaces are coated withspecialized multilayer dielectric materials, which are essentially 1 0 0 % reflectingat the incident wavelengths and are highly resis tant to optical damage.

The laser output i s e x trac ted through the output coupler, a multilayerdielectric-coated fused silica substrate with typically 5% transmission at thelasing wavelength. In a well-aligned rhodamine 6 G laser pumped by a 5145-Acw argon ion laser, 2 5 - 3 0 % of the optical pump power can be converted intodye laser output. Coarse wavelen gth tuning can be e ffected by incorporating aprism into the dye c avity (Fig. 9.4), or by including a diffraction grating as on e o f

the cavity elements. The current tuning eleme nt of choice is a Lyot plate, alignedwith its surface normal at the Brewster angle O B [4] with r e spec t to the lase rpropagation axis as shown in Fig. 9.11 . A dye laser beam polarized in the planeof the Lyot plate (which is made of birefringent single-crystal quartz) andincident on the plate at O B will be partly reflec ted off the plate and partlyrefracted into the plate. (This polarization, terme d th e S polarization, corre-spond s to polarizat ion along an axis perpen dicular to th e paper in Fig. 9 . 1 1 . )

Laser light with the orthogonal polarization (P polarization) incident at O B w i l l

be totally re fracted through the plate. Since a typical laser photon experiencesman y round -tr ip passes in the cavity prior to exiting through the output

coupler, the presence of these Brewster-angle surfaces therefore strongly favors



AXIAL MODE STRUCTURE AND SINGLE MODE S E L E C T I O N 297

P polarization in the laser beam. T he crucial property of the Ly ot plate is that itrotates the polarization of all wavelengths traversing the plate , with theexce ption of those wavelengths A obeying

d (n o — n e ) = nA= 1, 2, ...9 . 1 8 )

Here n., n e are the ordinary and extraordinary refractive indices [4] of thebirefringent plate , an d d is the plate thickness. Since a lase r beam that isconsistently P-polarized during each cavity pass is strongly favored, the Lyotplate effectively suppresses lasing at wavelengths that fail to satisfy condition(9.18). When the plate is rotated about its norma l, the wave lengths that fulfi ll Eq.9 .1 8 become shifted. Hence a Ly ot plate (or a stack of two or three Lyot plateswith different thickness) can afford continuous tuning of the laser o utput across

the entire dye gain bandwidth.

9.3 AXIAL MODE STRUCTURE AND SINGLE MODE SELECTION

Provided the loss coefficient y is accurately given (taking into account all cavityloss mechanisms), lasing will occur a priori for all frequencies fo r whichinequality ( 9 . 10 ) i s sat isf ied. T he resulting lasing bandwidth, shown graphicallyin Fig. 9 . 1 2 , extends over all frequencies for which the gain coeff ic ien t [ — a(v)]

exceeds 7/L. A new co nstraint is now posed by the physical bound ary conditionthat the laser oscillation must have n o d e s a t the surfaces of the cavity endmirrors. This means that an integral number of laser half-wavelengths must fit

Lasing B and wid th

7

Figure 9.12 Dye laser gain curve —a(v) versus laser frequency v. Lasing occurs for

frequencies satisfying —a(v)> ylL.



298 L A S E R S

into the cavity's optical path len gth, so that

= L(s)ds2-= 1, 2, 3, ...9.19)

Here n(s) is the refractive index along the light propagation path in the cavity,and the line integral is taken over a single traversal of the cavity (one-half roundtrip). For a lase r me dium with refractive index n and physical length L 1 situatedin air betwee n cavity mirrors separated by physical len gth L2 > L 1 , the opticalpath length is L = nL i + (L 2 — L 1 ) . Equation 9.19 can be rear ranged us ing

clv to give the allowed long i tud inal (or axia l) mod e frequencies [1]

v. = ncl2L= 1, 2, 3, ...9.20)

These axial mode frequencies are equally spaced with separationL lv v., — y ,,= cl2L = 11T,where T is the time required for one cavity round-trip. In the absence of tuning eleme nts, the actual lasing frequencies will be thosewhich simultaneo usly satisfy both Eqs . 9 .1 0 and 9 . 20 , and all axial mode s withfrequencies v fo r which — Œ(v) > y/L will lase (Fig. 9.13). T he shape of the gaincurve —Œ(v) d e p e n d s on the operat ive l ine -broadening me chanism: Dopplerbroadening in He/Ne or Ar ± lasers, la tt ice broade ning in solid-stateNd' :YAG lasers, and so forth.

The ax ial mode frequencies themse lves are not infinitely sharp. The widths ofthe individual mode s depend on ho w long a typical laser photon stays inside the

cavity. In consequence of lifetime broadening, we have

A E At ?, h/29.21)

linn+i

Figure 9.13 Dye laser gain curve —a(v) , with axial mode frequencies yr, = ncl2t

superimposed. Lasing occurs in axial modes yn for which —a(v) > y/ L. Axial mode

profiles are represented as Lorentzian profiles for realism, since they are broadened

by finite photon lifetimes in the cavity.



A XI A L MODE STRUCTURE AND SINGLE MODE SELECTION 299

and so we can deduce that since AE = h Av = hc Ai, for photons,

t p A v/4 z9.22)

where t p = A t is the photon lifetime inside the cavity. The photon lifetime t p iscontrolled by the cavity loss coef f ic ien t y. Cavi ty loss es tend to deplete the cavityphoton population by a factor of exp(— y) per cavity pass; by the definition of t i, ,this factor can be equated with exp(— TI2tp ) , where T = 2LIc is the round-triptime. (One round-trip equals two cavity passes.) Hence the photon l ifet ime is

t p = T/27 =9.23)

According to Eq. 9.22, the order of magnitude of z l i ) is then

1 / 4 n c t p = y 14n1'9.24)

for the width of individual axial modes in cm

As an example, we consider a laser with 1-m optical path length and cavityend reflectors with reflect ivit ies r 1 = 1.00, r 2 = 0.95 (these are typical values for arear (high) re f lec tor and an output coupler reflector respectively in a practicallaser). If cavity reflector losses dominate the loss coefficient,

y = — 11 n [(1 ) (0 .95) ] = +0.0259.25)

and s o the individual mode width will be the order of S 2.5 x 10-5 cm -1 .This sugges t s a potential for compressing the intense laser output into a

remarkably narrow (by classical spectroscopy standards!) bandwidth of< 10'cm', if the laser oscillation can be limited to just one of these axialmodes. In fact, this is readily achieved by incorporating an etalon, in effect anadditional pair of partially mirrored parallel reflectors separated by the etalonlength Le , into the cavity (Fig. 9.14). All lasing frequencies must then simulta-neously satisfy two sets of axial mode boundary conditions

v„ = ncl2L' ,= 1, 2, 3, ...

v n , =' = 1, 2, 3, ...9.26)

Le

s s •s

Eta Ion'Figure 9.14 Laser cavity of optical path length L ' augmented with an etalon,whose partially reflecting surfaces are separated by a precisely variable distance L e.



300LASERS

-y/L

Axial Modes

A

Etalon Mode

Figure 9.15 Laser gain curve —a(v), axial mode frequencies v r, = ncl2L', and an

etalon mode frequency vn . = n'cl2L e . Since L e « L', other etalon modes are too

widely spaced from v n . to appear in this frequency range. Lasing occurs selectively atthe axial mode which overlaps the etalon mode. Tuning the etalon separation L econtrols the axial mode selected.

The etalon spacing Le can be adjusted in practice so that only one axial cavitymode within the gain bandwid th — oc (v) > TIL satisfies both of Eqs. 9.18, as

shown in Fig. 9.15. Moreover, Le can be varied to select success ive cavity axial

modes at will.This discussion barely touches on the topics of cavity modes and high-

resolution laser technology. Laser light also propagates in transverse cavitymodes [5], which can introduce fine structure superimposed on the axia l mo def requenc i e s if lasing is not confined to the lowest-order TEM 0 0 t ran sver se mode .Ring d ye lasers are currently the most widely used frequency-stabilized high-

resolution lasers .



MODE-LOCKING AND ULTRASHORT LASER PULSES01

9.4 MODE-LOCKING AND ULTRASHORT LASER PULSES

W e now concent ra te on the t ime dependence of laser oscillation in a cavity

whose gain curve encompasses a large number of axial modes. Each axial modeamplitude will oscillate with a time dependence of the formexp[ico n ( t — x/c) + i ( / ) , ] , with circular frequenc y

= 2in, „ = nncIL= 1, 2, 3, ...9.27)

0 „ is the phase of oscillation for mode n. The total amplitude of laser oscillationwill then behave as

E ( t ) = _ - E Enekon(t-xm+i4)„

9.28)

where the amplitude factors E„ reflect the weighting of the laser gain curve at theaxial mode frequencies w n . Equation 9.28 describes the t ime dependence of arandomly spaced sequence of l ight pulse s, since E(t) is a superimposition ofdifferen t frequency compone nts adde d together with random phases

W e n ow consider what happens when a l l of the axial modes are forced toosc illate at the same phase, say 0 0 . For simplicity, we assume that (2k + 1)modes oscillate with identical amplitude E 0 , and that the equally s paced mode

frequencies run from w = w0 — k do) to w = w0 + k do), withdo) =2 n Av = nc/L = 27E I 'T. The total oscillation amplitude then simplifies into

E(t) = E0e0i ( 0 1 0 + niico)( t — x lc)

n=—

= E0el[..( t _ x/c)±00] sin[(k + 1-)Aco(t — x/c)]

s in[4dw(t — x /c)]

The intensity of the ass ociated light wave is then

s in 2 [(k + 1)L 1(0(t — xlc)]1 E ( t ) 1 2 = 1E0 1 2

sin'[dw(t — x/c)]

(9.29)

(9.30)

This function is periodic, with period T equal to the cavity roun d-trip time (Fig.9.16) ; it corresponds physically to the fact that on e light pulse is propagatingback and forth inside the cavity at all times. Since part of this pulse istransmitted outside the cavity everyt ime it strikes the output coupler reflector,the laser output consists of a train of pulses equally spaced in time by T Thezeros in I E ( t ) 1 2 on either side of the primary pulse peaks are separated by the

duration 2 T/(2k + 1), which gives an upper bound es t imate of the laser pulsewidth -c p . For a 1 -m optical path length cavity, the round -tr ip t ime T is2LIc = 6 . 6 7 n s . If 9 axial modes are forced to oscillate in phase with equal



•

k= 8

I -A. PI

k= 4

AN --V I k=25

All- lh it. I

302 LASERS

e•̂ ". 4-,

LU

0 AD)(t—x/c)

Figure 9.16 Graphs of the periodic time-dependent function lE(t)1 2 for k=4, 8,

and 25, corresponding to (2k + 1) = 9, 17, and 51 axial modes locked in the same

phase with equal amplitude. Note the laser pulse sharpening which occurs as the

number of locked modes increases.

amplitude, the pulse widths are on the order of 1.5 ns; for 51 phase-lockedmodes, the pulse widths would be reduced to 260 Ps. This trend is illustrated in

Fig. 9 .16 . Since the pulse widths depend on the number of locked modes 2k + 1via tp 2 T / (2k + 1 ) , generation of ex t r emely short laser pulses has become theprovince of mode-locked solid-state and d ye lasers, whose broad gain

bandwidths can permit simultaneous phase-locked oscillation in thousands of

axial modes.

Mode-locking does not occur spontaneously in a simple laser cavity. Either it

must be ac t ively d r iven by a cavi ty e leme nt which int roduces cavi ty losse s wi th aperiod of exactly T/2 (i.e., one-half the optical round-trip time), or it must b e



R E F E R E N C E S 303

passively induced by an intracavity nonlinear absorber [6 ] which discourageslasing at phases other than some phase 0 0 at which strong lasing initially occurs.In an active acousto-optic [7] mode-locker, coarse wavelength selection of the

laser output is provided by an interactivity triangular prism placed near th e rearmirror (Fig. 9.4). A thin layer of piezoelectric material is deposited on one of thetriangular faces. An oscillating voltage applied to th e piezoelectric creates amechanical s t r e s s , which is transmitted to th e prism in the form of a standinglongitudinal acoustic wave if the driven wave frequency matches a prismresonance frequency. The mechanical rarefactions and compressions induced inthe prism by the standing acoustic wave produce a spatial alternation inrefractive i ndex , creating a transient diffraction grating which deflects the laserbeam from its cavity path when the voltage is applied to the piezoelectric. Byproper synchronization of the applied voltage frequency with the cavity round-

trip time, the longitudinal modes can thus be forced to oscillate in phase.Acousto-optic mode lockers are commonly used in argon ion lasers, which canoperate in several strong visible lines (notably 4880 and 5145 A). The 5145-A linei s gene rally se lec ted wi th the tuning prism; Doppler broadening of this line in theargon plasma tube al lows som e 40 axial modes to lase under the gain bandwidthcurve in a 1 -m cavity. Active mode-locking of such a laser typically produceslaser pulses with — 5 0 0 P s fwhm.

It is beyond the scope of the present chapter to review th e technology andcapabilities of mode-locked lasers . The currently favored systems for picosecondpulse generation are mode-locked Nd 3 + :YAG lasers (which afford 1.06-iimpulses — 1 5 Ps fwhm and may be wavelength-converted by using their 5320-Asecond-harmonic pulses to pump tunable dye lasers) , although dye laserspumped by mode-locked A r + lasers are still widely used. Pulses f romN d 3 + :YAG-based s ys t ems have been compressed to less than 1 ps fwhm inoptical fibers. T he very shortest pulses now reported have been generated inpassively mode-locked Ar + -pumped colliding-pulse-mode r ing dye lasers , whichhave yielded pulses as short as 40 fs wide (1 f s = 1 0 -1 5 s).

REFERENCES

1 . B. A. Lengyel, Lasers, 2d ed. , Wiley-Interscience , New York, 1971 .

2 . K. H . Drexhage, in Dye Lasers, Springer-Verlag Topics in Applied Physics , Vol . 1 , D yeLasers , F. P. Schafer (Ed .), Springer-Verlag, Berl in , 1973.

3 . P . Avouris, W. M. Gelbart , and M. A. El-Sayed, C h e m . R e v . 77: 793 (1977).4 . E. Hecht and A. Zajac, Optics , Addison-Wesley, Reading, MA, 1976; M. V. Klein,

Optics , Wiley , New York, 1 970 ; M. Born and E. Wolf , Principles o f Opt ics , Pergamon,Oxford, 1970 .

5 . A. E . Siegman, An Introduction to Lasers and Masers, McGraw-Hill, New York, 1971 .6 . D. J. Bradley, in Springer-Verlag Topics in Applied Physics, V ol . 18, Ultrashort Light

Pulses , S. L. Shapiro (Ed.), Springer-Verlag, Berl in , 1977.

7 . A. Yariv, Q u an t u m Electronics, 2d ed., Wiley, New York, 1975.



304 LASERS

PROBLEMS

1 . As an exercise in evaluating criteria fo r las ing in an idealized system,consider the 3 2 1 3 3 1 2 - - ÷ 3 2 S 1 1 2 transition in a Na atom. The radiative lifetime ofth e 3 2 1 3 3 1 2 levels is 5 x 10 8 s; the photon ene rgy for the transition i s

16 ,978 cm'. It is proposed to explore the possibility of lasing in a u n iform 1 0-cm cavity bounded by end reflectors with r 1 = 1.00 and r 2 = 0 . 98 . As s ume thatth e translational temperature in the Na vapor is 300 K, and that no cavity losse sother than transmission losses at the end reflectors are opera t ive .

(a ) Determine the population inversion (g 1 /g 2 )N 2 — N 1 required fo r lasing in

this system; include units. How is th e answer changed if the translationaltemperature is increased to 6 0 0 K?

(b ) What are the most fundamental problems that limit the practicality o f sucha laser?

2 . Several compounds from th e limitless roster of organic species that cannots e rve a s useful laser d y e s are listed below. For each of these, indicate the mos timportant physical reason(s) why the molecule is an unsuitable laser dyecandidate. Consider only the S 1 — * S o transitions.

(a ) Naphthalene(b) An ilin e(c ) Ro s amin e 4

(d ) Dithiofluorescein(e ) Acridine(f ) Iodoanthracene

H2

N NH2

Rosamine 4

Di th jot I uorescein

Acrid Inc



P R O B L E M S 305

3 . A d ye laser 0 . 5 m long is operated in a single axial mode with end reflectorscharacterized by r 1 = 1.00, r 2 = 0.95. The single-mode output bandwidth is10 cm-1 . Is this bandwidth limited by end re f lec tor losse s? W hat e f fec t would

doubling the cavity length L have on the output bandwidth if the cavi ty losse sare dominated by r 2 ? If the cavity losses are uniformly distributed along L (e.g.,through diffraction losses?)

4 . An etalon i s us ed for s ingle-mode se lec t ion in a 1 -m rhodamine 6G laser. Ifthe dye gain bandwidth is commensurate with the width of the rhodamine 6 Gfluorescence spectrum shown in Problem 8.2, what etalon separations L e andetalon surface reflect ivi t ies would ensure that only one axial mode is selected atany time?

5 . A 0 .75-m solid-state Nd' :YAG laser is acousto-optically mode-locked to

yield ultrashort pulses cen t e red a t V L = 9416 cm'.

(a ) Assuming that the lasing bandwidth function is given by

1c0 ) + y/EI = c, .5 cm"

=0,V JJ > 0.5 cm -1

where C is a positive constant, how many axial modes will lase ? W hat pulseduration will result from perfect mode-locking in this laser? What will bethe time separation between adjacent pulses?

(b) Suppose now that the lasing bandwidth function is given by a Gaussianfunction of centered at 3 L with an fwhm of 1 cm -1 . How are the answersin part (a) qualitatively charged?



