Rotational Spectroscopy - University of Colorado Boulder · PDF fileRotational Spectroscopy ....

: Spectroscopy

eRG Handbook ~. Boca Raton,

~ew York, 1957.

'sity Press, Cam-

Pergamon Press,

1 '?d.. Springer

iod Elements, 1.

ipringer -Verlag,

r- Yerlag, Berlin,

Hill. ~ew York,

In.iw'rsity Press,

Chapter 6

Rotational Spectroscopy

6.1 Rotation of Rigid Bodies

The classical mechanics of rotational motion of a rigid body remains a relatively mysterious subject compared to that for linear motion. In order to dispel some of the mystery it is useful to note the extensive correspondence between linear motion of a point particle of mass m and rotational motion of the same particle (Figure 6.1 and Table 6.1). For simplicity the vector natures of most of the quantities are suppressed. The correspondences between the analogous linear and angular quantities in Table 6.1 are quite striking. The linear and angular variables are related by various equations,

x() =

r v

w - - (wxr=v) r

2va - (constant w)

r L - rp (L=rxp)

T - rF (T = r x F)

in which the full vector forms are listed in parentheses. For a single particle, the angular velocity wand the angular momentum L are vectors that point out of the plane of the rotation. In this case, the wand L vectors point in the same direction (Figure 6.2). If an extended object is rotating, then Land w need not point in the same direction (Figure 6.3). This behavior is represented mathematically by the matrix product

L=Iw, (6.1)

where I is represented by a symmetric :3 x 3 matrix with equation (6.1) written explicitlyas

xy xLX) (IXX I Ixz) ( w )L y Ixy Iyy I yz wy (6.2)

( Lz Ixz Iyz I zz Wz

The matrix I is called the moment of inertia tensor in classical mechanics.

161

'

6.1 Rotation 06. Rotational Spectroscopy162

Linear Motion Angular Motion

x

Figure 6.1: Linear and circular molion of a particle of ma.<;s m.

Table 6.1: The Correspondence between Linear and Angular Motion

Linear Motion "Property" Angular Motion

Distance, x

Velocity, v = x = dx/dt

Acceleration, a = x = d2x/dt2

Mass, m

Linear momentum, p = mv

Elc = ~mv2 = p2/2m2

Force, F

F = ma = dp/dt

Position

Velocity

Acceleration

Mass

Momentu.m

Kinetic energy

Force

Newton's 2nd law

Figll;' . Angle, ()

Angular velocity, w = iJ = dO/dt

Angular acceleration, 0= ij = d20/dt2

Moment of inertia, I = mr2

Angular momentum, L = lw

1 Ek = -Iw2 = L2 /21

2

Torque, T

T = 10 = dL/dt

~

L I F~

~

Ol

to the origlll ::. velocity w. ~.

in which ~

p

Figure 6.2: The circular motion of a particle of mass m.

The cross I,r· ,.j

The derivation of the form of the moment of inertia tensor for a collection of nuclei rotating together requires the usc of some vector identities and the definition of angular

givesmomentum. Consider a collection of nuclei of mass rna located at positions r a relative

,

163 ionaJ Spectroscopy 6.1 Rotation of Rigid Bodies

en

~

-."?:

- ~k.J00ll

='=d!6d.t.

Q = • = d"fj d: l

, = ooz:r-l

~ L = 1_

&:tYJll of Dud".1

nit.ion of angular itio:on;; r <l rdati\""{"

Figure 6.3: For an extended object wand L can point in different directions.

z

" yI

x Figure 6.4: A typical molecule with nuclei located by r", vectors.

to the origin in a Cartesian coordinate system (Figure 6.4) and all rotating with angular velocity w, so that the angular momentum is given by

L = L r", x p", = L m",r", x (w x r a ), (6.3) a '"

in which

W(}:=W and P'" = m",v = m",w x r.