10TWO-PHOTON PROCESSES

Up to now, we have been primarily concerned with one -photon absorption an demission processes, whose probability amplitudes are given by the first-orderterm

1c(t) =-LCOkrng 1 <ml /4741)1k>d ti

ih to(10.1)

in the time-ordered perturbation expansion (1.96). W e have seen that evaluationof the time integral (10.1) in the cw limit to —co,t +o leads to a statementof the one -photon Ritz combination principle E. — E k = hco, where c o is thecircular frequency of the applied radiation field (Eq. 1.112). The di scuss ions ofoscillator strengths and radiat ive l ifetimes in Chapter 8 proceeded from theassumption that

one-photonprocesses accounted

fo r all spectroscopic trans-itions of interest.Many radiative tran sitions cannot be treated under the framework of o n e -

photon processes. Raman transi t ions (which are two-photon processes) werediscovered by R a m a n a n d Krishnan in 1928; evidence for two-photon ab-sorption and more exo t i c multiphoton phenomena accumulated rapidly afterthe introduction o f lasers in the 1960 s . Some of the characteristics of two-photonproce s s e s are illustrated by the Ram an spectra of p-difluorobenzene (Fig. 10.1).These spectra were generated by exposing the pure liquid or vapor to a nearlymonochromatic cw beam from either a He/Ne or an argon ion laser, and

analyzing the wavelengths of light scattered by the sample at a right angle fromthe lase r beam. They are plotted as scattered light intensity versus the difference— co ' betwee n inciden t and scattered frequencies. p-Difluorobenzene exhibits

307



1600 200 10000 400

(a)

03 1

308 TWO-PHOTON PROCESSES

Figure 10.1 Raman spectra of p-difluorobenzene (a) pure liquid and (b) vapor,

recorded as light intensity /(a) scattered at frequency a versus the difference

(w — a) between incident and scattered frequencies. The spectra excited using an

argon ion laser (4880 A) and a He/Ne laser (6328 A) are nearly identical. Used with

permission from R. L . Zimmerman and T. M. Dunn, J . Mol. Spectrosc. 110; 31 2

(1985).

an S 1 4—So electronic spectrum with an origin band at 2713.5 A in the nearultraviolet, and is pra ctically transparent at the visible H e / 1 \ 1 e and Ar ± laserwavelengths (6328 and 4880 A , respectively). Photons at th e scattered freq-uencies co' are produced ess ent ially instantaneously (within 1 f s ) upon disap-pearance of incident photons at frequency c o . Consequently, this process cannotbe interpreted as a sequence of one-photon absorption and emission steps. A

one-photon absorptive transition with an oscillator s trength o f 1 in the UV-visible would populate an excited s tate with a radiative lifetime on the order of

ns (Chapter 8 ) . In a p-difluorobenzene molecule subjected to a visible laser , theemergence of photon w' would typically be delayed by a far longer time if i t

followed th e (extreme ly weak) one-photon absorption process at 6 3 2 8 or 4880 A .The frequencies w — co' of the Raman lines in Fig. 10.1 prove to be

independent of the excitation laser frequency c o , and analysis shows that they areequal to vibration al ene rgy leve l separations in S o p-difluorobenzene. This i s anexample of the energy conservation law h(co — = E,„ — E k in Raman s p e c -

troscopy: An incident photon with energy hco interacts with th e molecule; a

transition occurs from level 1k> to level 1 m > , and a scattered photon emerges witha shifted e nergy hol that com pensates for the ene rgy gaine d or lost by the

molecule (Fig. 10.2). Wh en c o > a, th e process i s cal led a Stokes Raman



••s•r 0 Omm

-6

T H E O R Y OF TWO-PHOTON PROCESSES 309

Figure 10.2 Energy level diagram for Raman scatter-

ing. A photon is incident at frequency w and a photon is

scattered at frequency cu'; the energy difference

h(cu — cu ' ) matches a molecular level separation Em — Ek.

The dashed line corresponds to a virtue/ state, which

need not coincide with any eigenstate of the molecule

(Section 10.1).

transition; when co < co', it is an anti-Stokes transit ion. W e will see that Stokest ransi t ions are gene rally s tronger than anti-Stoke s transi t ions , and only theStoke s port ions of the p-difluorobenzene spectra are reported in Fig. 1 0 . 1 .Detailed study of these s pectra reveals that some of the Raman lines (e.g., thelines at 3 0 8 4 , 8 5 9 , 6 3 6 , and 3 7 6 cm -1 in the liquid spectrum) are fundamentals invibrations of a g , b2 g , and b 3 g symmetry in the D2h point group. Suchfundamentals are symmetry-forbidden in one -photon vibrationa l spectros copy(Chapter 6 ). This illustrates the value o f vibrational Raman spectroscopy fo rcharacterizing vibrational modes that are spectrally dark in the infrared. Raman

spectra have also been used to probe rotational and (less frequently) electronicstructure.

T he other important two-photon process is two-photon absorption (TPA), inwhich two photons are simultaneously absorbed and a molecule is promotedf rom some s tate 1 k > to a higher-energy state 1m>. The selection rules in T P A aredifferent from those in one -photon absorption, an d T PA has proved fruitful iniden tifying electronic s tates that are inaccessible to conven tional electronicspectroscopy.

10.1 THEORY OF TWO-PHOTON PROCESSES

T he probabilities of two-photon 1 k > — > 1 m > t ransi t ions are controlled by thesecon d-order coefficients

cno 1 Ef en1W ( t i ) In> d t

f t .

e-iancnt2< (10 .2)

nofrom the Dyson expansion (1.96). W e may allow for the prese nce of two different



f e -r i

Xrroknr2

(Inke —iw1t2 + 6cnkei°)1 ` 2 )d t2to

( 1 0 . 7 )

310 TWO-PHOTON P R O C E S S E S

radiation fields with vector potentials A l (r, t) and A 2 (r, t) in the Coulomb gaugeby setting

ih qW O O = 2 (r , t ) • V

m c

ih qW (t 2) =A i (r, t 2 ) • V

m c

If one lets

A i (r, t 2 ) = A 1 cos(k i . r — w 1 t 2 )

A= —

2texp(i(k r — c o t 2 )] + exp[ —— c o l t 2 )]}

A 11=(r) ex p( iw i t 2 ) + xP(ko 1 - 2 )

(10.3)

(10.4)

and similarly treats A 2 (r, t 1 ) , we have

ih qhq<niW(t 2 )10=<nlik i (r)- Vik>e'" +nlAik>e'" 2

2mcmc

ih q=(an k e - ) 1 t 2 + i k e i")

2m c

and

ih q<ml W(t1)1n> =mIA 2 (r) V in> e

2m c

ih q=a e_ 2t1 + ri m ne)2m c m n

We finally obtain

q2

c(t) =4m2,2 i ( D n r a 1 (a„ m e i w2 t + .„e'' )d t

no (10.5)

ih q+nIA 2( — r ) • V I n > e + i ( D 2 t 1

2m c

( 1 0 . 6 )

as the second-order contribution to c m (t). This summation contains four crossterms fo r each n . Their interpretations will become clear as we de ve lop Eq. 1 0 . 7



THEORY OF TWO-PHOTON PROCESSES11

farther, and we list them fo r re ference below:

Termrocess

amnankwo-photon absorption

5Cmnankaman (Stokes if w1 > (0 2 )

amnankaman (anti-Stokes if w 1 < w 2 )

amnankwo-photon emission

These p roce s s e s can a l so be v i sua li zed in the same order using qualitative ene rgylevel diagrams in Fig. 10 .3 . The dashed lines in this figure denote virtual states,

which are not generally true eigenstates of the molecular Hamiltonian unlessone of the radiation field f requencies w 1 , w 2 happens to be tuned to one of the

molecular energy level differences. All of these two-photon proce s s e s a r eeffectively instantaneous, and the virtual states do not exhibit measurablel ifetimes. A second way [1 ] of visualizing these processes, which appears to becumbersome fo r displaying these (relatively) simple second-order phenomenabut which proves to be valuable in sorting out still higher order processes likesecond-harmonic generation (Chapter 1 1 ) , is to us e time-ordered graphs (Fig.10.4). The time coordinate in these graphs is vert ical , pointing upwards. Thephotons are represe nted by wavy lines . The ver t ical l ine s , which are divided intos egment s labeled k, n , and m, identify the molecular states that are occupied atvarious times; th e center portions of these lines denote the time intervals during

which the molecule i s in the virtual state labeled n . The state of the system at anyt ime t can thus be inferred by noting which portion of the vertical line and which(wavy) photon line(s) intersect the horizontal line representing time t. In the firstdiagram (corresponding to two-photon absorption), there a re two photons,(k 1 , w i ) and (k 2 , w 2 ) , and the molecule i s in the initial state I k > at time t a . By t ime

1k>

W I

(4)2

W I rn>1k>

Raman

0 , 2

k> I m >

Two- photonabsorption

Raman Two- photon

emission

Figure 10.3 Energy level diagrams representing the four contributionsto 4, ) (t) when the perturbation matrix elements are given by Eqs. 10.5 and 10.6.




( k 2 ,4,2)

k pur l )

k2 ,w2)

(

k ca dk2,02)

Two- photon

aman

aman

wo- photon

absorption

mission

Figure 10.4 Time-ordered graphs corresponding to the four contributions to

c („ 1 ) (t) when the perturbation matrix elements are given by E qs. 10.5 and 10.6. These

are shown in the same order as the energy level diagrams in Fig. 10.3.

t b , the molecule has undergone a transition to virtual state In> by virtue of theradiation-molecule interaction W (t 2 ), and only th e (k 2 , w 2 ) photon r emainsunabsorbed. At time t c the molecule has reached its f inal state 1 m > as a resul t ofthe interaction between virtual state In> and the (k 2 , (0 2 ) photon via coupling bythe W (t i ) term. The intersections of the photon lines with the vertical lines,which are labeled with interaction Hamiltonian terms like W (t 1 ) or W (t 2 ) ,

arecalled interaction ver t ices .The role implied by these time-ordered graphs for the virtual states called

" 1 n > " should not be taken too literally. In the treatment that follows, thesevirtual states are in e f f ec t expanded in inf ini te se r ies o f true molecular eigenstatesIn>, and no virtual state in any of the processes will coincide with any single,particular true In>. Hence, while the ene rgy conservation hco i + hco 2 = E n, —k

must be preserved in the overall two-photon absorption process, the first of thetime-ordered graphs is not intended to imply that hco = E n—Ek, whereE„ is the ene rgy of some true molecular eigenstate I n > . The absorption of photon

(k 1 , w 1 ) in this graph is called a virtual absorption, and it is not subject to th eenergy level-matching Ritz combination principle that is obeyed by one-photonabsorption (a r ea l absorption process).

W e have arbitrarily chosen to associate A 2 (r , t) with W (t 1 ) and A l (r , t ) withW (t 2 ) in Eqs. 10.3. If we allow in addition the reverse assignments [A 1 (r , t ) withW (t i ) and A 2 (r , t ) with W (t 2 )], we will genera te th e new energy level diagrams inFig. 1 0 . 5 and the new time-ordered graphs in Fig. 1 0 . 6 .

At this point, we have developed our theoretical framework sufficiently todeal explicitly with TPA and Raman spectroscopy. Spontaneous two-photonemission (which is depicted by the las t of each se t of time-ordered graph s in Figs.



TWO-PHOTON ABSORPTION 313

1 0

(0 2

W I

_

W,

W2

1 k

1m> 1k>

Two-photonamonamanwo- photon

absorptionmission

Figure 10.5 Energy level diagrams for four additional contributionsto c (n1 ) (t), generated by associating A, with W (t. , ) and A2 with W(t2 ).

Two-photonamanamanwoabsorptionmission

Figure 10.6 Time-ordered graphs corresponding to the energy level diagrams in

Fig. 10.5.

10 .4 and 10 . 6 ) exhibits transit ion rates far sm aller than those of El-allowed one-photon emission [1], and has no t been dete c ted. I t is l ikely to con tr ibute todecay in astrophysical systems in which one-photon decay is El-forbidden .

10.2 TWO-PHOTON ABSORPTION

We now develop the terms pert inent to T P A in Eq. 10 .7 . They become

2

TP ror )r210.8)Amq

crnnarikti e i(wnPr. + (1)241tA 2 2 to 2eto



TPA(t)m 4 2

4m 2 c 2 f tm n I n kt le i ( a ) km ± (4 ) 1 + W2 )t i

— oo

1

i (wk. + w1) (1 0 . 11 )


Se tting t o = — co in the cw limit, we have

142

C Tj(t) =(conm+2)ti0

4m2 c 2 E Œ m n Œ nk3 dt 1 e —

j•ti

—dt2e —+ wot2 (10.9)

To make the last integral on the right converge, we may replace c o k „ by (wk. +where c is small and posi tive , and then le t s — > 0 after the integration:

t i— e

—i(co k.+ cot + iE)t2 t

00

—e— i(.,„+coot'

(10 .10)

i(cokn + (0 1 + e)

—e —i(cok.+Wi+ioti

i(wk. + i(wk. + col)

This is more than just a mathematical artifice. T his substitution i s tantamoun tto replacing the ene rgy E n by (E„—i hE) , so that the intermediate state In>exhibits the t ime de pendence exp(— iEn tlh—£t) and hence physically decayswith lifetime 1/2e. The cons tan t s can be identified with y/4, where y is theLorentzian l inew idth (Chapter 8 ) . Such linewidths are gene rally much smallerthan level energies En , so that dropping e at the end of the integration yieldsgood approximations to c(t) in Eq. 10 . 10 . Nex t, we have

and so in the cw limit

c (co)2 7 r q 2m n a n k

(5 (Wicnt1 ± a 2 )4iM 2 C 2 n()knDI)

— 2 7 -c q 2mIA 2 (r) Vin><nlAl (r)- Vik>o(wk., + w, + a 2 )

4iM2C2kn W1(10 .12)

In the E l approximat ion (Chapter 1 ) , this is equivalent to

E2 < M I P i n > < n i l l i k > ElcIP A (co) oc E( c o k m + w, + 0 )2 )10 . 13 )

Wkn W1

where E l and E2 are the electric vectors of the inciden t light wave s ( 1 1 ( 1 , c o l ) and(k 2 , c o 2 ) . Since the roles of vector potentials A l and A2 can be reversed in T P Aand c m (t) must-ex hibit symm etry re flecting this fact, we finally conclude that

CO )• <mli/In><nlfilk> Ell < m l i t in > < n 1 P 1 k > • E 2 )IAC E 2

Wkn + w 1kn + w 2

X 6(CA, ± 0 h C ° 2) (10 .14)




Since the delta function in Eqs. 10.13-10.14 is proportional to

(5[E k -Em + h(co l + co 2 )] , it yields the obvious energy-conserving criterion(E. - E k ) = h(co i + c o 2 ) relevant to TPA. It is thus clear that the terms included

in E q. 1 0 . 8 are the ones associated with TPA, and furthermore that theparticular physical processes connected with the other terms in E q. 1 0 . 7 may beidentified by examining the signs of the time-dependent exponential arguments.

Equation 1 0 . 1 4 descr ibes the TPA transition amplitude for a moleculesubjected to two light beams with arbitrary electric field vectors and propa-gation vectors. A particularly useful application of TPA in gas phase spec-troscopy employs two counterpropagating laser beams with k 1 k - iki11k2I-In this case, a molecule traveling with velocity vx parallel to k 2 will experienceDoppler shifts

(01 —

0x/CC O I

(0 2 - 0 3 = — V x /C(0 2

( 1 0 . 1 5 )

in the f requencies c o i , co 2 relative to the f r equenc i e s ai? , co? experienced by amolecule at rest (Fig. 10 .7) . The total energy absorbed in a transition involvingphotons (k b ai l ) and (k 2 , co 2 ) will then be proportional to

V x‘(0 12 = (0 1. + (0 21(0 +(2

0+

c02)

v x+a 1 (4) ( 1 0 . 1 6 )

or

0),,W12 — w12 = kwl w2/ ( 1 0 . 1 7 )

T he Gaussian absorption profile that results from Doppler broadening of the

TPA transit ion probability as a function of co 1 2 will then be

P(co i 2 ) . . - _ - p 0 e -mv,12kT = po e - inc 2 (co 1 2 - co?2) 2 /2k 71,4 _4) 1 0 . 1 8 )

k 2, (02

kI

c ol

Figure 10.7 Two-photon absoription in a molecule subjected to counterpropagat-

ing light beams (k 1 , (0 1 ) and (It, (0 2 ) directed along the x axis.



CA(x)

E2 • (5 2 S 1 1 1 1 3 2 PX3 2 P I P 1 3 2 S> • E1l • <5 2 S 1 / 1 1 3 2 P ><3 2 P 1 / 1 1 3 2 S> • E2G C

E3p E3 hco

+E2 • <5 2 S 1 1 1 1 4 2 PX4 2 1 3 1 1 1 1 3 2 S > • E l + E 1 • < 5 2 S 1 / 1 1 4 2 P>O2 P 1 / 1 1 3 2 S> • E2

E4 P E35 ho)

E2 • < 5 2 S 1 P 1 5 2 P>O2 P 1 / 1 1 3 2 S > • E11 • < 5 2 S 1 1 1 1 5 2 P>K5 2 P I P 1 3 2 S > • E2

E5p E35 h c o

316 TWO-PHOTON P R O C E S S E S

T his profile exhibits a full width at half-maximum

2 1 w ° — w° 1 (2kT ln 2Y /22fwhm = 1c (10.19)

where m is the molecular mass. If photons of identical frequency are u sed(w7 = w?), the Doppler broadening cancels between the counterpropagatingphotons, and one nominally obtains zero fwhm. (In practice, one will stillobserve l ifetime and possibly other residual broadening effects.) Doppler-freeTPA spectroscopy is the most practical means of obtaining high-resolutionabsorption spectra in thermal gases .

T he 3 2 S — 0 5 2 S t rans i t ion in N a vapor provided the now-famous prototypes ys t em for observing Doppler-free TPA [2, 3]. The one -photon 3 2 S — 0 5 2 S

transit ion, which would occur at 3 0 1 . 1 1 n m, is El-forbidden (A l = 0 ) . If tw o

counterpropagating photons with identical wavelength = 60 2 .23 nm are u sed ,we have c o l = C O 2 C O an d k 2 = —k 1 (Fig. 10.8). T he leading contributions tothe 3 2 S — 0 5 2 S TPA transition amplitude will then be

(10.20)

T he intermediate states In> are restricted to the m 2 P states (with m 3) by theEl selection rule Al = + in each of the matrix elements of p . Contributions

42

P

32 Pigure 10.8 Energy level diagram for

3 2 S — > 5 2 S two-photon absorption in Na

2P - 3

2S Fluorescence

vapor. The two-photon process is moni-

tored by detecting 4 2 P-3S fluorescencefrom the 42 P level, which is populated by

cascading from 5 2 S atoms created by

two-photon absorption.




from the 6 2 P , 7 2 P states, etc. will be smaller than those in Eq. 1 0 .20 , because the

energy denominator (E n i p — E35 - hw) increase s with tn. In one of the earliest Navapor TPA experiments , a N2 laser-pumped rhodamine B dye laser provided

linearly polarized pulses at 60 2 .23 nm. These laser pulses were passed through athermal Na vapor cell, and then reflected backward by a mirror, causing them tocollide inside the cell with later pulses passing through the cell for the first t ime .The 3 2 S — > 5 2 S T P A transition was de t ec t ed by monitoring 4 2 P — > 3 2 S flu-orescence from the 4 2 P Na atoms, gene rated by cascading from the 5 2 S atomscreate d by T PA; this particular fluorescence transition in Na occurs at a visiblewavelength that is easily monitored by conventional phototubes.

The elimination of Doppler broadening in this e xpe rimen t allows the clearobservation of hyperfine structure that arises from the interaction of electronicand nuclear an gular momenta. The total atomic angular mom en tum is

F=L+S+I10 .21 )

where I is the nuclear spin angular mom entum. In 2 S states of 'Na, L = 0, I =1and S = 4, so that the possible F values are

F =I + S ,I — S I = 2 , 110 .22)

in both the 3 2 S and 5 2 S states. The splitting betwee n the F = 1, 2 sublevels is

larger in the 3 2 S than the 5 2 S state (as might be ex pected because the 5 2 S orbitalis more diffuse and has less e lectron probability dens ity near the nucleus). It canbe shown that the se lection rule on A F i s AF = 0 in T P A [2]; thus twotransitions will be obse rved (F = 1 1and F =2 - > 2 as shown in Fig. 10 .9) at

52

S2

32

S

Figure 10.9 Detailed energy level diagram for

3 2 S — > 5 2 S two-photon absorption in Na, showingsplitting of the n 2 S levels into hyperfine comp onents

with F = 1, 2 . Dashed lines ind icate virtual states.




single-photon frequencies separated by

2 dco = A3 — 2 1 5 or dw = 1 -(A 3 — A 5 ) =2 .6 x 10-2 cm' (10.23)

(the factor of 2 is required here because this is a two-photon transition). T heactual TPA spectrum obtained this way is shown in Fig. 10 . 10 . The hyperfinecomponents exhibit approximately a 5 : 3 intensity ratio, because 5 and 3 are thedegene rac i e s of the F =2 and 1 sublevels. The resolved hyperfine peaks resultf rom Doppler-f ree TPA of photons travelling in opposite directions. The broadbackground in which these Doppler-free peaks are superimposed ar i ses f romTPA of pairs of photons traveling in the same direction, since nothing in thisapparatus can present TPA of copropagating photons. This background can beremoved using circularly polarized photons, however (Fig. 10.11) .

It is instructive to touch briefly on the TPA spectroscopy of benzene [4 ] s ince

we have discussed its one-photon absorption spectroscopy in Chapter 7 . ForTPA from th e l A i g benzene ground state to some final vibronic state f,

(E 2 • ( f IPIn><nlplA„> • E 1i<flilin><MPIAl g > • E2

CinrrA(C°) CC

E — E Aig

h colEn — E — t 1 (0 2 )

For El-allowed TPA, it is then necessary that both

F(n) 01-(p) 0 A ig

F(f) 0 r(p) 0 F(n)

(10.24)

simultaneously contain A i g for some intermediate state In>. Since (x , y) and ztransform as E iu and A 2 „ in D 6h, respect ively , th e intermediate states In> m u s thave E l „ or A 2 „ symmetry. Consequently, th e allowed symmetries of the finalvibronic states If > are A ig , A2 g , Ei g , and E 2 g . This exemplif ies the obvious factthat th e selection rules in TPA a re anti-Laporte in centrosymmetr ic molecules ,and that TPA can be used to study excited states that are inaccessible to one-photon absorption f rom the ground state.

While the 'I3 2 u S 1 state in benzene has inappropriate symmetry for an

intrinsically El-allowed S i4—

So TPA transition, S i 4 — So TPA is still observeddue to vibronic coupling. Since

B2u

_ b2u

1 3 1u

e2u

elu -

-A 1 g -

A2 g

E 2g_ -

(10.25)

in D 6h, there are four symmet r i e s of normal modes (b 2 u , b i „, e 2 , e i „ ) that can

serve as promoting modes in vibronically induced S 1 4 — So transitions. This



P 2

XPANDED SCE

LINEAR

POLARISATION

Figure 10.10 Two-photon absorption profiles in Na vapor, obtained using linearlyand circularly polarized light beams. Profile (a) was obtained using only copropagat-

ing beams. Profiles (b) and (c) were obtained using counterpropagating beams thatwere linearly and circularly polarized, respectively. Reproduced by permission from

F. Biraben, B . Cagnac, and G. Grynberg, Phys. Rev. Lett. 3 2; 643 (1974).

319



C H A R TU A L

R E C O R D E R1 2O X C A R

I N T E G R A T O R •

PMT

- X/4 P L A T E

=O W P A S S.=›

S W I M C E L L

A N D

H I G H•V E N

P A S S

F I L T E R

- X 1 4 P L A T E

N 2

LASER

F I L T E R

PM T

(a)

o 1. 0

05.0(GHz

DYE LASER

Figure 10.11 Apparatus for measurement of two-photon ab sorption profiles in Na

vapor using counterpropagating circularly polarized beams. PMT denotes photo-

multiplier tube. The d ye laser is wavelength-scanned by rotating an intracavity

Fabry-Perot etalon. Profile (a) was obtained by two-photon absorption from one

l inearly polarized beam. Profile (b) shows the Doppler-free F = 1 1 and 2 —*2

hyperfine peaks, obtained using counterpropagating circularly polarized beams.

Used with permission from M . D. Levenson and N . Bloembergen, Phys. Rev. L et t. 3 2,

645 (1974). Note that this work arrived in a d ead heat with that of Biraben et al., Fig.

10.10.

320



RAMAN SPECTROSCOPY 3 21

situation contrasts with that in the one-photon spectrum, where vibrationalmodes of e 2 g symmetry are required for the intensity-borrowing (Section 7.2).

10.3 RAMAN SPECTROSCOPY

In ordinary Raman scattering, we a re concerned with the two-photon proce s swhereby photon (k 1 , c o 1 ) is annihilated, photon (k 2 , co 2 ) is created, and themolecule undergoes a transition f rom s ta te 1k> to s ta te Int>. Energy conservationrequires that h(co i — co 2 ) = E. —Ek. The possible time-ordered graphs satisfy-ing these conditions are shown in Fig. 1 0 . 1 2 . These two graphs should not beregarded as physically distinct in the sense that the first graph depicts"absorption" of photon ( 1 , c o 1 ) followed by "emission" of photon (k 2 , c o 2 ) , while

the second graph depicts these events occurring in reverse sequence. They ares imply a bookkeeping method fo r keeping track of different terms in theperturbation expression (10.7). (For that matter, the portions of the time lineslabeled "n" in Fig. 1 0 . 1 2 are unresolvably short, and the interaction vert iceslabeled "W(t i )" and "W(t 2 )" coincide fo r practical purposes.) Using th e termsassociated with these two diagrams in E q. 1 0 . 7 , it is straightforward to show by aprocedure similar to that carried out for TPA in Section 1 0 . 2 that the s econd-order probability amplitude for Raman scattering in the cw limit is

c(cc)

(E • < m l P l n > < n l P l k > • E

+nliiInXn1P1k> • E2)

oc E WknCIknD 2

x .5 ( co k „+ (0 1 — ) 2) ( 1 0 . 2 6 )

For a given intermediate state In>, the first and second right-hand terms in E q.1 0 . 2 6 correspond to th e first and second time-ordered graphs in Fig. 1 0 . 1 2 ,respectively. When c o l > co 2 , th e scattered radiation f r equency c o 2 is said to be

(k 1 , cod

Figure 10.12 Time-ordered graphs for the Raman process with incident frequency

co l and scattered frequency cu2.




Stokes-shifted from the incident frequency w ; anti-Stokes scattering is obtaine dwhen co l <(0 2 . The energy difference h ( c o 2 — c o i ) normally matches either amolecular vibrational— rotational or rotational level difference , and the incidentfrequency c o is usually some readily gen erated visible frequency (e.g., an Ar orH e / N e laser line) in conventional Raman spectroscopy. In such cases1 c 0 2 — 1/co « 1. W e may specialize Eq. 10.26 to chemical applications ofRaman spectroscopy [1 ] by using Born-Oppenheimer states for the molecularzeroth-order s ta tes ,

lk> =)zo„(Q)>

I n > = 101 ( q ,10 .27)

=) z o v , ( Q ) >

This implies that the initial and final states1 k >

and1 m >

are differentvibration al le vels within the same (normally the ground) electronic state, andthat the in te rmed iate s ta tes In> , in terms of which the virtual s tates areexpanded, are vibrationa l levels in electronically excited man ifolds (Fig. 10.13).Us ing this notat ion, the Ram an transi t ion am plitude becomes

E2 • <ZOCIPOniZnv"›<Znv"IiinOlZ0v> • ElCRm( CC) CC E

n,v"On ± W0v,nv" +

where

+E1 <xov , Ilionix„„->< znclonolzov> • E 2

WOn ± W Ov,nv" — W2

/ion =on(Q)

(10.28)

(10.29)

•

>

cu 2

exovi>

exov>

Figure 10.13 Energy level scheme for chemical appli-cations of ordinary Raman scattering. The first allowedelectronic state Ike> lies well above the energy tu a l of the

incident photon; the energy separations h(cu l — cv 2 ) cor-

respond to rotational/vibrational energies in the ground

electronic state le>.



RAMAN SPECTROSCOPY 323

i s a transit ion mome nt funct ion for the electronic transition I n > <-10>. Inanalogy to the R-dependent diatomic trans ition moment function M e (R) in Eq.4 .49 , p 0 (Q) depends on the vibrational coordinates Q in polyatomics. T he

quantity h won (E 0 — E n ) is the electronic ene rgy difference betwee n states l k >and I n > , and ho) 0,,,s the vibrational ene rgy differen ce betwee n states 1 k > andI n > . In a conventional Raman exper iment where 1 0 ) 2 — cod/w1 « 1 and1 ( 1 3 0 v % n v "1 « ICON' (Fig. 10.13), one may use the approximation c o 2 co condneglect the vibrational energy difference terms c o o ,, , n ,” in the energy denomi-nators , with the res ult that the Ram an trans i t ion ampli tude takes on thesymm etrical form

E2 * <ZovilioniXnv-><XncilinoiXov> E1c(co) cc E

n, y"On +

(E2 . /0nOn0 •El)+

(El' l io n X i in o • E 2 )< x . „ , 1 Ec o v > (10 .30)wOn ±o o n —

T he polarizability tensor Œ ( a ) ) for a molecule in electronic state 1 0 > subjected to asinusoidal electric field with circular freque ncy a) has com pone nts

(liOnViinA

t iOnVi inO)iŒij(W)hco on ±hW hCOOn (10.31)

This bears a family re sem blance to expression (1.35), which was derived fo rthe polarizability of a molecule subjected to a static electric field ( a ) = 0).Equation 10.31 is a generalization of Eq. 1.35 for the dynamic polarizability inthe presence of applied fields with non zero f requency a). (The dynamicpolarizability i s reduced when the applied frequency is increase d, because theelectronic motions in molecules cannot react instantaneously to rapid changesin the external electric field.) A comparison of Eqs . 10 .30 and 10.31 then shows

that

d ( 00) CC E2 <ZOIYIMMIX0v> "El10.32)

which is to say that the transi t ion am plitude is proportiona l to the matrixelement of the frequency-depende nt m olecular e lectric polarizability takenbetween the initial and final vibrationa l states . T he polarizability tensor *co)generally depend s on Q as we ll, through the Q-dependence of , i 0 in Eq. 10 .29—otherwise, Eq. 10.32 tells us that no vibrational Raman transi tions would takeplace betwee n the or thonormal Born-Oppenheimer states 1 k > and 1 m > .

W e may o btain working selection rules for vibrational Raman transit ions byexpanding the molecular polarizability t ensor in the norm al coordinates Q.

)1 . <X0v1POniXnv"›<Xnt,"10nOlZ0v> . E2+

W OnI)




about the mo lecular equilibrium ge om etry,

a t x a 2 o tiQ ; ± • •- - - c t o + EQi ± 2Qi 0 Q.;)0

so that

a t :<xocIcelx0v> = 0(06„,, +x0„ , 1i 1x0 , )

12ot+x o t , , 1 4 ,3Q,a.2)QiQ i1x> + • • •

ii

(10.33)

(10.34)

Hence , the R aman selection rules on Av depend on the re lative magnitudes ofthe derivatives of a in different orders of

Q.The Raman fundamentals

(Av = + 1 ) arise from the terms l inear in Q i . Symme try selection rules may bederived by noting that quantities like (aa/0Qi ) 0 Q i and (a 2 a /Niagdos2e2 ;t ransform unde r point group operat ions l ike a i t s e lf ; any co mpone nt a i i of at rans forms like x i x ; (i.e., c like xy, etc). Hence , Raman funda-me ntals generally arise only from normal mode s Q. which transform unde r pointgroup operat ions l ike l ine ar com binations of x i x i such as (x 2 — y 2 ) , z 2 , xy .

Allowe d vibrationa l Raman transi t ions involve only modes that change themolecular polarizability (the right sid e of Eq. 10 .32 would otherwise reduce intoE2 71((0) • El<X0v1X0v>5 which is proportional to ö v ) . Case s do ar i se in which

Raman fundamentals are symme try-allowed, but very weak. For ex ample, alkalihalide m olecules consist of pairs of oppositely charged ions whose electronicstructures are nearly insen sit ive to the internuclear separation R fo r R nea r Re .

T he alkali halide molecular polarizability

a a(M + ) + Œ ( X )10.35)

i s thus near ly R-indepen de nt , and alkali halide s e xhibit very we ak Ramanscattering e ven though their Raman fundamentals are symmetry-allowed. (Theirdiatomic vibrational motion is of course totally symmetric in C., and the E

irreducible represe ntat ion t ransforms as both (x 2 + y2 ) amd z 2 . )For a nontrivial example of Raman symme try selection rules, we consider the

molecular vibrations in p-difluorobenze ne . W e list in Table 10 .1 , next to each ofthe irreducible representations in D2h, the number of vibrations o f that speciesex hibited by p-difluorobenze ne , the vector and tenso r compone nts that t rans-form as that repres en tation, and the corresponding infrared (one-photon) andRaman activity expected for fundamental transit ions in the pertinent vibrationalmodes . Table 1 0 .2 gives an analysis of the p-difluorobenzene Ram an spectrashown in Fig. 1 0 .1 . These tables illustrate the point that in centrosymm etricmolecules, sym metry-allowed Raman fundamen tals appear only in modes with

g symm etry, while the in frared-active fundamen tals occur only in modes of u




Table 10.1 Molecular vibrations in p-difluorobenzene

Irreducibleumber of

representationibrations

Raman

ag 2 , y 2, Z 2

big yb 2 g b 3 g za u 1 3 1 u b2 u b 3 u Table 10.2 Assignments of p-difluorobenzene Raman bands

Liquid Vapor

v (cm - ')Relativeintensity Assignment v (cm

Relative1 )intensity

1 6 1 9 ( 11) v2(a g ) 1 6 1 5 (3 )1 6 0 6

( 3 ) [2 v9

(b ig)], A g1 3 8 8 ( 1 ) [2 v 1 6(b2g)], Ag

1 2 8 5 ( 3 ) V25(b3g)

1 2 4 4 (42) v 3 (a g ) 1 2 5 7 (30)114 1 ( 14 ) va(a g ) 1 1 4 0 ( 5 )

1 0 1 9 (2 ) Ev17(b2 g ) + V26(b3g)], Big 1 0 1 3 (2 )89 5 ( 1 ) [2 v 6 (a g )] , Ag 8 97 (2 )

[21'270 3ga Ag 8 6 7 ( 16 )8 5 9 (68) v 5 (a g ) 8 5 9 (73)8 4 0 (89) [2v8(a u )] , Ag 84 0 (35)7 99 (2 ) v9(bi g )

6 3 6 (23) v26(b3 g ) 6 3 5 (6 )451 (84) v6(a g ) 450 (37)3 76 (47) v17(b2 g ) 3 7 4 ( 9 )

symm etry. Hen ce, vibrat ional modes ex hibiting in frared and Raman funda-me ntals be long to m utually exclusive sets if the molecule has a center of

symm etry; this fact has bee n used as evidence fo r assigning molecular geometry.Laser Raman spec troscopy has found wide applications in environmentalchemistry, biochemistry (e.g., conformational analysis of proteins), and medicine

as well as in chemical physics.A specialized situation called resonance R a m a n scattering ar i ses when the

laser frequency co l is tuned close to r e sonance wi th one of the molecular




eigenstates I n > =(Q)>, as shown in Fig. 10 .14 . T he energy denomi-nator ( c o - I - c oOn • — Ov,nv" w1) in E q. 10.28 becomes very small relative to its valuein ordinary Raman scattering, and the t rans i t ion probability (which is pro-portional to j e „ , 1 2 ) becomes anomalously large. In this limit, the vibrationalenergy differences w 0 , cannot be ignored next to the other terms in the ene rgydenominators, with the result that the trans i tion amplitude c(co) no longerreduces to the symmetrical form (10 .30) . Hence , the assumptions leading up toE q. 10.32 (which gives the Ram an t rans i t ion ampli tude in terms of a matrixe l e m e n t of Œ(w)) fall through: T he ordinary Raman selection rules are notapplicable to resonance Raman transi tions. It turns out that some transit ionsthat are forbidden in ordinary Raman become allowed in resonance Ramanspectroscopy (Problem 10.4). Time-resolved resonance Raman (T R 3 ) scatteringhas been developed into a useful technique for m onitoring the populat ions oflarge molecules in electronically excited states (Fig. 10.15). Such excited-state

populations might be more conventionally probed by studying the evolution ofS „ 4- S 1 or T„ 4-T 1 one-photon transient absorption spectra on the ns or ps t imescale. These transient spectra tend to be featureless (due to spectral conges t ion)in photobiological molecules and transi t ion metal complexes. T R 3 scattering isa more advantageous probe, because the resulting spectra exhibit sharpvibrational structure similar to that in Fig. 10.1 . T he enhanced sensitivityinherent in T R 3 can be rendered specific to an excited state of interest by tuningthe probe lase r f requency co l , because each excited state will be uniquely spacedin energy from higher-lying electronic states.

In Chapter 8, we characterized the strengths of one -photon transi t ions in

terms of Einstein coeff ic ien ts and oscillator strengths. According to Beer's law, aweak light beam with incident intensity / 0 will emerge from an absorptivesample of con centrat ion C and path length 1 with diminished intensity / = / 0

e x p ( —C1), where a is the molar absorption coeff ic ient . Bee r 's law can be recas tin the form

I=10.36)

•••

Figure 10.14 Energy level scheme for resonance

Raman scattering. The incident photon energy ha is innear resonance with a vibronic level in some electroni-cally excited state Iv>; the energy differences

h(tu i — (2)2) correspond to rotational/vibrational energies

in the ground electronic state 100>.