The cross product identity

P x (Q x R) = Q(p. R) - R(P· Q) (6.4)

gives

6. Rotational Spectroscopy164

L L mo(w(ro . r o ) - ro(ro . w)) o

L mo (w(x~ + y~ + z~) - ro(xowx + YoWy + zowz)) . (6.5) o

Writing out the vector components gives

L ""' (( 2 2 2)" (2 2 2)- (2 2 2)"L m o Wx X o + Yo + Zo eJ +Wy X o + Yo + Zo e2 +Wz X o + Yo + Zo e3 o

2 - - " - xowxeJ - xoyowyeJ - xozowzeJ - 2 - " - yo.1:oWxe 2 - YoWye 2 - yoZoWze 2

" - 2· )- ZoX oWxe3 - zoyoWye 3 - ZoWze 3 , (6.6)

which can also be expressed in matrix form as

LX) (LmO(y; + z;) - Lmoxoyo - Lmoxozo L y = - L moyoxo Lmo(x~ + z;) - Lmoyozo( )(~: )L z - Lmozoxo - Lmozoyo Lmo(x; + y;)

(6.7) Let us now identify the diagonal matrix elements of the matrix I as

I xx = L mo(y; + z~) = L mor;,-l' (6.8a) 0 0

I yy - L mo(x~ + z;) = L mOr~,-l' (6.8b) 0 0

I zz - Lmo(x~ +y~) = LmOr;,-l' (6.8c) 0 0

These elements are referred to as the moments of inertia. Similarly, let us identify the nondiagonal matrix elements as

I xy = - Lmoxoyo, (6.9a) 0

I xz - L m o·1:o zo , (6.9b) 0

I yz - - Lmoyozo. (6.9c) 0

These elements are referred to as products of inertia. Notice that the moment of inertia with respect to an axis involves the squares of the perpendicular distances of the masses from that axis, for example, r; -l from the x-axis.

In classical mechanics the ~otion of a collection of objects can be broken into the center of mass translational motion (see below) and the rotational motion about the center of mass. If a rigid rotor is assumed, the 3N - 6 internal vibrations are ignored.

6.1 Rotation of

The natural ': ,~i

molecule, The locan :,

made up of a ... j

If the origin e,: ':'

The mOlll' :~.

to find an orr:,.:: tensor I (in ",,' the coordina T· -",

The column" .:: As discll"--:

matrix I b\" ,t -,:1

or

The If matri:·: ,0

coordinate S\'·'·

I In most work' :", be dropped ,tl1:

write

very simple 1,,:11 or Lx = I.r-: 1

rtl Spectroscopy

_.~,} . (6.5)

2 2)A- '10. - Z", e3

(6.6)

)(~J (6.7)

(6.8a)

(6.8b)

(6.8c)

us identify the

(6.9a)

(6.9b)

(6.9c)

mfC'nt of inertia s of the masses

wken into the :ion about the 15 are ignored.

6.1 Rotation of Rigid Bodies 165

The natural origin for the molecular coordinate system is the center of mass of the molecule.

The location of the center of mass (given by a vector R) for a system of total mass

M=Lm", (6.10)

'" made lip of a collection of particles is given by

MR= Lm",r",. (6.11)

'" If the origin of the coordinate system is at the center of mass, then R = 0 and

Lm",r", =0. (6.12)

'" The moment of inertia tensor is a real symmetric matrix, so it is always possible

to find an orthogonal transformation matrix X that transforms the moment of inertia tensor I (in equation (6.7)) into diagonal form. The matrix X represents a rotation of the coordinate system, which can be written as

r' = X-1r or r =Xr'. (6.13)

The columns of the matrix X are made up of the normalized eigenvectors of I. As discllssed in Chapter 3, the diagonalized matrix I' is related to the original

matrix I by a similarity transformation -that is,

IX =XI' (6.14)

or

X-1IX = I'. (6.15)

The I' matrix is a diagonal matrix whose elements are the eigenvalues of I. This new coordinate system is called the principal axis system and I' has the form

Ix'x' o I' = 0 Iylyl oo ) . (6.16)

( o o Izlzl

In most work the use of the principal axis system is assumed so that the primes will be dropped and Ix = lx,x" Iy = Iy'y" and Iz = Izlzl. In the principal axis system we write

x

( LX) (Ix 0 0) ( W )Ly = 0 Iy 0 wy (6.17) Lz 0 0 Iz W z

or Lx = Ixwx, Ly ~~ Iywy, and I z = Izwz. The kinetic energy expression also has the very simple form

6.1 Rotation of166 6. Rotational Spectroscopy

z

•. '-"

'.

y

Figure 6.5: The lbO molecule.