Figure 10.15 Resonance Raman detection ofpopulations in electronically excited states S, andT., following creation of S, state molecules by laserexcitation at frequency W e . Probing S., molecules

at frequency ca l will generate intense resonanceRaman emission at t2/2 if the S2 4— S, transition is

El -allowed, because Li), is in near-resonance withthe energy separation between vibrationless S, andsome vibronic level of S 2 . Probing at frequency c u . ,

will generate similarly intense emission only ifappreciable population has accumulated in T., byintersystem crossing from S,, since wis in re-

sonance with the T, — T, energy gap. This excited-

state selectivity of resonance Raman scattering hasrendered it a useful tool for monitoring time-resolved excited state dynamics.

where N is th e molecule n umber dens ity in cm -3 and a is the absorption crosssection. This cross section, which has units of area, is related to the absorptioncoefficien t by

a(A 2 ) = 1.66 x 10 -5cx (L mo1 -1 cm -1)

= 3.83 x 10 -5e (L mo1 -1 cm -1 )10 . 3 7)

The physical picture suggeste d by the concept of a cross section may beappreciated by visualizing the photons as point particles impinging on a samplecontaining N molecules per cm 3 . Eac h molecule is imagined to have a well-defined c ros s sectional area a. A photon is absorbe d when it "hits" a molecule,but is trans mitted if it traverses the path length 1 without sco ring a hit. Underthese conditions, the fraction N o of photons that are transm itted will beexp( — Nul) . For strongly allowe d one-photon transitions, a is some what smaller

than the molecular size: The absorption cross section for rhodamine 6G at5 3 0 0 A ( c m a x = 1 0 5 L mo1 -1 cm 1 , Problem 8.2) is 3 .8 A 2 .

Raman scattering intensities (which are proportiona l to 41 2 ) are commonlyexpressed in terms of cross sections. The differential cross section d u l d S 2 2 isdefined as

du dN„Idf2 (10 .38)6 1 1 2 2N inc idA

where di i i n c is the number of photons (k 1 , co 1 ) which traverse the area elemen td A normal to wave vector k 1 in the incident beam, and dNs c is th e number ofphotons scattered into the solid angle element c/[22 = s in 0 2 04 (Fig . 10 .16) .




Laser

Raman

Figure 10.16 Raman scattering geometry. The laser is incident along the z axis,and the Raman emission is scattered into the volume element ci f -1 2 = sin 0 2 c/ 6 1 2 d02 .

The total Raman cross section

2 Is ( a )

a=f in 0 2 d02 dcP

0(10 .39)

ha s an interpretation s imilar to that of the one-photon absorption cross section:a is related to the fraction N o of incident photons that rem ain unscattered aftertraversing path length 1 in the sample, vi a ///0 = ex p( — No- 1 ) . I t may be shownthat th e differential cross section for Raman scattering is [1]

E (E2 <m111 10<nlfilk> Ei ±E 1 . Onli l in>07 1 f l i k > • t2)

Wkn W1

kn (0 2(10 .40)

d o -. ( 0 3

df2 24ne 0 hc 2 ) 2

2

where Ê 1 , f are unit vectors in the directions of polarization of the electricfields associated with photons (k 1 , c o l ) (k 2 , w 2 ) . A special case called Rayleighscattering occurs when oi l = w 2 (i.e., the initial and final molecular states are thesame). The differential cross section for Rayleigh sca t ter ing i s obta ine d byreplacing c o 2 by co l in Eq. 1 0 . 4 0 , with th e res ult that the cross section becomesproportional to the fourth power of the incident f requency w. This pheno-me non is respons ible for the inimitable blue color of the cloudless sky , becauseth e shorter wavelen gths in the solar spectrum are preferentially s cattered by th eatmosphere . According to Eq. 1 0 . 4 0 , the scattered photon may propagate intoany direction k'2 in general. The angular distribution of Raman scattering(relative intensities of l ight scat tered into different directions k 2 ) may be ob-tained by averaging expressions like (10 . 40) over the orientational distributionof molecules in the sample. For conve ntional vibrational Raman transitionsexcited by visible light, a typical cross section doldr2 2 is on the order of 10 14A2,with th e result that only one photon in 10 9 is scattered in a sample with moleculenumber dens i ty N = 1020 cm

—3 and path length 1 = 1 0 cm. (The cross sectionsfor resonance Raman transitions are, of course, far larger). This is why in tense



PROBLEMS 329

light sources (preferably lasers) are required for Raman spectroscopy. Visiblerather than infrared lasers are norm ally used in vibrational and rotationalRaman spectroscopy, because the cross sections (which are proportional to

icol) and the photon detector sen sitivities are more advantageous in the visiblethan in the infrared.Raman line intensities are proportiona l to the number dens ity N of molecules

in the initial state 1k>, which is in turn proportiona l to the pertinent Boltzmannfactor for that state at thermal equilibrium. C onse quently, the relative intensitiesof a Stokes transition 1k> 1m> and the corresponding anti-Stokes transition1m> 1k> are 1 and exp( — h c o ,„ k /kT) , respectively. (The factor c o i col varies littlebetween the Stokes and anti-Stokes lines , because the Raman frequency shiftsare ordinarily small com pared to cop) Hence the anti-Stokes Raman transitions(which require molec ules in vibrationally excited initial states) are considerably

less intense than their Stokes counte rparts, particularly when the Raman shiftc o „ i k is large. In much of the current vibrat ional Raman literature , only theStokes spectrum is reported (cf . F ig. 1 0 .1) .

REFERENCES

1. D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics,

Academic, London, 1984.2 . M . D. Levenson and N . B l oemb erg en , Phys. R e v . Lett. 32: 645 (1974).

3 . F. Bi raben, B . Cagnac , and G. Grynberg , Phys. R e v . Lett. 32: 643 (1974).4. D. M . F ri e d r ich an d W . M . McClain, C h e m . Phys. Lett. 32: 541 (1975) .5 . M . M. Sushchinsk i i , Raman Spectra of Molecules and Crystals, Is rael Program fo r

Scientific Trans la t ions , New York , 197 2 .

PROBLEMS

1 . Two-photon 3s —) 5s absorption i s observed in Na vapor at 400 K withcounterpropagating laser beams whose frequencies co? and c o ? are no t quite

ident ical. H ow large must the frequency difference 1 c o ? — 041 be so that theDoppler contribution to the linewidth equals the Lorentzian contribution if the5s radiative lifetime is 1 0 n s ?

2 . What states in K can be reached by two-photon absorption from 44'1 1 2

level in the El approximation? To what term s ymbols are the intermediate statesrestricted?

3 . Show that expression (10.40) for the differential cross section in Ramanscattering has units of area as required.

4 . The acetylene molecule C 2 H 2 has five . vibrational modes, three nondegen-erate and two doubly de generate (Chapter 6 ).




(a) Several of this molecule's lowest energy vibrational levels are listed below.The polarizability function i s assume d to have the form

5 a i Q i + E biNiQ; + E

i= 1

j

jk

with no other terms . Which of the levels will be re ached by El-allowedRaman transition from th e 0 0 0 0 0 0 0 level?

aQ, • • • , Q5) =EV3

141 ,4

/55

V Symmetry

0 1' 0 H g0 0 1 1 fl u0 1 1 1 1 /U F

0 2 ° 1 1 nu0 0 2 ° Ag

0 0 0 E g+

0 0 3 1 1 1 „

0 0 33 (u

0 0 1 1 flu

1 0 0

0 1 1 1 1

0 0 0 E s+

0 2 ° 1 1 H u

0 0 0 E g+0 0 0 I+

0 0 0 E g+0 3 1 0 H g0 2 2 0 Ag

1, 12o 0 0 0

0 0 0 0 0 0 0 1 0 3 3 0 0 0 (b) Consider the hypothetical case in which the laser frequency co l i s tune d

close to th e lowest E l-allowe d e lectronic transition in C 2 H 2 , so thatresonance Ram an emission occurs and the polarizability expression (10.32)for the Raman transition probability amplitude is no longer applicable.

Which of the vibrational leve ls listed above c an be re ached from 0 0 0 0 0 0 0ace tylene by El symme try-allowed res onan ce Raman transitions, eventhough they cann ot be reached by conve ntional Raman transitions?

(c) T he frequency of mode 2 in acetylene is 1974 cm -1 . W hat will be theapproximate ratio of intensities in the Raman fundamentals of this mode i nthe Stokes and anti-Stokes branches in a 3 0 0 K sample?



1 1NONLINEAR OP TICS

The discussion of Raman and Rayleigh scattering in Section 1 0 . 3 was based on

the t ime-dependent perturbation theory of radiation—matter interactions devel-oped in Chapter 1 . The scat tered l ight inten si ties we re found to be l inear in the

incident laser intensity; the scattered Raman frequencies we re shifted from the

laser freque ncy by mo lecular vibrational/rotational freque ncies . Identical con -clusions may be reached using a contrasting theory which treats the polarizationof bulk media by electromagnet ic f ields classically. S uch a classical theoryprovides an insightful vehicle for introducing the nonlinear optical phenom enadescribed in this chapter, and so we begin by recasting the familiar Raman andRayleigh scattering processes in a classical framework.

The total dipole moment p of a dielectric material con tained in volume V i s

given by the volume integral

p=d r11.1)Jv

of the local dipole moment density P [1 ] . Def ined in this way, P (which isnormally called the polarization) ha s the same units as electric field. In a materialcomposed of nonpolar molecules or randomly or iented polar molecules, P

vanishes in the absence of perturbing fields. In the presence of an externalelectric field E , the polarization becomes

P s o x E11.2)

where x is the dimensionless electric suscep tibil ity tensor. This expression givesthe correct zero-field limit P = O. However, the polarization is not necessarily

331



332ONLINEAR OPTICS

linear in E, because the susceptibility itself may depend on E. We shall see thatthis nonlinear i ty fo rms the basis for the phenomena in this chapter. Forisotropic media (gases , l iquids , and most amorphous solids), the susceptibil i tyreduces to a scalar function z(E), and the polarization P points in the samedirection as E. In many crystalline solids, the induced polarization does not

point along E, and z is an anisotropic t ensor . Equation 11 .2 close ly rese mblesthe expression for the dipole moment induced in a molecule by an electric field,

Pind = a • E . For atoms and isotropically polarizable molecules at low densitiesN (expressed in molecules/cm 3 ) , the bulk susceptibility is clearly related to themolecular polarizability by

= N a/go11.3)

At general number densities where the total field expe r i enced by a molecule maybe influenced by dipole moments induced on neighboring molecules, thesusceptibility is given instead [1 ] by x = (Nale0 ) I (1 — N a /3 0 ) .

We now consider an electromagnetic wave with time dependenceE = Eo cos co o t incident upon a sys t em of isotropically polarizable molecules.For s implici ty, we assume the molecules undergo classical harmonic vibrationalmotion with frequency w in some totally symmetric mode Q. The normalcoordinate then oscillates as Q = Q0 cos(cot + 6 ) , where 6 is the vibrationalphase and Q 0 is the amplitude. If the molecular polarizability a is linear in Q (asa special case of Eq. 10.33), the vibrational motion will endow the molecule withth e oscillating polarizability

Œ = a o + Œ 1 cos(cot + 6 )1 1 . 4 )

Ignoring the vibrational phases 6 (which will be random in an incoherent lyexc i t ed sys t em of vibrating molecules), th e polarization induced by th e ex t e rna lfield will be

P = eo x E

=N(Œ 0 +a l cos wt)E 0 cos c o o t

{N E o { Œ o cos c o o t + —2 Lcos(o o + w )t + cos(w o — co)t] ( 1 1 . 5 )

According to the classical theory of radiation [1] , an oscillating dipole m o m e n tp will emit radiation with an electric field proportional to its s e c o n d timederivative P. Equations 11.1 and 11 .5 then imply that radiation will be sc at teredat the f requencies co o , w o — w, and co o + co, corresponding to Rayle igh, Stoke sRaman, and anti-Stokes Raman scattering, respectively. The sca t te red e lec t ricfields are proportional to E 0 , so that the Rayleigh and Raman in tens i t i e s arelinear in the incident laser in tens i ty. Expressions similar to Eq. 11.5 are



NONLINEAR OPTICS 333

frequently cited in classical treatments of Raman scattering [2, 3]; they em-phasize the central role of Q-dependent molecular polarizability, and theydemonstrate heuristically how Raman-scattered light frequencies are shifted bymolecular frequencies a) f rom the laser f requency co o .

As lasers with high output powers became accessible to spectroscopists in the1 9 6 0 s , conspicuous nonlinearities emerged in the polarization P ( E ) induced byintense fields. The components Pi of P may be expanded in powers of

components E» Ek, E i of the electric f ield E via

3 (

V „(1)E _j_ V ,(2) X (3 ) E- -Pi 1 - - 0 Z ., Aij 1-'j -1 - Z., ItEjE +E Ek 1+ikkl

(11 .6)

The linear susceptibility x" ) gives rise to the Raman and Rayleigh proce s s e s

treated in Chapter 10; it dominates the polarization in weak fields. As the lightintensity is increased, the responses due to the nonlinear susceptibilities d ,d,•.. gain prominence. A discussion analogous to the one culminating in E q.11 .5 shows that scattering may occur at frequencies that are multiples of thelaser f requency. A process controlled by the second-order nonlinear sus-ceptibility d is second-harmonic generation (SHG), whereby two laser photonsat frequency c o o are converted into a single photon of f requency 2(0 0 . (A relatedprocess is sum-frequency generation, in which laser photons c o l and c o 2 arecombined into a single photon with frequency c o l + co 2 .) The third-ordernonlinear susceptibility d is responsible for third-harmonic generation,

c o o + c o o + coo —> Ro o . It also leads to coherent anti-Stokes Raman scattering(CARS), which is treated in Section 11 .3 . Generation of nth-harmonic freq-uencies is governed by the nth-order nonlinear susceptibility x ( " ) ; ninth-harmonic pulses have been generated by 10.6-prn CO 2 laser pulses in nonlinearmedia.

B e c a u s e the scattered light intensities occasioned by the second- and higherorder terms in Eq. 11.6 increase nonlinearly with the incident light intensity,higher order contributions to the susceptibility become important at suff ic ien tlaser powers. SHG conversion eff iciencies of 20 % from 1064 to 532 nm areroutinely achieved in pulsed Nd Y AG lasers, and were unimaginable prior to

the invention of lasers.Explici t formulas for the nonlinear susceptibilities x ( P , xV4, . . . may be

derived by working out the coefficients c L3 ) (t), c (nt ) ( t ) , , respectively, in the t ime-ordered expansion (1.96). Straightforward evaluation of the integrals in the t ime-ordered expansions rapidly becomes unwieldy, and an efficient diagrammatictechnique is developed in Section 11 .1 for writing down the contributions toc(t) that are pertinent to any multiphoton process of in te res t . In Sections 1 1 .2and 11 .3 , we apply this technique to obtaining the nonlinear susceptibilities fortwo important nonlinear optical processes, SHG and CARS. Experimentalconsiderations that are unique to such coherent optical phenomena are alsodiscussed .



1c ( 3 ) t =( ih) 3

e -iwPm"<mlif(t i)IP>dtnt <pl W(t2)In>dt2i t t

3 3 4 NONLINEAR OPTICS

11.1 DIAGRAMMATIC PERTURBATION THEORY

T o illustrate the simplifications int roduce d by d iagramm atic perturbationtheory [4], we consider the three-photon process es corresponding to the third-order term in the Dyson expansion of c m ( t ) ,

ft,x0 k- t 3 <nIW(t 3 ) 1 k > d t 3

to(11.7)

W e may a s soc ia te the perturbations W (t 1 ), W(t 2 ) , and W (t 3 ) with vec torpotentials for elec t romagne t ic waves with frequen cies c o i , w 2 , and (03

respectively:

ih q<mlw(t 1 )1p>mc (ccmPe

-i(itimpe iwoi)

ihq< P I M t 2 )In > = 2m c

(cc'e - iw2t2ne i t )2t1

ih q<niW(t 3 )1k> --=

2mc(o c

" ke - i w 3 t 3 + ak e i 0 ) 3 t 3 )

(11.8)

Substitution of these matr ix e lements in to Eq. 11 .7 then yields terms in c(t)proport ion al to the eight products

Œmp°IpnankmpapnC(nk

Implpnt5Cn kpŒpnank

oempapnank

Xy n pa pn Cl n k

ampŒpnank

mpaptAnk

These correspond to the eight time -ordered graphs (a ) through (h), respectively,in Fig. 11 . 1 . These are only a small fraction of the possible third-order graphs,because the arbitrary assignment of vector poten tials to perturbations W(t i ) inEqs . 11 .8 is only one of six permutations of c o l , co 2 , (0 3 among the W(t). Hence ,c(t)contains 48 t ime -ordered graphs, of which only eight are shown in Fig.1 1 1 .

W e n ext evaluate the terms in C(t) correspon ding to the first two time -ordered graphs. The contribution from graph (a) in the cw limit is

q31m

3.3E

a m p Œ p n C C n kxp[ — i(co p m + co l ) t i ]dt ixp[ — i(co n p + co 2 )t 2 ]dt2

p noto



DIAGRAMMATIC PERTURBATION THEORY 335

x expE- to(WIcn ± W3)t3Nt3

— 27 -c q 3

8M 3 C 3

ampŒpnInk

Ph (Wkp ± W2 ± (03)(Wkn + (0 3)( (cokm + Wj + (0 2 + (0 3) (11.9)

The delta function in (11.9) implies the ene rgy conservationE.— Ek = h(0)1 ± (02 + (03) pertinent to three-photon absorpt ion . This is con-sistent with graph (a), which shows photons c o l , 0 ) 2 , c0 3 incident at early times,and no photons scattered at long t imes . T he co ntribution from graph (b) is

t2

8m 3 c 3 E ampŒpAkxp[—i(co p . +co i)tidt i

Iexp[ —i(co n p + co 2 )t 2 ]dt 2xp[ — i ( w k o o 3 ,t3 _.dt 3

too— 2 n q 3a m p ° ( pnkk

6(Wk m + W1 ± W2 —W3) ( 11 . 10 )8M3C 3PhWkp (0 2 W3)(Wkn — (»3)

pn t

q 3

(b) (c) (d)

w2

(e) (f (g) (h)

w2 c o

Figure 11.1 Time-ordered graphs representing the eight contributions to 4) (t) in

Eq . 11.7 when the perturbation matrix elements are given by Eq. 11.8 .



336ONLINEAR OPTICS

( 0 3

Figure 11.2 Energy level diagram for the process representedby graph (b) in Fig. 11.1. The process is sum-frequency

1 m > generation when states 1 k > and 1 m > are the same, and hyper-1 k > Raman scattering when they are different.