T - lwtlwEk - 2

Ixwx )~(wx wy wz) Iywy

( Izwz 1 1 1 2- Ixwx

2 + - Iywy 2 + - Izwz2 2, 2

L 2 L 2 L 2

---..£ + --.JL + _z . (6.18)2Ix 2Iy 2Iz

The x-, Y-, and z-axes are chosen by some set of geometrical conventions. For example, the z-axis of a molecule is always chosen to be the highest order axis of rotational symmetry, and the x-axis is out of the plane for a planar molecule. For example, the moments of inertia for the H 20 molecule (Figure 6.5) are

I z = 2mHf2, (6.19)

I y = moh2 + 2mHg2, (6.20)

and

Ix = I z + Iy = m o h2 + 2mH(l + f2). (6.21)

For any planar molecule the out-of-plane moment of inertia is equal to the sum of the two in-plane moments of inertia. There is another labeling scheme for the axes in a molecule based upon the magnitude of the moments of inertia. In this case, the axes are labeled a, b, and c with

IA :::; 10 :::; Ie (6.22)

so that Ie io ',' and c-axc~ ,~,> :-J

For CXdC; ,

mo = 16,011 .. :the center (,f ' , ~,

obtain

and z = b. i - •1

into (a, b, r 1 .:' :.:

z-axes arc pL ~: .•:

rotational 5\T;.::"

Moleculco " ~::

The five Ca5(o "

1. Linear n·,'

2. Sphcricr.

3. Prolate < "]

4. Oblate '.

5. Asymm,' '.'

Group rh,· ~1

spherical Ui!'O

are readily r', .~

example, th, ,':1

asymmetric t

symmetry al: ,~

aa.l Spectroscopy 6.1 Rotation of Rigid Bodies 167

x (e)

(6.18)

Itions. For examLUs of rotational ;oor example, the

(6.19)

(6.20)

(6.21)

• tbe sum of the .c the axes in a s case, the axes

(6.22)

> N z (a)

y (b)

Figure 6.6: Linear molecule.

so that Ie is always the largest moment of inertia and I A is the smallest. The a-, b-, and c-axes are chosen in order to ensure that this inequality holds.

For example, using r = 0.958 A, e= 104.5°, mIl = 1.00 atomic mass unit (u), and ma = 16.00 u for H20 results in f = 0.7575 A, g = 0.5213 A, and h = 0,fJ65 2 A using the center of mass definition (6.12). Thus from equations (6.19), (6.20), and (6.21) we obtain

I z 1.148 U A2 (= Ill)

I y 0.6115 u A2(= I A )

2 Ix 1.760 u A (= Ie)

and z = b, Y = a, and x = c. There are six possible ways that (x, y, z) can be mapped into (a, b, c) depending on the particular values of the moments of inertia. The x-, y-, z-axes are picked by a customary set of rules, such as z is along the highest axis of rotational symmetry, but a, b, and c are chosen to make equation (6.22) true.

Molecules can be classified on the basis of the values of the three moments of inertia. The five cases are as follows:

1. Linear molecules, IH = Ie, IA = 0; for example, HCN (Figure 6.6).

2. Spherical tops, IA = In = Ie; for example, Sf\ and CH4 (Figure 6.7).

3. Prolate symmetric tops, IA < In = Ie; for example, CthCI (Figure 6.8).

4. Oblate symmetric tops, IA = In < Ie; for example, BF3 (Figure 6.9).

5. Asymmetric tops, IA < In < Ie; for example, H 2 0 (Figure 6.10).

Group theory can be used to classify the rotational properties of molecules. The spherical tops (Oh, Td' and h point groups) and linear molecules (Coov and Dooh) are readily recognized. All symmetric tops have a Cn-axis, with n greater than 2. For example, the symmetric tops CH3Cl and benzene have C3- and C6-axes, while the asymmetric top H20 has only a C2-axis. But what about allene (Figure 6.11)'1 By symmetry allene has IE = Ie, and hence it must be a prolate symmetric top. Allene