T he ene rgy conse rving condition here is E. — Ek = h(Wi + W2 — (0 3), corre-spond ing to absorption of photons c o l , c o 2 and scattering of photon c o 3 (Fig.11.2). When the init ial and f inal molecular states 1 k > , 1 m > are identical, theprocess is sum-frequency generation [5 ,6] , an important gating technique used in

time-resolved laser spectroscopies. When the two states differ, the process ishyper-Raman scattering [4]; the ene rgy difference E m — Ek usually correspondsto a rotational/vibrational ene rgy se paration in gases, or to phonon frequenciesin lattices.

It is clea rly ted ious to e valuate 48 such integrals. T he number of graphsmushrooms as 2 n n ! with the pe rturbation order n : the four- and five-photonprocesses are ass ociated with 384 and 3840 graphs, respe ctively. The dia-grammatic technique 's great ut i li ty con sis ts in that it quickly isolates thos egraphs that contribute to any given multiphoton process. It also provides simplerules fo r generating express ions like Eqs . 1 1 .9 and 1 1 . 1 0 directly from the graphs,

without recourse to e xpl ic i t integra t ion. T hese rules (which should be s el f -evident by induction to readers who have retraced the steps leading to Eqs . 1 1 .9an d 1 1 . 1 0 and studied the accom panying graphs) are:

1 . For a given multiphoton process, decide which frequenc ies are inciden tand which are scattered. In sum-frequency generation, fo r example, one canstipulate that frequenc ies o h, c o 2 are incident and frequency ( 0 3 is scattered; thefrequencies obey the conse rvation law (0 3 = c o l + co 2 .

2 . W rite down all of the graphs consis tent with these as s ignme nts offrequencies . Thes e graphs will exhibit n interaction vertices (Section 10 . 1 ) in ann-photon process . In our sum-frequency generat ion exam ple , there are sixdistinct graphs having incident frequencies oh, c o 2 and scattered frequency c 0 3(Fig. 11.3). Only one of these graphs is contained in the set shown in Fig. 11.2.

3 . Each interaction vertex between s ta te s 1 p > and l > in any diagramcontributes a factor a i x i for a photon inciden t at that vertex, and a factor d e pq for aphoton scattere d at that vertex. (These quantities are defined in E qs . 10 . 5 -10 .6 . )T he diagrams (a) through (f) in Fig. 11 . 3 yield cumulative factors of 0 C mpapn

Impcxpnank, ŒmpElpnŒnk 9 ampapnŒnk, CimpŒpnInk, and i m p Œ p n Œ n k 5respectively.



Wl

o2

c o 3

0.)2u i

c o3

DIAGRAMMATIC PERTURBATION THEORY 337

(a)b )c)

c o 2o2

'

( d )e)f )Figure 11.3 The set of time-ordered graphs that can be drawn for the sum-

frequency generation process in which w, an d c .u 2 are the incid ent frequencies and

tu 3 is the scattered frequency. Six graphs result from permuting the three frequencies

among the three interaction vertices.

4 . Reading from bottom to top of the time line in each graph, each of the first( n — 1) interaction vertices contributes a factor of ( c o l a + a)) to the energydenominator. Here co k i = (Ek — E 1 )/h, 1 / > is the state lying above the vertex onthe time line, 1 k > is the initial state, and c o is the total photon e nergy "absorbed"at all vertices up to and including that vertex. The energy denominators for thesix diagrams of Fig. 1 1 . 3 are in order

(wkp + w1 — (0 3)(wk. — w3), (wkp + 02 — (0 3)(wk. — 3)1

(W k p (0 3 +1)(a kn W 1)5 ( W k p —(03±W)(W kn (02),

(Wkp ±2± (9 1)(Wkn W1), and (Wkp

2)(WknW2)



338ONLINEAR OPTICS

5. The contributions from th e graphs are summe d to give the total proba-bility amplitude. For the sum-frequency generation w 1 + co 2 (O3 , we have

c, n (oo) =— 2 7 rq 3C m pn5Cn k

(2M03C) k p W —(931‘W kn03)

CX m pn 6- n k

m p .5 ( p n Œ n k

▪ (a k p — (03 + W 2)(W kn (03) k(Dkp —W3 +WAD) kn W1)

Œ m p O E p n Œ n k( mpatpnank▪ ,

k (thp°3 d- (02)( 0)kn d- 0)2)c ok p -E 00 2 -E 00(04. -E ()1)

m p Œ p n n k

▪ ( W k p1 W 2)(W kn ± (0 2)

T hese rules provide an enormous labor-saving device fo r evaluating nonlinearsusceptibilities, as we s hall see in the last two sections of this book.

11.2 SECOND-HARMONIC GENERATION

Secon d-harmonic gene rat ion (SHG), th e special case of sum-frequency gen-eration where c o i = c0 2 co and w 3 = 2w, is an invaluable frequency upconver-sion technique in lasers [5,6]. Most near-UV lasers are frequency-doubled

beams originating in visible dye lasers, and Nd' :YAG-pumped dye lasers a reex ci ted by the 532-nm SHG rather than the 1064-nm fundamen tal from th eYAG laser. Autocorrelation diagnoses of pulse durat ions gen erated by mode-locked lasers also rely on frequency doubling.

It is clear from Eq. 1 1 . 1 1 that for SHG (in which lo> represents both the ini t ialand final state)

C(3irq) (CO) = Z. 4

(2/710 3 pn

o p C C ',nanoto p n Œ n o

(Wop — (*Don —2w) +[

(coop — 04(w +

E .( o pnŒno

(Wop ±ao)(co,„ + co)

(11 .12)

Since th e se cond -order no nlinear sus ceptibil ity is proportional to c ( 3 ) (co), w ehave in the El approximation

<obuilp><piiiiin><nigkio>oc E

p„ [ (w o w)(w o n — ao)

+ < 0 1 k + < o l t t i l P > < P I t t k I n > < n 1 P i l o > ]1 1 . 1 3 )(co o p — c o ) ( c o o n + co)c o o p + 2 c o ) ( c o o n + c o )



SECOND-HARMONIC GENERATION 339

where y i is the ith Cartes ian component of the electric dipole operator. T hesummations in (11 .13) are carried out over all s tates In>, I p > other than theground s tate to> . Efficient freque ncy -doubling will naturally occur on ly inmaterials that are t ransparent at both w and 2w, so on ly terms fo r which co p ° ,c o n , > ao will contribute to AP ,.

T he practical problem s asso cia ted with S H G may be appreciated byconsidering a class ical theory fo r wave propagation in the medium. Bycombining the Maxwell equations (1.37c) and (1.37d), we o btain the homog-en eous wave equat ion [1 ]

V x V X E + tt o s n E = 011 . 14)

fo r electromagnetic waves propagating in free space.inceV x V xE V(V E) — V 2 E and since V E = 0 in vacuum, the wave equationbecomes

V 2 E — p o s n E = 011 .15 )

This ha s solutions (Section 1.3) of the form

E(r, t ) = E e i( k11 .16 )

Wh en an elec t romagne t ic wave propagates through a frequency-doubling

medium, the homogene ous equation (11 .15) becomes superseded by [1 ,5]V 2 E — itog oE = itoP

where the source term in P reflects the fact that electromagne tic waves areradiated from re gions with os cillating polarization P. Owing to the source term,the plane wave (11 .16) is not a s olution to the wave equation inside the medium.Wh en the optical nonlinear i ty is dominated by second -order terms due to S H G ,the polarization is

P = coXE= 8 0( X W E X 2 ) E2 )

where we have assumed that the sus ce ptibili ty is isotropic to s im plify ouralgebra. T he wave equat ion then becom es

(0 2 E 2

VE — /1 0 8,,E = eol-to X (n E ± X

( 2 1

at2)

= - p o i'

The source term in (11.19) now con tains two contributions P 1 an d P2 due to thelinear and nonlinear susce ptibilities x ( 1 ) an d X ( 2 ) , respectively. As a zeroth-order



340ONLINEAR OPTICS

approximation to E, we may take the plane wave (11 .16) . The nonlinearcontribution to the source term is then

1 52 =0X(2)E?0,4e2i(k ir - wit)

1 1 .20 )

This equat ion asse r ts that as the incident wave of f requency c o l propagatesthrough the medium i t stimulates the radiation of new waves with frequencyc o 2 = 2co 1 a t every point r along the optical path. The SHG electric field d E 2

gene rated by the incident wave travelling through distance d r near r = r willthen have an amplitude proportional to [5]

(2) u2_2 _2((k - w")dro X

Note that the amplitude varies as E ?, so that the SHG intens ity is quadratic inthe laser intensity. Since the SHG radiation will itse lf propagate with wavevec-

to r k 2 , the infinitesimal field genera ted n ear ro will beEox(2)E? w i e 2i(kiro-col()e i[kAr-ro)]d E 2 c c

after i t has propagated from r o to som e arbitrary point r down the path. Thequantity k 2 (r — r o ) is the phase change accompanying SHG beam propagationfrom r o to r. The total SHG field observed at point r is the res ultant sum

f2 oc Eox(2)Ei co ie i (k2r-w2t) r ei ( 2 1 c l - k2)rod ro

J o

=Eo x (2)Eiwiei[k2r LOA r eiPki-k2,_ 1

L2) 1 1 .22)

of interfering waves generated betwee n positions 0 (correspond ing to th e edge ofthe medium) and r. T he wave vectors k l , k 2 are related to the fundamental an dSHG frequencies co l , w 2 by

= n i c o i /c1 1 .23 )

and

k 2 = n 2 a) 2 /c = 2n 2 0 .) 1 /c1 1 .24)

where n 1 , n 2 are the refractive indices of the medium at frequencies c o l , c o 2 . Sincemos t mater ials are dispersive (n 1 én 2 ) , one ordinarily finds 2 k 1 — k 2 0 0.He n ce the SHG amplitude will osc illate as exp[i(21c 1 — k 2 )r] — 1 , withperiodicity

i c = 7r /12 k i — k2I1 1 .25 )



COHERENT ANTI-STOKES RAMAN S C A T T E R I N G41

along the optical path. This oscillation, caused by interference between SHGwaves generated at different points along the path, se verely limits the attainableSHG intensity unless special provisions are made to achieve the index-matching

condition 2 k 1 — k 2 = 0. The distance l c , called the coherence length, is typically

several wavelengths in ordinary condense d media.In an index-matched medium, the coherence length becomes infini te , and

complete conversion of fundame ntal into second harmonic become s theoretical-ly possible. Index matching is no t possible in isotropic media with normaldispersion, where n 2 > n 1 . (In dilute isotropic media, the molecular polariza-bility a is proportion al to (n 2 — 1 ) according to the Clausius-Masotti equation.It is apparent from Eq. 10 .31 that the f requency-dependent polarizability a(w)increases when the optical frequen cy is increase d, in the normal dispersionregime where w i s smaller than any freque ncy w o n for El-allowed transitions

fromthe ground state 1 0 > to excited state In> of the medium. It then follows that

n 2 > n 1 in normally dispersive media.) It is possible in principle to achieveindex-matching in anomalously dispersive isotropic media, in which w is largerthan some w o n , and in which n 2 is not ne ces sarily larger than n 1 . Howe ver , suchmedia are like ly to absorb prohibitively at c o l (not to mention w 2 ) . Practicalindex matching is ins te ad achieved in birefringent crystals [6], in which therefractive indices differ for the two linear polarizat ion s and depend on thedirection of propagation. The incident laser fundam en tal is l ine arly polarized ,and the SHG emerges with orthogonal polarization. The direction of propa-gation is adjusted by aligning the crystal in a gimbal moun t so that the refractive

indices at c o l and c o 2 become equal for the respective polarizations. Large SHGconversion efficiencies can then be obtained.Still another criterion for an SHG medium is implicit in Eq. 1 1 . 1 8 , where the

presence of the nonze ro se cond-order n onlinear suscept ibi li ty X ( 2 ) implies thatthe polarization P cannot simply change sign if the electric field E is reversed indirection. (The inclusion of only odd-order terms varying as E, E3 , E 5 , ... in Pwould e nsure that P( — E ) = — P(E).) He n ce SHG is impossible in any mediumfor which reversin g E produces an equal but opposite polarization P. Suchmedia include iso tropic media (liquids, gase s) and crystals belonging tocentrosymmetric space groups. For these materials, X ( 2 ) vanishes by sy mme try.

By way of contrast, efficient third-harmonic gen eration is possible in isotropicmedia, and was dem onst ra ted man y years ago in Na vapor. A common SHGcrystal for frequency-doubling Nd' :YAG lasers is potassium dihydrogenphosphate (KDP), which belon gs to the noncentrosymmetric space group 42m.

11.3 COHERENT ANTI-STOKES RAMAN SCATTERING

Coherent anti-Stokes Raman scattering (CARS) is one of several four-photonoptical phenom en a that can occ ur when a sample is expose d to two intense laser

beams with frequencies c o l , w 2 . Some of the other phenome na, two of which areshown in Fig. 1 1 .4 , are the harmonic generat ion and frequency-summing



342ONLINEAR OPTICS --w

col

co l

col 3 w

i

a/2 20) 1 +0)2

1 k > 1k)

Figure1.4 Energyeveliagramsorhird-harmoniceneration

+ w1 +3w(left) and frequency summing c u . , + W , + W 2 --0 (2w, + w 2 )

(right), two of the four-photon processes which are possible in a material subjectedto two intense beams at frequencies w 1 and w2 .

processe s which yield th e scattered frequencies 3 w i , 2 co l + co 2 , c o l + 2w 2 , and3W 2 . In CARS, two photons of frequency co l are absorbed, one photon offrequency c o 2 i s sca t tere d via st imulated em ission, and a photon at the newfrequency c o 3 = 2co l — c0 2 i s coheren tly scat tered (Fig. 11.5) . It is apparent inthis figure that when the frequency difference w 1 — co 2 is tuned to match amo lecular vibration al/rotational en ergy level differen ce, c o 3 becomes ident icalwith an anti-Stokes frequency in the conventional Raman spectrum excited by a

laser at w . In this special cas e (called res onant CAR S), the scattered intensity atc o 3 exceeds typical intensities of Stokes bands in ordinary Raman scattering by

c o 3

— n>

l o >

P >

4 , 3

Nonresonantesonant

CARSAR S

Figure 11.5 Energy level diagrams for nonresonant CARS and resonant CARS.

The latter case occurs when (w., u . , 2 ) matches some molecular energy level

difference E„ ,— E o .



4- (0)0q -F 20)1 -- 0) 2) ( 0)0p -F 0) 10 2)(c000 -- 0)2 )

(0)0 0 - - 00 2 -- ( 0 3) 1)((00p -- (0 3 A- 0 ) 1 )(0)00 - E 0)1)

oq C lcqp(x pnano

C O H E R E N T ANTI -STOKES RAM AN SCA TTERING 343

several orders of magnitude. Resonant C A R S thus offers greater sensitivity and

improved spectral resolution over conventional Raman spect roscopy.As a first step in der iv ing expressions for CARS transition probabilities, we

list in Fig. 1 1 . 6 th e 1 2 time-ordered graphs corresponding to absorption of tw ophotons at co l and scattering of photons at c o 2 , c o 3 . Using the diagrammatic

techniques introduced in Section 1 1 . 1 , we immediately get for the CARS

contribution to th e fourth-order coefficient in the time-dependent perturbation

expansion (1 .96)

2irq 4 v,qa qp° p n C X noA RS( co) =(27)2c) 4 npq((0qo 2 A- 2co l )(co o p + 2 (0 1 )( 0 ) 00 -F 0) 1 )

6 - ( oq5Cqpapnano

▪ (Wog —(03 2(0A W o p +200(0n0)

5CoqŒqp&pnŒno

((00 q -- (02 - E 20) 1)(01 0p -- 0) 2 - E 0) 1 )( 0)0 0 - E col)

aoq; pnŒo

▪ (Wog 7- c o 3co l )(coopo 3 -F co l )(co o , -E c o l )

loqaqpapnŒno

(0)0 q4 ) 2 -- ( 0 3 -E 0) )(w -- 0)2 -F 0 0 1 )( (000 - E col)

loq6Cq p 6tpnŒno

- -ioqa(qp2pi: 6 6 n°

▪ (Woqth--()3)( 0) 0p d- 0 )1 - - 0) 3) (0)00 -- () 3)

C ito45( qP2pn6tno

( 1 1 . 2 6 )

( 0)0 q -- (0 3 - F 00 1 -- 0) 2)(coop A - 0 ) 1

loqaqpapnano

0)2)(()00 -- 0)2 )

(cooq) 2 d - o) 1 -- (9 3 )(0) 0p A -

c c o g a gi a p , r i „ „

( 0 3 )( c0 o o ( ) 3 )

(c)oq A- co,) 302)(C0 op -- ( 2 ) 3

loq2qp6Ipn -Cino

o) 2 )(c000 0) 2 )

((00 q -F 0) 1 -- 0 ) 2 - - 0) 3)( 0)0p -- 0 ) 2 - - 0 ) 3)( (0 00 ( 0 3)

This coefficient is proportional to th e third-order nonlinear susceptibilityresponsible for CARS. It may be simplified somewhat by defining



344

(e) (f)(g) (h)

() (k) (I)

cy 3

(b)

W2 W 3

(a)

ca l2.) 3

o

w iIFigure 11.6 The 12 time-ordered graphs that can be drawn for CARS with two

incident photons at frequency ca l and scattered photons at frequencies co 2 and w3 .