I

168 6. Rotational Spectroscopy

z (e)

I F

'. I .... po--- y (b)

r Figure 6.7: Spherical top.

x (e)

H.'.-:,. E I z (a) H~C

H/

Figure 6.8: Prolate symmetric top.

z (e)

F\I F I y (b)

x (a)

Figure 6.9: Oblate symmetric top.

has only a C2-axis, but it does have an S4-axis. The complete rule is, therefore, all molecules with a Cn-(n > 2) or an S4-axis are symmetric tops. Note that the presence

6.2 Diatomic a

of an Sn-rt':lc '.;.l

be explicit i" c - j

The SYll.::--l

axes. For eXl::·1

along it (e:, H inertial axcc j."

6.2 Diat

For a rigid ;u.· 1

expression : ~

al Spectroscopy 6.2 Diatomic and Linear Molecules 169

. therefore, all It the presence

Z (b)

'1""< Y (a)

H H

x (c)

Figure 6.10: Asymmetric top.

x (b)

H"I._r / H

z (a)-----:f.'r~= t_, H

y (e)

Figure 6.11: Allene, a symmetric top.

of an Sn-axis with n > 4 implies the presence of a Cn-axis, n > 2, so this case need not be explicitly stated.

The symmetry properties of a molecule are also helpful in locating the principal axes. For example, if there is a Cn-axis with n > 1, then one of the principal axes lies along it (e.g., H20). Any molecule with a plane of symmetry has one of the principal inertial axes perpendicular to the plane (e.g., H20).

6.2 Diatomic and Linear Molecules

For a rigid linear molecule with no net orbital and spin angular momentum the classical expression for the rotational kinetic energy is, from equation (6.18),

1 1 1 2 Ek T = 21xwx

2 + 21yWy2 + 21zwz

1 1 2 21xwx

2 + 2 1yWy

J; J; _ ,J2 (6.23)21 + 21 - 21

6.2 Diatomic all(170 6. Rotational Spectroscopy

-" X J

z

y

Figure 6.12: Rotational angular momentum in a linear molecule.

since Iz = 0, Ix = Iy = I for a linear molecule, and the customary symbol J is used to represent the total angular momentum (exclusive of nuclear spin) (Figure 6.12). For a rigid rotor in isotropic (field-free) space the rotational Hamiltonian operator for a linear molecule is

A }2 (6.24)H= 21"

The Schrodinger equation can be solved immediately, since 1jJ must be one of the spherical harmonics, 1jJLM = YJ M. The specific Schrodinger equation for this case is

}21jJ = E1jJ, (6.25)2I

so that

}21jJ = J(J + l)li21jJ = BJ(J 1)1jJ. (6.26)2I 2I +

Thus we see that the energy eigenvalue F(J) is

F(J) = BJ(J + 1), (6.27)

in which

li2 h2

B = 2I = 8rr2I' (6.28)

with B in the SI units of joules. In spectroscopy it is customary to use F (J) to represent the rotational energy-level expression and the value of B is usually given in MHz or cm- 1 rather than in joules. Since E = hv = he/>. = 102hcii , the value of B in Hz is

h (6.29a)B[Hz] = 8rr2 I'

or in MHz,

B[MHz] = h x 10-6 , (6.29b)8rr2I

Convenient ex I)··' .

and

For a diatomic 1:/

with J1 the rc,1 .' .,

The usc of a -.:.~:

may be in Ulll-'

convention \Y: .. =..

Selection R u.

The inten~ir·.

moment

For a linear n.· ..•

and the dip"l. ::, its compolJ('l.'· .1

so that eq\lar>:~

M

or in em-I,

at Spectroscopy

lCUloe _

~-mbol J is used Figure 6.12). For III operator for a

(6.24)

N

b

(6.25)

be one of the t his case is

(6.26)

(6.27)

(6.28)

! J, to represent j,,-cn in MHz or

of B in Hz is

(6.29a)

(6.29b)

6.2 Diatomic and Linear Molecules 171

B[cm-1J = h

81f2cI X 210 . (6.30)

Convenient explicit expressions for Bare

B/cm- 1 = 16.85762908 I/(u A2

) , (6.31)

and

B/MHz = 505379.006

I/(uA2

)

(6.32)

For a diatomic molecule A---B we have

21= JLr , (6.33)

with JL the reduced mass

JL = mAmB . (6.34) mA+mB

The use of a single symbol B for the rotational constant to represent a number which may be in units of joules, ~Hz, or cm- 1 is an unfortunate but common practice. This convention will nonetheless be followed in this book.

Selection Rules

The intensity of a pure rotational transition is determined by the transition dipole moment

M = J'l/JJ'M'I-t'l/JJMdT. (6.35)

For a linear molecule the wavefunction 'ljJJM can be written explicitly as

'l/JJM = YJM ((), ¢) = 8 JM (())eiM¢/~ (6.36)

and the dipole moment is oriented along the internuclear axis of the molecule, so that its components in the laboratory axis system can be expressed in the form

I-t JLxel + JLye2 + JLze3

JLo (sin () cos ¢e1 + sin () sin ¢e2 + cos ()e3) , (6.37)

so that equation (6.35) becomes

/10 21r l1r -iM' . iMA .M - el 8 J'M,e ¢sm()cos¢8JMe ¢sm()d()dep( 121f 0 0

(21r r+e2 io io 8J'M,e-iM'¢sin()sin¢8JMciM¢sin()d()dep

+e3121r l 1r 8J'M,e-iM'¢COS()8JMeiM¢Sin()d()d¢). (6.38)


If we now employ the relationships cos ¢> = (cut> + e- i<l»!2, sin ¢> = (ei<l> - e- i<l»!2i and a recursion relationship for the associated Legendre polynomials, namely

(2l + l)zpr(z) = (l + m)Pi'~l (z) + (l - m + l)pr;l (z), (6.39)

in which z = cos() and 8 JM (()) = NPr(cos()), the selection rules tlM = 0,±1 and tlJ = ±1 are obtained.

In addition, if the molecule has no permanent dipole moment (,.LO = 0), then there are no allowed transitions. Thus, symmetric molecules O=C=O, ClCl, H C=G· H have no pure rotational transitions, if only one-photon electric-dipole selection rules are considered. Molecules such as O2 (X3I:;) undergo weakly allowed magnetic-dipole pure rotational transitions. Molecules such as H-C=GD or H . D, for which the center of mass is displaced from the center of charge when the molecule is vibrating, possess a small dipole moment (8 x 10-4 D for HDl) and also undergo weak rotational transitions.

The above derivation of selection rules has also assumed that there is no additional vibrational, orbital, or spin angular momentum present (i.e., I: vibronic states are assumed). If there is additional angular momentum, then Q branch (tlJ = 0) rotational transitions are possible, such as for a II vibrational or electronic state, in which case the Q transitions are between the two nearly degenerate levels with the same J value but opposite total parity. The energy levels of II vibrational states of linear molecules are considered later in this chapter.

The selection rule tlJ = ±1 for a linear molecule results in transitions with frequencies

I/J+h-J F(J') - F(J")

B(J + l)(J + 2) - BJ(J + 1)

2B(J + 1). (6.40)

Customarily, transitions are written with the upper state, indicated by primes (J'), first and the lower state, indicated by double primes (J"), second with an arrow to indicate absorption J' f-- J" or emission J' ---> J". The first transition J = 1 f-- 0 occurs at 2B, and the other transitions are spaced by multiples of 2B from one another (Figure 6.13). This is illustrated by the pure rotational transitions of hot HF (Figure 6.14) and the far-infrared absorption spectrum of CO (Figure 6.15).

The intensity of a rotational transition is determined both by the dipole moment and the population difference between the two levels (Chapter 1). The rotational populations can be calculated from statistical thermodynamics. If the total concentration of molecules is N, then the concentration of molecules NJ with the rotational quantum number J is

e-BJ(J+l)/kT NJ = N(2J + 1) = NPJ, (6.41 )

qr

where qr is the rotational partition function

qr = 2,)2J + l)e- lJJ(J+l)/kT ;:::: kT, (6.42) J aB

with a, the symmetry number, equal to 2 or 1 for a symmetric or nonsymmetric molecule, respectively. The expression (6.41) assumes that only the ground vibrational and

6.2 Diatomic aJ

v- •

~

5C=

Figure 6.1.1 j' .r

lines due t() 11."

electronic,' -,' .• 6.16 for Cfl !3 with maxinL.:1J. This gi\.'