W I



COHERENT ANTI-STOKES R A M A N SCATTERING 345

c o l — co 2 = co3 — w 1 . In the El approximation, repeated application of theidentity 2 w 1 — co 2 — co 3 = 0 then leads to

[ter,u7P + tirti,,P)tifitir,,q 1 . 2 . (c o o q + w 3 ) ( w 0 p + 2(0101 0 +(0)

(Prir + / 7 n /Og g Prrifk" 1 174 + 2( 0).q + (0 2)(coop+ 2 (0 1)(won+ w 1)

+(wo g + 0 3)((0 o p + A)((00„ + (01)

tir tt7P pfq prin 1 4!4r+ (w o g + 0 2)(0 4„ — AX(0 o n + (0 1)

+(w o g — (0 i)((00p + zi)(0 )0. + (D1)

/IT i i i t P Prpqi l l °n y r PrPr

+

(0)0q— (01)(wL )p—Awi)+

(wo g + w3)(w0p + AX woo — w2)o n p r Prprr 7PPrur

+ (w o g +0 2X(00 p — zi)((0 o n — (03)

+

(Woq — CO 1)(aop+1 1 )(W0n — C°2)

Priglirill°llritr(1 174 ± fire)+ (Wog 7-- (4) 1 X W o p — A)((oon — ( D 3 )

+2 (Woq — W1)(W0p — 2w1)(w0n — w2)

i ir P lcI P (fir g liT 9 ± tirPr) - I ( 1 1 .27 )+ 2(co o q — wi)(wop — 2w 1 )( w 0 — W 3 )]

where we have used th e abbreviation 12 7" = <olpi ln > and sim ilarly throughout.The terms in Eq. 1 1 . 2 7 have been listed in the same order as the graphs ( a ) — ( 1 ) inFig. 1 1 . 6 . When the frequencies co l , c o 2 are tundd so that h(co i — w 2 ) hAmatches some molecular energy level difference Ep — E 0 (Fig. 11 .5) , the terms inxr : A I R sw proportiona l to ( c o o p + AY 1 become large. Thes e terms, which correspondto graphs (c), (e), (g), an d (i), a re responsible for the re sonant C A R S phenomenon:When the laser frequency c o 2 is swept across th e resonance conditionh(co l — co 2 ) = Ep — E o while w i is held fixed, the CARS intensity peaks sharplyat the value of c o 2 at which Stokes Raman scat tering off the laser frequency w iwould be observed. The

remaining(nonresonant)

terms inar contr ibute a weak background intens ity which varies little w ith c o 2 . Itmay be shown that the total scattering intens ity at w 3 is [4]

12 2 1 k 4 N21 13 =

6n 2 E6c2 if#03%;(c0ak(w2)Êi(0)1)fiRs1 2ijki

( 11 . 2 8)

H ere /1 , / 2 are the incident light intensities at co b co 2 ; N i s number of scatteringmolecules; Ê(w3 ) is the projection of the unit electric field vector at frequenc y 0 ) 3

along C artesian axis i, and similarly fo r f ( c o l ) , etc.; andaR s is averaged overthe molecular orientational distribution. The inten si ty is proport ional to thesquare of both the number of scattering centers and the third-order nonlinearsusceptibility.

,CARS. -3Aijkl



4 1 4 — k3 + k3cos u =

4k i k 2

4co 1 w 2 n3 — 4 w?(n3 — n?) — w3(n3 — n3)(11 .30)

4w 1 w 2 n i n 2

346ONLINEAR OPTICS

Generat ion of a CARS signal requires that the momentum conservationcondition

2 k 1 = k 2 + k 311 .29)

be satisfied for the incoming and scattered photons. T he magnitude k i of eachwave vec tor is given by n i c o i lc , where n i is the sample me dium's re fractive i ndexat frequency c o i . In dilute gases, where th e refractive index dispersion is low (i.e.,n i is relatively inse ns itive to w i ) , one automatically satisfies (11.29) by using acollinear beam geometry in which all three wave vectors are parallel. Therefractive indices depend appreciably on the frequencies w • in liquids, however.In this case , i ndex matching may be achieved by crossing the incident beams atan angle 1 9 which achieves m omen tum conservation (Fig. 11.7). From th e law ofcosines, the required angle 1 9 is given by

O wing to this m ome ntum conservation, the CARS signal with wave vector k 3 isdirectionally concentrated in a laser beam with a divergence of typically 1 0 -4

his contrasts with conventional Raman spectroscopy, in which th es ignal is dispersed over 4n s t e rad ians . C A R S is thus an advantageous techniquefo r studying vibrational transitions in samples where the scattered signal ofin te res t i s accompanied by fluorescence background, a problem frequentlyencountered in biological systems . Its directional se lectivity, combined with th ein tens i ty enhancem ent encountered in re sonant C A R S , r ender s i t sensi t iveeno ugh to de t ec t ga ses a t pressures down to — 1 0 - " atm. Disadvantages ofCARS include the need for a tunable laser to sweep w 2 (a s ingle-wavelength lasersuffices in convent ional Raman spectroscopy), and the sensitive alignmentsrequired for momentum conservation in condense d samples . The ultimate

k2

Figure 11.7 Index-matching geometry for conservation of photon momentum inCARS, 2k 1 = Ic2 + k„ The experimental angle between the laser beams at freq-

uencies 0., 1 anda/ 2 must be adjusted to the value 0 given in E q. 11.30 for observationof CARS; the scattered anti-Stokes signal emerges in the well-defined direction k3.



02 (1Ag )diss

=266nm

r n jAt

1425445465 1485

COHERENT A N T I -S T O K E S R A M A N SCATTERING 347

I 1335355375395415

Raman sh -Ft (cm-1 )

Figure 11.8 Vibrational Q-branch CARS spectrum of a 1 A 9 02 produced by 03

he bands originating from V = 0, 1, 2, and 3 are

centered at 1473, 1450, 1428, and 1403 cm -1 , respectively. Fine structure arises

from rotational transitions with AJ = 0. Reproduced by permission from J. J.

Valentini, D. P . Gerrity, D. L. Phillips, J. C. Nieh, and K. D. Tabor, J. Chem. Phys. 8 6 ;

6745 (1987).

limitation on CARS sensitivity is imposed by b a c k g r o u n d s c a t t e r i n g a r is in g

f r o m th e nonresonant t erms in E q. 11.27.

An exam p l e o f CAR S d e t ec t ion o f mo le cu le s a t l ow d e ns i t ie s i s g ive n in Fig.

11.8 , which shows the CARS spectrum of a l A g 0 molecules created by

p h o t o l y s i s of ozone [7],

030(a l A g ) + 0( 2 P)

The ozone is photolyzed by a 266-nm (near u l t ravi o l e t ) la s e r pu l s e . Th e fixed

pump fre quency co l fo r C A R S d e t e c t i o n o f t h e n a s c e n t 02 m ol ecul e s i s pr o v id ed



348 NONLINEAR OPTICS

by 532-nm second-harmonic pulses from a Nd:YAG laser. Tunable pulses (57 2—5 7 8 n m ) from a dye laser provide the variable probe frequency c o 2 . The Ramanshif t 4= co l — o) 2 is plotted as the horizontal coordinate in Fig. 11.8. TheCARS transitions (which obey the selection rules dv = 1, ziJ =0 between states

J o > and I n > ) originate from vibrational leve ls y" = 0 through 4 in a l A g excitedstate 02 molecules produced in the photodissociation. The rotational Q-branchlines appear at different frequencies for different Jowing to differences in B " , B 'for the lower and upper vibrational levels in each CARS transition. The lineintensities may be analyzed to yield the rotational/vibrational state populationsin the 02 protofragments, which reflect on the dynamics of the photodis-sociation process. (CARS line intens ities are not proportional to s tate popula-tions (cf. Eq. 11 .28) ; peak intensities for transitions connecting levels (f', J") and

(y', J') vary as [N(v" , J" ) — N(v ' , .1 )1 2 .) The alternations in CARS rotational lineintensities shown in Fig. 11.8 are not a consequence of nuclear spin statistics in

1602 , since a l A g 02 (unlike )( 3 E0, Section 4.6) can exis t in either even- orodd-J levels . Rather , they ind icate a propensity for selective 03 photodis-sociation in to even J leve ls . M easurement of these s ta te populations byconventional Raman spectroscopy is not feas ible, since the initial 0 3 pressure i s

only 1 torr. Laser-induced fluorescence (in which the photofragment moleculesare excited to a higher electronic s tate, and the res ulting rotationally re solvedfluorescence band intensities are analyzed to determine the state populations) i s

more sensitive than CARS. H owever, this technique requires an El-accessibleelectron ic state that can be reached by a tunable lase r . There is no such s tate in02 , which begins to absorb stron gly only in the vacuum UV. Hence, this

ex ample illustrates the generality as well as se nsitivity of CARS.

REFERENCES

1 . M . H . Nayfeh and M. K. Brus se l , Electricity and Magnetism, Wiley , New York, 1985.

2 . G . W. King, Spectroscopy and Molecular Structure, Holt , Rinehart, & Winston, NewYork, 1964 .

3 . J. I. Steinfeld, Molecules and Radiation, 2d ed. , MIT Press, Cambridge, MA, 1985.

4 . D. P.Craig

and T. Thirunamachandran ,Molecular Q u an t u m Electrodynamics,

Academic, London, 1984.

5 . G . C. Baldwin, An Introduction to Nonlinear Optics, Plenum, New York, 1969; N .Bloembergen, Nonlinear Optics, W . A. Benjamin, Reading, MA, 1965 .

6 . A. Yariv, Quantum Electronics, 2d ed., Wiley, New York, 1975.7 . J. J. Valentini , D. P . Gerrity, D. L. Phillips, J.-C. Nieh , and K. D. Tabor, J. Chem.

Phys . 86 : 6745 (1987) .



Appendix A

F UN D AM E N T AL C O N ST AN T S

Atomic mass unitElectron rest m a s sProton rest mass

Elemen tary charge

a m u = 1 . 6 6 0 5 6 5 5 x 1 0 -2 7 kgm e = 9 .1 0 9 5 3 4 x 1 0 -3 1 kgn ip = 1 . 6 7 2 6 4 8 5 x 1 0 -2 7 kg

e = 1 . 6 0 2 1 8 9 2 x 1 0 -1 9 coulomb (C)Speed of light c = 2 .9 9 7 9 245 8 x 10 8 m/sPermittivity of vacuum E o = 8 .8 5 4 1 8 7 8 2 x 1 0 -1 2 C2 /J • mPermeability of vacuum = 1 .2 5 6 6 3 7 0 6 1 4 x 1 0 -6 henry/mPlanck's constant h = 6 . 6 2 6 1 7 6 x 1 0 s

h = 1.0545887 x 10' J • sFree electron g factor g e =2 .00231931Rydberg constant R = 1 . 0 9 7 3 7 31 x 1 0 5 cm -1Bohr radius = 0 . 5 29 1 7 7 0 6 x 1 0 ° mFine structure constant Œ = 1 / 1 3 7 .0 3 6 0 4

Avogadro 's numbe r NA = 6 . 0 2 2 0 4 5 x 1 0 2 3 /molBoltzmann constant k = 1 . 3 8 0 6 6 2 x 1 0 -2 3 J/K

Data taken from E. R . Cohen and B . N. Taylor, J. Phys. C h e m . R e f Data 2 : 66 3 (1973).

349



Appendix B

ENERGY CONVERS ION

FACTORS

1 J = 6.24146 X 10 1 8 eV

= 5.03404 X 10 2 2 cm -1

= 1.43834 x 10 2 3 cal/mol

1 eV = 1.60219 X 10 -1 9 J

= 8 0 6 5 . 4 8 cm -1

= 2.30450 x 104 cal/mol

1 cm -1 = 1 . 9 8 6 4 8 X 10 -2 3 J

= 1 . 2 3 9 8 5 X 10 -4 eV

= 2.85724 cal/mol

1 cal/mol = 6.95246 x 10 -24 J

= 4.33934 x 10 -5 eV

= 3.49989 x 10 -1 cm-1

1 erg = 10 J

351



Append ix C

MULTIPOLE EX PANSIONS OFCHARGE DISTRIB UTIONS

The general problem is to evaluate the electrostatic potential OW at some pointr due to the presence of a molecular charge distribution p(r). Accordin g toclassical electrostatics, it is given in S I units by

1 C p(r)de4)(r) —4n e 0 j ir — ri

(C.1)

In consequence of the identity (J. D . Jackson, Classical Electrodynamics, Wiley,New York, 1 9 6 2 )

19 = 4 7 r E mi-i+ 1 Yte, 0 1 )/71 .0 9 , )C.2)I r —øm—i L i ,,

the charge distribution OW may be expressed as

1 /Ef r(0' , Or'p(r)dr'l Yon(0,

r1 + 1C.3)E0 im + 1

where r = (r, 0, 4)), r ' = (r', 0', 4l), and r, (r< ) is the greater (lesser) of r and r' .Equation C.3 is the well-known multipole expansion of the electrostatic potentialOr). It is particularly use ful for evaluating the electros tatic poten tial at distancesr which are large compared to the distances r' over which the molecular charge

distribution is apprec iable (Fig. C.1). In this long-range limit, the expansion (C.2)converges rapidly with r, = r a nd r < = r'.We may define the multipole moments

353



354ULTIPOLE EXPANSIONS OF CHARGE DISTRIBUTIONS

Figure C.1

q t of the charge distribution as

q 1 Y,(0',(r)deC.4)

To appreciate the physical significance of the q h „, we evaluate som e of the lowerorder moments explicitly. The moment q 0 0 is

Roo = Y%(0 ' OP(Ildr'

. \ 1 1 4( e )de = q l . \14 .7r. 7 r

. (C.5)

where q is the total molecular charge. In l ike manner , the moments q 1 0 and q 1 1

are

i1 0 = f Yt o r'p(r)dr = — f r' cos O' p(r')dr'

z i r c

=' p(r)cle =4 n

(C.6)



MULTIPOLE EXPANSIONS OF C H A R G E DISTRIBUTIONS 355

11 1 = f Yt l rip(e)de = - — f ' sin O ' e - 'If p ( r ) d e

8n

and

(C.7)

where px , py , pz are the Cartesian components of the molecule's electric dipolemoment p . The multipole moment q 2 0 is

q 20 = f 2' 0r '2 P(r)de

2—4n

(3z'' - r' 2 ) p ( r ) c l e1

(C.8)

(C.8)

where Q z z is the zz component of the molecule's e lectric quadrupole momenttensor.

Substitution of the moments back into Equation C.3 finally yields the

electros tatic poten tial

( r )1 i.( 0, 0 )= — Ei

E 0/ + 1 mr 1 + 1

1„qlr + I t • O . - +5

L• •)4ne 0 (C.9)

which transparen tly shows the charge, dipole, and quadrupole contributions toO W . Such multipole expansions yie ld valuable insights into m olecule-radiation

interactions and long-range inte rmolecular forces .



Appendix D

B E E R 'S L AW

Discussions of light absorption in a homogene ous sam ple with molar concen-tration C and path length b are usually couche d in t e rms of Beer's law, whichstates that the light intensity I t ransmit ted by such a sam ple is re lated to th e

incident intensity / 0 in a parallel light beam by

I = I oe œ b CD.1)

Here a is the molar absorption coefficient, which has units of L/mol cm.Experime ntal spectroscopists often work with the decadic absorption coefficient

e, defined by

I = I 0 (10) — g b CD.2)

Clearly E = a/2.303. Absorption spectra are f requently recorded as the quantityebC (which is terme d the absorbance or optical density) versus incident lightwavelength A ; knowledge of the absorption cell geom etry and sample concen-tration then allows extraction of the wavelength-dependent absorption coeffi-cient e ( A ) . Be er's law is applicable only when the incident light inten sity / 0 is loweno ugh that the molec ular state populations are essentially unperturbed by it ;the redistribution of molecules into ex cited s tates by the light beam wouldotherwise materially change e for different photons within the beam. In addition,Be er's law is strictly valid only in very thin optical samples.

357



Appendix E

ADDITION OF TVVOA N GU LA R M O M E N TA

He re we cons ider the nature of the angular mome ntum states that result fromthe vector addit ion of two angular mome nta J1 and J 2 to form a resultant J,

= J 1 ±

E . 1 )

T he Cartesian components of J 1 and J2 obey the usual angular mom en tumcommutat ion rules

[fix, fly] = ihfl z (E .2)

[12x f2y] = 6 1 2 z (E .3)

z] (E .4)

[II 2]

= o (E .5)

and the cyclic permutations x y, y z, z x . Since J1 an d J2 are independentangular momenta , all components of J1 comm ute with all componen ts of J 2 ,

.r2A = 0E. 6 )

fo r all (i, j). Using the commutation rules (E.2) through (E.6), i t is e asy to showthat

[fx ` T y ] 12x, fly ± f2y]

= ih(ji2z) = ihjzE.7)

359



360 ADDITION OF TWO ANGULAR MOMENTA

and that

f]= o

E.8)

H e n c e , the vector J = J 1 + J2 itself behaves quantum mechanically like anangular momentum. According to (E.8), it is possible to set up a complete set ofeigenstates that are simultaneously eigenstates of P and Z.

The question now arises as to whether such eigenstates can also beeigenstates of J , J , J1 , and f2 2 . It follows from (E.2) through (E.8) that

f.]= [f3, fz] = o

but that

[f2 , J1 z ] = 2ih(J 1,0 2 y —y ) 0 0

[f2J22 ] = 2ih(f2xf1y f1 xJ2y ) 0

H e n c e two possible commuting sets of observables are

(E.9)

( E . 1 0 )

( E . 1 1 )

( E . 1 2 )

and

Ji, Ji, J , Jz

The eigenstates j i m i j 2 m 2 > of the first commuting set obey the e igenvalueequations

i11i1M1:12 1 1 1 2> = 1 A2 li1n1i2n1 2>E . 13 )

Î121i1M1:12 M 2>

E . 14)

i2C12

A2 U1M1 : 1 2M2>

E . 15)

/22 1 i1 m 1 3 2 m 2 > = M2h1i1M1i2M2>E . 1 6 )

and are referred to as the uncoupled' representation of the resultant angularmomentum states. Since P does not commute with fl z or f2 z , these uncoupledstates are not eigenfunctions of Pin general , and the value of J2 is indefinite.For the second commuting set of observables, it is possible to construct acomplete set of eigenstates Iiii2 im > which simultaneously obey

+E.1 7)

=j20 2 +

E.1 8)



ADDITION OF TWO ANGULAR MOMENTA 361

P I A . / 2im> = ( E . 1 9 )

fzliihim > = mhlii h im> ( E . 2 0 )

This is the coupled representation. For a given combination of ji and 1 2 , thepossible values of the resultant angular momentum quantum numberj are givenby the Clebsch-Gordan se r ie s

= (il +12), (11 +12 - 1 ) ,21E.2 1 )

(P . W . Atkins, Molecular Quantum Mechanics, 2 d ed . , Oxford Univ. Press,London, 1983) . For given j, the quantum number m can assume one of (2j + 1)values ranging between +m and —m. The coupled states 1 1 1 1 2 1 m> are notgenerally eigenstates offz or /z (cf. E q s . E .1 1 , E .1 2 ) and s o the values of J1 z

and J2, are generally indefinite in these states.The pictorial vector models of the uncoupled and coupled representations

(Fig. E . 1) embody the physical consequences of the angular momentumcommutation rules. In the uncoupled representation, the vectors J 1 and J2 canli e anywhere on their respective cones with their tips on the edges of the cones.Since these cones are invariant to rotations about the z axi s , they represent

states with fixed 1 . 1 1 1 = 1 ) and f ixed 421 = h02(i2 + 1). The pro-jections of all vectors J 1 on the lower cone have the definite value m l h. Since all

1

1Figure E .1 Vector models for the uncoupled representation (left) and the coupled

representation (right). J;, J,,,J, and . . 1 2 , are constantsof the motion in the

uncoupled representation; J.;, J, J2 , and J, are constants of the motion in the

coupled representation.



362 ADDITION OF TWO ANGULAR MOMENTA

orientations ofJi on the cone a re equally l ikely, i ts projections JL „ and J1 y alongth e x and y axes are indefinite. Similar observations apply to J2: J2, i s f ixed ,while J2, and J2 y are indefinite. Since the projections of J 1 and J2 on the xy

plane are uncorrelated (i.e., the phases of J 1 and J2 fluctuate independently),neither the orientation nor magnitude of their vector sum J is a constant of the

motion in the uncoupled representation. (However, J does exhibit a defini teprojection J =lz J2 = (nil M2)h along th e z axis.)

In the vector model of the coupled representation, the vector J can be found

anywhere on the large cone. It exhibits fixed length IJI = hO(j + 1) and f ixedprojection J = mh along th e z axis. The individual angular momentum vectors

J 1 and J2 have the fixed lengths PO I ( ji + 1) and h.02 (j 2 + 1), respect ively.Since their resultant IJI must also be of constant length, the coupled vectormodel depicts J1 and J2 precessing together, head to tail, to produce a resultantvector J of f ixed length. (In quantum mechanical language, the relative phase of

J 1 and J2 is f ixe d.) It is clear from the coupled vector model that the motions ofJ 1 and J2 on their respective c o n e s do not yield fixed projections J1 z and J2 z

along th e z axis; their sum J = mh is, of course, definite. Such vector modelsprove to be useful in discussion of the anomalous Zeeman effect in atoms(Section 2 . 6 ) and angular momentum coupling in diatomics (Chapter 4).

Since th e uncoupled states I iimi i 2 m 2 > form a complete s e t o f eigenstates, th ecoupled states I jij2 jm> may be expressed as the linear combinations

i11i21m> - E

E.22)m2

where the coeff ic ien ts <i1m1i2m21/1/2/m> are known as the Clebsch-Gordan

coefficients . Techniques for obtaining these coefficients using theraising/lowering operators j =fi ± + J2+ are described in Section 2.2 .



X 2 + y2 , z 2X 2 - 2 , x y

(xz, yz)

z, R z

f (x, y)l( R x ,

C s C 5 ci

A

E '

E"

1{ 1 1

fi

1

e 4

e 2

E3

18 2

E 3

E A

1e 3

£ 2

4

e =exp(2ni/5)

z, R z

J (x, y)1(R, R)

x2 ± y 2 , z2

(xz, yz)

(x2 y 2 xy)

Appendix F

G R O UP C H A R A CT E R TABLES

AND DIRECT P R O D U C T S

C2 E2A

C3 C3i z, R z

, R x , R y

x2, X y , z , xy

xz , yz

E = exp(2ni/3)

A z, R z

( x , y )

(R x , R)

X 2± y 2 , z 2

( X Z , yz )(x 2 _ y 2 , xy)

C 4 C2 C4

A 11

— 1

— 1

1— 1

i

— i

1—1

363



D3 E 2C3C2

A1 1 1 A2 1 112 — 1 D4 E C2 2C4C2

A 1 1 1 1 A2 1 1 11B 1 1 1 — 1 B2 1 1 — 112 —2 0 D 2C 5 2C

A 1 1 1 1

A2 1 1 1

E l 2 2 cos 0 2 cos 20

E2 2 2 cos 2 0 2 cos 40

1— — 1

1

0

5C '2

z, R z

(x , y)

l(Rx , R y )

X 2 + y 2 , Z2

X 2 — y 2

Xy

(xz, yz)

= 27r/5

1— 1

0

0

z, R z(x, y)

l(Rx , R y )

X 2 ± y 2 , Z 2

(xz, yz)

(x 2 — y 2 , xy)

z, R z

f (x , y)1(R, R y )

X 2± y 2 , z 2

( (xz, yz)

1 (x 2 — y 2 , xy)

36 4 GROUP CHARACTER TABLES AND DIRECT PRODUCTS

C6 C3 C2 Ci C Z E = exp(27ri/6)

1— 1

E

Es

82

e4

11

£2

E4

e 2

1— 1

E 3E

3

11

11

8 4

E2

E2

£ 4

1— 1

e 5

E

6 4

e2

z, R z

(x, y)l( R x , R y )

X 2 + y 2 , z 2

(xz, yz)

(x 22 5 xy)

D2 E Cz2 C Y

A 1 1 1 1 1 X 2, y 2 , Z 2

B 1 1 1 — 1 — 1 z, R z xyB2 — 1 1 — 1 Y , R y xzB3 1 — 1 — 1 1 X, R x yz

C6

A

E '

E"

1

1

1 1 11 11



D6 E C2 2C3 2C 6 3C2

A1 1 1 1 1 1

A2 1 1 1 1 -1B 1 1 - 1 -1 1

B2 1 -1 1 - -

El 2 — 2 — 1 1 0

E 2 2 2 —1 —1 0

_ X2 + y 2 , z 2

z, R z

(x, y)

t(Rx , R)(xz, yz)

(x 2 —y 2 , xy)

C3 v E 2C 3u,

A 1 1A2 1 11

2 — 1 C4v E C2 2 G 4a„

A1 1 1 1 A2 1 1 11B 1 1 1 — 1 B 2 1 1 — 112 — 2 0 C5v 2C 5 2Ci

A1 1 1 1A2 1 1 1

E l 2 2 cos 0 2 co s 20

E 2 2 2 cos 20 2 co s 40

5av 2n/5

2 + y 2 , z 2

f(x 2 — y 2 , xy)

1 (xz, yz)

2 0 - d

R z

(x , y)

l(R x , R)

2 ± y 2 , z 2

x 2 — y 2

(xz, yz)

_ X2

± y 2 , z 2

(xz, yz)

(x 2 — y 2 , xy)

1- 0

0

0

GROUP CHARACTER TABLES AND DIRECT PRODUCTS 365

C2, E2(z) a ,

A 1 1 11

x 2 , y 2 , z 2A2 1 -1 - R z xyB 1 —1 1 — 1 x, R y xz

B 2 - -1 1 y, Rx yz



C 1 1 , E ah

A' 1 . X, y, R z

A" 1 —1 R x , R y , z

C 2

1 1 1 -11-1 -1 -1 11C2h

A g 1A„ 1B g 1B u 1

2 2 2X , y , z , xyxz , yz

R z

R x , R y

X, yxz , yz

C3 Ci Œ h S3 ah C i

1 1 82E2 £2

E2 1

— 1

1

1

—

— 1

1

— 1

E

e 2

— £ 2

—

1

— 1 .

8 2

E

— E

A'A "

E '

E"

c = exp(27ri /3)

X 2 + y 2 , z 2

(x 2 — y 2 , xy)

(xz, yz)

3 6 6 GROUP CHARACTER TABLES AND DIRECT PRODUCT

E C2 2C3 2C6 3Cid 30v

11

11

2

2

11

— 1— 1

—2

2

1

1

1

— 1

11

—1

1

—

1— 1

— 11

0

0

1— 1

— 1

0

0

R z

(x , y)

l(Rx , R)

X 2 + y 2 , z 2

(xz, yz)

(x 2 _ y 2 , xy )

C6 v

A 1

A2

B1B2

E l

E2

C41, =j X C4

S2 C51., = (Th X C5 C6h = i X C6

s4

A

R x , R y , R z

X, y, zA g A u 1E241 1 11-1

ii -1

2 2x 2 , y , z 2 , xy, xz, yz

X 2 4- y 2 , z 2

(xz, yz)

( x 2 _ y 2 , xy)



D2d E C2

GROUP CHARACTER TABLES

S6 = i x C3

2S 4C2ad

AND DIRECT PRODUCTS 367

A 1

A2

B 1

B2

1111

2

111

1

— 2

11

— 1— 1

0

1— 1

1

— 1

0

— 1— 1

1

0

R z

(x , y)

l(R„, R y )

X 2± y 2 , Z 2

x 2 — y 2

(xz, yz)

D3h E ah

D3d = j X D3

2C 3S 3

D2 h — i X

3 G '2o - t ,

D2

A', 1 1 1 1 1 1 X 2± y

2, z 2

A' 1 1 1 1 — 1 — 1 R z

A'; 1 — 1 1 — 1 1 — 11 — 1 1 — 1 — 1 1

E' 2 2 —1 — 1 0 0 (x, y) (x 22 , xy)

E" 2 — 2 — 1 1 0 0 (R, R) (xz, yz)

E

D4h XD4

3C 2C 3

D5h = ah

4C '3

X D56h = i X D 6

= exp(2ni/3)

A 1

{11

3

1

1

1

— 1

1

82

0

1

82

0f(R x , R y , R z )

(x , y, z)

Td E 8C 3 3 G 2 6 adS 4

1 1 1 1 A2 1 1 1 — 112 — 1 2 0 T 1 3 0 — 1 — 1 (R x , R y , R z )

T2 3 0 — 1 11 (x, y, z)



2C 4 , cv

1 1 1

1 1 — 12 2 cos 4 ) 02 2 cos 2 4 ) 0

C o o ,

x+

z -HA

Rz), (R x , R y )

X2

+ y 2 , z 2

( X Z , yz)(x 2 — y 2 , xy)

2 y 2 , xy)

368 GROUP CHARACTER T A B L E S AND D I R E C T PRODUCTS

0 E 8C 3 3 C 2 6 C 2 6 C 4

A 1 1 1 1 1 1A2 1 1 1 — 1 — 1

2 — 1 2 0 0

T 1 3 0 — 1 — 1 1

T2 3 0 — 1 1 — 1

22 , 3z 2.2)

f(R x , R y , R z )

1 (x, y, z)(xy, yz, zx)

z: 1

11

H g 2fl u 2

Ag 2A u 2

2C 4 ,

1111

2 cos 4 )2 cos 4 )

2 cos 2 4 )

2 cos 2 4 1

C 2 S 4 ,v1 —1 —11 — 1 1z1 —1110—2 cos 4 ) (R x , R y )xz, yz)0 —2 cos 4 ) x, y)0 cos 2 0 0 —2 —2 cos 2 4 9 ± y 2 , z2

A set of rules for obtaining the direc t product of two irreducible represen-tations in any point group was s e t down by E. B . Wilson, Jr., J. C. Decius, and P.C. Cross in their classic book, Molecular Vibrat ions, McGraw-Hill, New York,1955. The multiplication properties are as fol lows:

AOA=13013=A

AOB=B

AOE=BOE=E

AQT=BOT=T

gOg=u0u=g

gOu=u

I 0 f=f, c )



GROUP CHARACTER TABLES AND DIRECT PRODUCTS 369

A®E1 =E1

OEi = E 2

A 0 E2 =E2 B 0 E2 =

Fo r subscripts on A and B representations,

10 1 = 0 = 1

1 0 2 = 2

in all groups except D2 and D2h; fo r these,

102=3

203=1

103=2

In all groups except C4, C4,, C4h, D 2d, D4, D4h, and S4,

E i 0 Ei = E2 E2 = Ai 0 A2 0 E2

E l E2 = B1 0 B2 0 El

In the exce ptions noted above,

E 0 E = A 1 0A2 0 Bi B2

In the point groups Td , 0, Oh,

EC) = E 0 T2 = T1eT2

T 1 ®T 1 = T 2 0 T 2 = C) ET 0 T 2

T i 0 T 2 = A 2 0 E Ti C ) T 2

In the linear point groups C o h,

E ± 0E+ =E+

E + 0E- =E -

E + 011=E- on =

E + 0A=E- 0A=A

11011=E+0E 0A

A0A=E+ 0E- 01-

110A=1-100



Appendix G

TRANSFORMA TION B ETWEEN

LABORATORY-FIXEDAND C ENTER-OF-M AS SC OO RDINATES IN A

DIATOM IC MO LECU LE

The positions of the nuclei with masses MA, MB in a diatomic molecule may bespecified by the vectors RA, R B with respect to an arbitrary space-fixed origin (cf.Fig. 3.1). They may also be specif ied using the vectors

R e. (MR A + M B R B ) /(M A + M B )3.1)

R RB R A3.2)

where M = (M A + M B ) is the total nuclear mass. The nuclear kinetic ene rgy inthe diatomic molecule is

T = I M A 1 k ,2k + 1- M B 1 1 43.3)

Using the inverse

M BR A = Rcni

mR

R B = Re m ± (M A) R

(G.1 )

(G.2)

371



372 TRANSFORMATION IN A DIATOMIC MOLECULE

of the laboratory to center-of-mass transformation (Eqs. 3 .1 -3.2), the kineticenergy becomes

BT =-1-MM B 1 k e m ± WA(M2±s M A)

= M1 m + 1 1 N 1 2G.3)

whereA M B /(M A + M B ) is the nuclear reduced mass of the diatomicmolecule . The f irs t term in the kine t ic ene rgy is asso cia ted wi th translation of themolecule's center of mass. The second term - 1 - , u 1 k 2 may be separated into1 /A2 + , 1 1 R 2 h 2u representing the kinetic energies of molecular vibration(changes in the length of the internuclear axis) and molecular rotation throughan angle 0 about an axis perpendicular to the molecular axi s R .



AUTHOR INDEX

Albrecht, A. C., 219-221Atkins , P. W., 27-28, 361Avouris, P., 217 , 220 , 250 , 258 , 261 , 293

Baldwin, G. C ., 3 3 6 , 3 3 8 - 3 4 0 , 3 4 8Bederson , B ., 1 0

Behlen, S., 24 0Berg, R . A., 280Berry , R. S., 126, 161

Biraben, F. , 316 , 319 , 329Birks, J. B ., 278, 280Bixon, M., 252-254 , 261Bloembergen , N . , 316 , 320 , 329 , 336 , 338-339 , 348Bloor, D. , 7 5

Bradley, D. J. , 30 3Brussel, M. K., 3 , 11 , 1 6 , 28 , 44 , 71 , 269 , 280 ,

3 3 2 , 3 3 9 , 3 4 8

Cagnac, B ., 316 , 319 , 329Chernoff , D. , 237Chisholm, C. D. H., 2 09 , 2 2 0Christofferson, J. , 236-237 , 260C o h e n , E . R., 349Condon, E. U ., 40 , 47 , 71 , 1 78 , 180 , 277 , 280Craig, D. P., 3 1 1 , 3 1 3 , 3 2 2 , 3 2 8-3 2 9 , 3 3 4 , 3 3 6 ,

345 , 348Cross , P . C . , 196 , 220 , 368

Dalgarno, A., 29Davydov, A. S., 2 0 - 2 1 , 2 8Decius , J. C ., 1 9 6 , 2 2 0 , 3 68Demtr6der, W ., 1 5 2

Dicke, R . H., 17, 28Drexhage, K. H., 2 81 , 2 91 -2 93 , 2 95Dunn, T. M., 30 8

Ellis, J. W ., 217, 221El-Sayed, M. A., 217 , 220 , 250 , 258 , 261 , 293England, W ., 1 2 9

Feinberg, M. J., 126 , 162Fischer , G ., 244 , 263Flygare, W. H., 53 -56 , 71 , 1 3 4 -1 3 5 , 1 3 7 , 1 6 2 ,

1 7 2 , 1 7 6 , 1 8 0 , 2 2 6 , 2 60Fors t , W . , 259 , 261

Friedrich, D. M., 318 , 329Furry, W. H., 45, 71

Gantmacher, F. R . , 187 , 189 , 192 , 199 , 220Gelbart, W . M ., 217 , 220 , 250 , 258 , 261 , 293Gerri ty, D. P., 3 47 - 3 48Gilmore, F. R., 1 59 - 1 60Gole, J. L., 1 0 9

Gordon , R . G ., 2 6 9 , 2 80 , 2 82Gottfried, K., 44 , 71 , 268 , 280Green , G. J. , 1 0 9

Grynberg, G ., 316 , 319 , 329

Haarhoff , P . C. , 260-261Hecht, E., 297 , 303Heicklen, J. , 2 3 2 -2 3 4 , 2 6 0H e n r y , B . R ., 22 1H erzberg , G. , 4 3 , 6 3 -6 4 , 71 , 94 , 98 , 99 , 1 0 2 -1 0 3 ,

144 , 147-149 , 150 , 156 , 162 , 173 , 176 , 180 ,211 , 214 , 220

Hesse l , M. M., 1 0 2 , 1 5 8Hinze , J. , 1 1 6

Hirayama, F. , 243Hollas, J. M., 236 , 237 , 260Hougen, J. , 146, 150 , ' 162Howell, D. B ., 232

373



374U T H O R INDEX

Huber, K. P., 9 4 , 1 56 , 1 6 2

Jackson , J. D., 4 , 11-12 , 16 , 28 , 353J ammer , M., 156 , 162Jortner, J. , 252- 254 , 261

Kelly, N., 2 3 2 - 2 3 4 , 2 6 0King, G. W ., 1 0 0 - 1 0 1 , 1 0 3 , 2 0 1 , 2 1 3 , 2 2 0 , 333 ,

348Kirby, G. H., 236 , 237 , 260Klein, 0 . , 1 6 2Konowalow, D., 1 2 1

Kruse , N. J., 26 2Kusch, P ., 1 0 2 , 1 5 8

Landau , L. D., 82 , 102 , 112 , 161Lawson , C. W , 2 4 3Lee , E . K. C., 26 1

Lengyel, B. A., 28 4 , 28 6 , 28 8 - 28 9 , 29 8 , 3 0 3Levenson , M. D., 316 , 320 , 329Lewis , J. T ., 29Lifschitz, E. M., 82 , 102 , 112 , 161Lipsky, S ., 243Long, M. E., 219-221

McClain, W. M., 318 , 329McClintock, M., 1 5 2

McDonald, D., 2 40M a h a n , B. H., 126 , 161Marion, J. B., 1 4 - 1 5 , 2 8 , 7 4 , 1 0 2 , 1 66 , 1 8 0 ,

185 , 220Mehler , E . L., 126 , 162Merzbacher , E., 3, 24, 28-29, 44-45, 51-52,

71 , 89 , 103Miller, T. M., 1 0

Mulliken, R . S. , 117, 140 , 162

Nayfeh, M. H., 3 , 11 , 16 , 28 , 44 , 71 , 269 ,2 80 , 3 3 2 , 3 3 9 , 3 4 8

Nering, E. D. , 187 , 189 , 192 , 199 , 220Nieh, J.-C., 3 47 - 3 48

Olson, M., 1 2 1

Orlandi , G., 247-248 , 261

Pace, S. A., 1 0 9

Panofsky, W. K. H., 1 2 - 1 3 , 1 6 , 2 8Partymiller, K., 2 3 2 - 2 3 4 , 2 6 0Pauling, L., 173, 180Phillips, D. L., 3 47 - 3 48Phillips, M., 1 2 - 1 3 , 1 6 , 2 8Pitzer, K. S., 8, 29P ly ler, E . K., 76