For CO at ;, ,. '::I

~

173 111 Spectroscopy

"" - E- I <1»/2i and Deh'

,. (6.39)

.l..\/ = O. ±1 and

= oj . then there CI. H -C=C-H Ie ~k'Ction rules I magnetic-dipole D. for which the rok is Yibrating, I IJ:·ak rotational

~ no additional rome states are , = ij rotational 1'_ in which case Iw:- same J yalue linear molec ules

Iilions llii t h fre

;6.-10

~ J' . first T':"," t~, indicatE- ,) (,(-cu~ at

lIIoJo( hEr Figun-:

~ 6.l-t and

liipok:- rooIIlf.'D t

ror.at ional porr[ O:'{}('Pntration j,-...n.:J quanturn

I).·ll·

6.42,


J

126 .. 3

66

66 J. 2

26

o 26

46

! 1

o

Figure 6.13: Transitions of a linear molecule.

v=o 15 1

20 I

25 -,

HF

v=1 14 16 8 I I T

.I .I .1 I

500 700 900 Wavenumber /cm- 1

Figure 6.14: Pure rotational emission of hot HF molecules. The spectrum also contains weaker lines due to H2 0 and LiF molecules.

electronic states are populated at temperature T. This distribution is plotted in Figure 6.16 for CO (B = 1.9225 em-I) at room temperature (298 K). The rotational state with maximum population J rnax is determined by setting dNJ / dJ = 0 and solving for J. This gives

J rnax = (~~) 1/2 1

(6.43)2

For CO at room temperature the state with maximum population has a J value of 7.


I I , [I I Ie120"

80

701

g 60 ... ... i sol

....c... o

~ °30

201

101

9's 20 2S 30 3S 40 WO¥CIIUftlbcr ClII-1

Figure 6.15: Far-infrared absorption spectrum of CO showing transitions J = -1 <-- 3 at 15.38 em- l to J = 10 <-- 9 at 38.41 em-l.

Centrifugal Distortion

A molecule is not strictly a rigid rotor. As a molecule rotates, the atoms experience a centrifugal force in the rotating molecular reference frame that distorts the internuclear positions (Figure 6.17). For a diatomic molecule one can obtain an expression for the stretching of the internuclear separation r by allowing the bond to stretch from re to rc under the action of the centrifugal force

'W 2

2 j2Fe = -- = I"W r = -. (6.44) r I"r3

The centrifugal force is balanced by the Hooke's law restoring force

Fr = k(re - rc ) (6.45)

in the bond, and after some algebra (Problem 14) one finds that

F(J) = BJ(J + 1) - D(J(.J + 1))2 = (B - DJ(J + 1))J(J + 1). (6.46)

The constant D is called the centrifugal distortion constant and, in fact, there are additional higher-order distortion corrections that lead to the rotational energy expression

F(J) = BJ(J +1) - D(J(J +1))2 +H(J(J +1))3 +L(J(J +1))4 +M(J(J+1))5 +.... (6.47)

A useful expression for D is given by the Kratzer relationship (Problem 14)

6.2 Diatomic

PJ

Figure 6 >

FiS:I:>

in which _ _ L;

(6.46) all'; ,

(6.47) apl)j>:'

Centrii js:a

effectiv(' r, ~ d t

that the T:- dr~

••

at Spectroscopy

, IC'JOI6

40

'= I - 3 at 15.38

m:- experience a the internuclear :pression for the retch from re to

(6.44)

(6.45)

(6.46)

there are addi;'I'g,Y expression

00IJ-1))5+ (6.47)

I 1.,1)


0.09

0.08

0.07

0.06

PJ 0.05

0.04

0.03

0.02

0.Q1

0 0 5 10 15 20 25 30

J-

Figure 6.16: Distribution of population among rotational states of CO at room temperature.

..lo.

J

..lo. ..lo.

Fcent Fcent

Figure 6.17: Nonrigid diatomic rotor with ml and m2 connected by a spring.