Preuss, D. R., 1 0 9

Rajaei-Rizi, A. R., 1 0 6

R e e s , A. L. G., 158 , 162Reif , F. , 276 , 280Rice, S. A., 126 , 16 1 , 237 , 240Rosenkrantz, M., 1 2 1

Ross, J., 126 , 161

Ruedenberg, K., 1 2 6 , 1 2 9 , 1 6 2Rydberg, R., 1 6 2

Sabelli, N . H., 1 1 6

Sachs, E. S. , 1 1 6

Salmon, L. S., 1 2 1

Schaefer , H . F. III, 1 3 5 , 1 62 , 2 2 6 , 2 60Schawlow, A. L., 179-180Schonland, D. S., 111 , 161 , 199 , 200 , 228 , 260Sethuraman, V ., 240Short ley , G . H., 40 , 47 , 71 , 178 , 180 , 277 , 280Siebrand, W ., 221 , 247-248 , 261

Siegman, A. E., 3 0 0 , 3 0 3Small, G. J. , 2 6 2 - 2 6 3Sommerfeld, A., 156 , 162Steinfeld, J. I., 333 , 348Strickler, S. J., 23 2 , 28 0Stwalley, W. C., 1 0 6 , 1 6 1 - 1 6 2Sushchinskii , M. M., 329Swofford, R. L., 219-221

Tabor, K. D., 347-348Taylor, B. N., 349Thirunamachandran, T ., 3 11 , 3 13 , 3 2 2 , 3 2 8 -3 2 9 ,

334 , 336 , 345 , 348Tinkham, M., 198 , 200Townes, C. H., 179-180Tully, J. C., 79 , 83 , 102

Valentini, J. J. , 3 47 - 3 48Varsanyi, G., 2 3 6 , 2 6 0Verkade, J. G ., 130 , 162Verma, K. K., 1 0 6 , 1 6 1 - 1 6 2

Wallace, R., 22 1Walsh, A. D.,

2 3 0 , 2 6 0Williams, D. F. , 2 2 1

W ilson , E. B . , 1 7 3 , 1 8 0 , 1 9 6 , 2 2 0 , 36 8Wittke, J. P, 17,29

Yariv, A., 3 03 , 3 4 0 , 3 4 8

Zajac , A., 297 , 303Zare , R . N., 152 , 157 , 162Z e m k e , W . T . , 1 0 6 , 1 6 1 - 1 6 2Zimmerman, R . L., 30 8



SUB JE CT INDE X

Absorbance, 35 7Absorption coe fficient:

decadic, 357molar , 357

Absorption spectra:atomic, 34diatomic electronic, 10 8far-infrared, rotational, 74 -75, 8 5-87nea r-infrared, vibrational-rotational , 76 -77

Acetylene, see C 2 H 2 (acetylene)Acridine dyes , 294Allowed transitions, 22Angular mom entum:

diatomic rigid ro tor , 83-10 4orbital, 25, 3 7rigid ro tor , 16 6-176

body-fixed , 167 , 170-172spaced-f ixed , 170-1 72

spin, 45vibrational, 211

Angular momentum coupling:a tom s , 47 - 51 , 5 8 - 62diatomics, 141 -146

Anharmonic oscillator, 100Aniline, see C H 5 NH2 (aniline)Anthracene, see C1 4 H 1 0

Anti-Stokes Raman transi t ions , 309, 32 9, 332Ar atom, 10Ar 2 , 86 , 94Aufban principle, 5 8Avoided crossing, 82. See also Noncross ing ruleAxial modes , 297-30 3

B2, 131 , 132Balmer s e r i e s , 34 - 35

Bandhead , 108 , 15 3BaO, 86 , 94Be 2 , 131-132Beer ' s law, 284 , 326 , 3 57Benzene , se e C 6 H 6 (benzene)BF 3 , 2 09Birge-Sponer extrapolat ion, 101-102Blackbody distribution, see Planck blackbody

distribution

Born-Oppenheimer approximation, 77-83Born-Oppenheimer s ta tes , 79 , 250-260 , 322Born-Oppenheimer theory of vibronic coupling,

2 4 5 -2 4 9Bosons , 149

C atom, 14 9C2, 131-132CARS (Coherent Anti-Stokes Raman scattering),

3 4 1 -3 4 8momentum conservation, 34 6resonant , 342

Cascading, 287Center-of-mass coordinates, 74, 371Cen tr ifugal distor t ion, 98 -10 0Centrifugal potential, 87CH radical, 218C 2 1 1 2 ( ace ty le ne ) , 2 03- 2 0 7 , 2 09 , 32 9 - 330C 2 H 4 (ethylene), 264C 6 H 6 (b en zen e), 21 7 - 21 8 , 242- 245 , 24 9 - 250 ,

2 59 , 2 65 , 3 1 8C 1 0 1 1 8 (naphthalene) , 21 8 , 22 5, 2 39-242C 1 4 H 1 0 (anthracene), 21 8

C 1 8 1 1 1 2 (tetracene), 263CH 3 C1, 7, 179-180Chemiluminescence, 33

375



376UBJE C T INDEX

C 6 H 4 F 2 , 307-309, 324-325(s- tr iazine) , 26 3

C 6 H 5 N H 2 (aniline), 235-245C H 2 0 (formaldehyde) , 16 5 , 26 1Clebsch-Gordan coeff ic ients , 47 , 362Clebsch-Gordan se r i e s , 36 1

C1 F 3 , 199-203C O , 7 , 8 6 , 9 4 , 2 8 2C O 2 , 2 1 0 - 2 1 3 , 2 31 , 2 6 4Coherence length, 34 1Combination levels, 2 0 8Configurational interaction, 57 , 135Correlation of diatomic and electronic states,

1 1 3 - 1 2 1

Correlation error, 55Correlation functions, see Electric dipole

correlation functions

Coulomb gauge, 1 3

Coulomb integral, 1 2 5 , 1 6 4Coumarin dyes , 294-295Coupled representation, 4 6 , 3 6 1Creation operator, 9 1

Cross sections:absorption, 327Raman scat ter ing, 3 2 7 - 3 2 8

Cs atom, 1 0

CS 2 , 221 , 231

A sta tes , 1 1 3

Dens i ty of s tates , 258-260Deslandres analysis , 141 , 16 3Destruction operator , 9 1

Diagrammatic perturbation theory, 334-338Diffuse se r i e s , 35 , 36Q -Difluorobenzene, see C6H4F2Dipole moment, see Electric dipole moment;

Magnetic dipole momentDirac delta function, 27 , 268

"energy-conserving" , 27integral representation, 27

Dissociation ene rgy , 1 0 1

Doppler broadening, 2 7 3 - 2 7 4Doppler-free spect roscopy, 275 , 315-321A-doubling, 1 4 3

Dyson series, 2 3

Effect ive vibrational potential, 8 7

Einstein coefficients, 27 5Electric dipole correlation functions, 2 6 7 ,

281-282Electric dipole moment, 2, 355

induced, 7i ns t an t aneous , 7

permanent, 7

Electric dipole (El) t ransi t ions, 22, 26Electric f ield , 3

Electric quadrupole (E2) transitions, 2 2 , 2 6Electric susceptibility t en so r , 33 1Electromagnetic spectrum, 34Electronic-rotational interact ions, 1 4 3

Electrostatic potent ia l , 3Emission spectra, atomic, 33-36Eosin, 2 93E ta lon, 299-300Ethylene , s ee C2 I- 1 4

Euler angles, 170-171Exchange integral , 1 2 5 , 1 6 4

F2 , 129, 131Fermions , 1 4 9

Fermi r e s o n a n c e , 2 2 0FG matrix method, 1 96Figure axis, 16 9Fine structure constant, 72Fluorescein, 293Fluorescence excitation spectra:

aromatic hydrocarbons, 236-237, 240-241diatomic , 1 0 8 - 1 0 9

Fluorescence spectra, diatomic , 1 0 5 - 1 0 9FNO, 182Forbidden transitions, 22Force constant, 88Force constant matrix, 195

Formaldehyde , see C H 2 0Franck-Condon factor , 1 3 7

Fundamental series, 35-36

Gauge transformat ion, 1 3

Gauss i an l ineshape, 2 73Gaussian type orbitals ( G T0 s ) , 1 35Generalized coordinates, 14, 192g-factor , 4 4G (kinetic energy) matrix, 1 96Golden rule , 2 6 8Grotr ian diagrams:

He atom, 6 3Hg atom, 6 4K atom, 4 3Na atom, 5 0

H atom, 1 0 , 2 9 , 34 - 35 , 4 1 , 1 4 9H 2 , 8 6 , 9 4 , 1 4 9H 2 + , 1 2 2 - 1 2 7 , 1 3 0 - 1 3 1Hamiltonian:

classical, 1 5

molecules in radiation field, 1 7

quantum mechanical :

diatomic electronic, 1 2 2 - 1 2 3



S U B J E C T INDEX77

diatomic molecule, 78diatomic rigid rotor , 83diatomic vibrational, 8 7

harmonic oscillator, 89, 90

hydrogenlike atom, 36many-elec t ron a toms , 5 1

molecule in radiation field, 1 7

polyatomic molecu le , 25 1polyatomic vibrational, 1 9 7

rigid rotor (nonlinear) :oblate top, 1 7 0

prolate top, 1 7 0

spherical top, 1 6 9

Harmonic approximat ion, 1 8 5

Hartree-Fock equat ion, 5 6 , 1 34HBr, 7

HC 1, 7 , 74-77 , 86 , 93-94 , 97 , 100HCN , 221He atom, 1 0 , 52 -58 , 6 3 , 1 4 9Heisenberg r ep resen t a t ion , 2 6 9 - 2 7 0Herzberg diagrams, 147-155Herzberg-Teller theory of vibronic coupling,

245-249H F, 86H g atom, 35-36, 64H 2 0, 7Homogeneous broadening, 27 3Homogeneous wave equation, 3 0 , 3 3 9Hot bands, 2 1 0

Hund's coupling case s , 141-146Hund's rule, 6 1

Hyperf i ne structure, 3 1 7

Hyper-Raman scat ter ing, 336

1 2 , 8 6 , 9 4 , 9 9 -1 0 0 , 1 1 6 -1 1 7 , 1 4 5 , 1 53 , 1 61 , 2 8 0ICI , 8 6 - 8 7 , 9 3- 9 4 , 15 4 - 15 5Index-matching, 3 4 1

Inhomogeneous broadening, 27 4Intensi ty-borrowing, 24 7Internal conversion (IC), 249-250Internal coordinates, 1 9 4

Intersystem crossing ( ISC) , 249 -250

jj-coupling, 61

K a tom, 1 0 , 3 5 -3 6 , 4 3

A, 1 1 4

A-doubling, 1 4 3

Laboratory coordinates, 74, 371Lagrangian equat ions, 14, 186

Lagrangian funct ion, 14, 186Landé g factor , 70Land & interval rule , 47

Laporte selection rule, 1 3 9

"Large-molecule" behavior , 249 , 260L a s e r cavity, 285Lase r s :

argon ion, 2 8 6 , 2 9 1 , 3 03CO2 , 333dye , 291 , 320H e/Ne , 28 6 , 28 7 -29 7krypton ion, 1 0 5

N2, 161 , 320N d 3 +:YAG, 287, 291 , 303, 333 , 338

L C A O-M O , 1 2 2 -1 3 6 ,Li atom, 1 0

Li 2 , 131-132Li 2 +, 131-132Lifetime broadening, 27, 271-273

Lineshape function, 26 9Local modes, 21 8 - 220Longitudinal modes, see Axial modes

Lorentz fo rce , 1 6

Lorentzian lineshape, 272Loss coeff ic ient , 285Ly man se r i e s , 34-35Lyot plate, 2 9 6 - 2 9 7

Magnetic dipole moment, 26Magnetic dipole (MI) transition, 26Mass-weighted coordinates, 1 8 5

Maxwell's equations, 1 1M C S C F calculation, 1 3 5

Me tas tab le s t a te s , 42 , 28 8Microwave spect roscopy, 178-180

Stark-modulated, 1 7 9

Minimal basis se t , 1 3 5

Mode -lock ing, 301 -303acoustooptic, 30 3passive, 30 3

Molecular orbital, 1 3 4

Molecule-fixed coordinates, 1 4 6

Morse potential, 1 0 0

Multipole expans ion , 3, 353-355

Na atom, 10 , 50 , 149 , 316 -318N atom, 10 , 149N a 2 , 86 , 94 , 105 f f , 120-121 , 1 51-153 , 155 , 158 ,

2 4 9NaCl , 86 , 94N a H , 8 6 , 94 , 1 0 0 , 1 1 4 -1 1 6Naphthalene, see C

Ne atom, 10NF3 , 1 7 9

NH 3 , 1 8 2NO, 86, 94, 154

NO2 , 7, 231



378U B J E C T INDEX

Noncross ing rule, 62 , 8 2No nlinear s usceptibili ty , 333Normal coordinates , 191-194

symmetry classification, 1 98-2 0 7N2, 8 6 , 9 4 , 13 1 - 13 2 , 1 6 0

N 2 +, 160 -161N2- , 1 6 0 - 1 6 1Nuclear exchange symme try , 1 4 8

Nuclea r spin angula r m omen tum, 148, 317Nuclea r spin statistics, 1 4 9

Number operator , 92

(I, 11 50 atom, 10 , 149OCS, 182 , 231Octupole moment, electr ic , 4O H , 86 , 94

One-photon proces se s , 22-28Optical densi ty, 357Os cilla tor s trengths, 27 7 - 28 02, 86 , 94 , 131-132 , 134 , 149 , 159 , 16102+ , 159, 16102 - , 159 , 1613, 2 3 1 , 3 4 7-3 4 8Overlap integral, 123 , 164Overtone levels , 20 8Over tone transitions, 93Oxazine dyes , 2 94

H states, 1 1 3

Pair coupling, 2 88Parallel bands, 139, 213P-branch, 95PC1 3 , 181-182Perpendicular bands, 139, 213Perturbation theory:

stationary-state, 6t ime-dependen t , 1 7-2 2

P F 3 , 181-182Pfund ser ies , 34-35Phenolphthalein , 293Phosphorescence, 25 0Planck blackbody distribution, 27 6Polarizability, electr ic:

dynamic , 323static, 7 , 10

Polar ization , 33 1Population inversion, 2 8 4 - 2 8 7Poten tial ene rgy curves:

alkali halide M X , 8 0 - 8 1H2 + , 1 2 3

harmonic oscil la tor , 88

1 2 , 1 1 7

Li 2 , 121N2, N 2 + , N2- , 1 6 0

N a 2 , 97 , 107 , 121 , 158N a H , 1 1 60 2 , 0 2 + , 02 - , 1 5 9

Preston' s l a w , 70Principal rotational axes, 1 6 7

Principal ser ies , 35-36Progressions, 233

Q-branch, 95Quadrupole moment, electric, 4 , 355Quantum yield, fluorescence, 27 9

Racah notation, 290Radiationless relaxation, 249-260Raman spectroscopy, 3 21 - 3 3 3

Rayleigh scattering, 328 , 332R-branch, 95Recur r ence time, 25 9Reduced mass:

atomic, 36nuclear , 74

R e s o n a n c e Raman scattering, 325time-resolved, 326

Rhodamine 6G, 280 , 281Ritz com bination principle, 2 7

Ritz-Paschen series, 34-35Rotational constants:

diatomic, 84polyatomic, 168

Rotational con tours , 10 8-10 9 , 237-238Rotational ine r t ia tensor, 1 6 6

Rovibrational structure, 1 0 5

Russe l l-Saunders coupling, 59Rydberg-Klein-Rees calculations, 156-161

E , 114Scalar potential, 3, 11Schrbdinger equation:

t ime-dependen t , 5t ime- independen t , 5Schrödinger representat ion, 26 9Second harmonic generat ion (SHG), 333 , 338-341Second quantizat ion, 90Selection rules:

El, M l , E 2 , 2 8electronic transitions:

alkali atoms , 42 , 51diatomic molecules , 13 6 - 13 8 , 13 9 - 14 0

rotational fine structure, 1 4 6- 1 55vibrational bands, 137-138

hydrogenlike atoms, 40 , 51



SUBJECT I N D EX79

many-electron atoms, 6 5pure rota t ional t ransi t ions:

dia tomic , 84polyatomic, 1 76-17 8

vibrational transitions:

diatomic, 93,polyatomic, 208-213Raman, 323

vibration-rotation transitions:

diatomic; 94polyatomic; 213-22

Self-consistent f ield , 56 , 1 3 4 -1 3 5Sharp series, 35-36SI (International Standard) uni t s , 1 1Slater determinant, 55

Slater diagram, 6 0Slater-type orbitals (ST0s) , 53 , 135SO 2 , 231Sommerfeld condition, 1 5 6

Space-fixed coordinates, 1 4 6

Spherical top, 1 6 8

Spin-orbit coupling:

a toms , 43diatomics , 78, 82-83, 11 5, 120 , 141

Spontaneous emission, 2 7 5

E s tates , 1 3 3

( /) s tates , 1 1 3

Stimulated emission, 275-276Stokes Raman transition, 3 08 , 3 3 2s-Triazine, se e C 6 H 3 N 3Sudden approximation, 3 1

Sum frequency generation, 336Sum rule, 1 3 8

Supersonic jets, 108-109 , 141 , 241 , 275Symmetric product representations, 20 9Symmetric top:

accidental, 1 6 9

oblate , 168 , 173-188prolate , 168 , 174-188

Symmetry-adapted linear combinations (§ALCs),228-234

Symmetry coordinates, 20 1

Term symbols:a toms , 47

diatomic, 1 1 3

Tetracene , see C 1 8 1 - 1 1 2Third harmonic generation, 333Three-photon absorption, 335"Time— ene rgy uncer t a in ty p r inc ip le" , 27Time-ordered graphs, 311-313Triplet—triplet absorption, 292Turning points , classical, 1 56- 1 58Two-photon absorption, 313-321Two-photon emission, 311-313Two-photon processes, 2 2 , 3 0 9 - 3 1 3

Uncoupled representation, 4 5 , 3 6 0

Variational theorem, 52V ec tor model s fo r coupled and uncoupled

representations, 3 6 1

V e c tor opera to r s , 1 7 7

V ec tor potential, 1 1Vibronic coupling, 23 3 , 241 , 244- 245

quantitative theories, 245Virtual states, 311-312Voigt lineshape, 274

Walsh 's rules , 23 1Wavefunctions:

diatomic rigid rotor, 84harmonic oscillator, 89hydrogenl ike, 37-39polyatomic vibrational, 1 9 7

symmetric top, 1 7 3

Xanthene d y e s , 294

Zeeman effect:

anomalous, 67-71normal, 6 6

Struve Fundamentals of Molecular Spectroscopy

Documents

Transcript of Struve Fundamentals of Molecular Spectroscopy