D= 4B~ 2

(6.48) w ' e

in which We is the equilibrium vibration frequency. The negative sign in front of D in (6.46) and (6.47) has been introduced in order to make D a positive number. Equation (6.47) applies to both diatomic and linear polyatomic molecules.

Centrifugal distortion increases the internuclear separation r, which decreases the effective rotation constant "Beff" = B - DJ(J + 1) of a pure rotational transition, so that the transition frequency can be written as

39.0


V= 3 83 .. 'V 2 8( > 2

8 1

80

Figure 6.18: Each vibrational level of a diatomic molecule has its own rotational constant Bv .

V./+lo--.l = F(J+1)-F(J) = 2B(J+1)-4D(J+1)3 = 2(fl-2D(J+1)2)(J+1). (6.49)

For example, the values of fl and D for CO in the vibrational ground state are B(v =

0) = 57.6359683 GHz, D(v = 0) = 0.1835055 MHz, H(v = 0) = 1.725 X 10-7 MHz, and L(v = 0) = 3.1 X 10- 13 MHz. 2

The rotational constant also depends on the vibrational and electronic state (Figure 6.18). For a diatomic molecule, as v increases the molecule spends more of its time at large r where the potential energy curve is flatter (Figure 7.5), Thus, the average internuclear separation (r) increases with v while

2 h /1) (6.50)

B v = 81r 2 /L \ r 2

decreases. This vibrational dependence is customarily parameterized3 by the equations

flv = Be - D:e(v +~) + 'Ye(v + ~)2 + ... (6.51)

and

Dv = De + !3e(v + ~) + .... (6.52)

The rotational energy level expression also becomes dependent on v, namely

Fv(J) = BvJ(J + 1) - Dv(J(J + 1))2 + .... (6.53)

At room temperature the pure rotational spectrum of a small molecule will not usually display the effects of vibration because the excited vibrational energy levels have little population. For a more floppy molecule with low-frequency vibrations, "vibrational satellites" appear in the pure rotational spectrum (Figure 6.19) since each vibrational level has its own set of rotational constants. Including the effects of centrifugal distortion and the vibrational dependence of the rotational constants results in transition frequencies given by

6.2 Diatomic

J-14

TFigure () ~,

lites.

VibTation~

The totitl ,:',;::

in which R angular Ii, <:.

standard - ,:11

(02(XJ~1

angular ::,':0

R for 1J I.' -.~ spect rc,", . ; .:; technok.;,' S not E'f! . ,. : scope (,t .: .:'

\'ibr,' ,:~

tional dI.;' .. ,1

gcncrdF ' , :,'

6.201, r,,- . x,

177 al Spectroscopy

ional constant Bv .

,2 J-l). (6.49)

state are B(v =

2.5 , 10- 7 11Hz,

Ilio: state (Figure Ill:' (Jf its time at 1Lt'. the average

(6.50)

..,." the equations

(6.51)

(6.52)

lffielv

(6.53)

,""ill not usually ",..-cIs have lit~. '\ribrational lC·h vibrational trifugal distor


Frequency I GHz

39.0 37.0 35.0 33.0 31.0 29.0 27.0 I I I I I I

J-14 13 12 11 10

... .~ ~ J

I I I

9

.J

Figure 6.19: The microwave spectrum of H-C==C-G=G--C==N showing vibrational satclIi tes.

VJ+l.-J Bv(J + 1)(J + 2) - Dv((J + 1)(J -t- 2))2

- BvJ(J + 1) + Dv(J(J -+- 1))2

2Bv(J + 1) - 4Dv(J + 1)3. (6.54)

Vibrational Angular Momentum

The total angular momentum j in a linear molecule is given by

j=R+L+S+l (6.55)

in which R, L, S, and 1 are the rotational, electronic orbital, spin, and vibrational angular momenta, respectively. In spectroscopy it is customary to associate different standard symbols4,5 with different types of angular momenta. Most common molecules (02(X3~~) and NO (X 2ll) are exceptions) have no unpaired spins or electronic orbital

angular momenta (L = S = 0) and only 1 needs to be considered in addition to R for linear polyatomic molecules. In recent years the sensitivity of pure rotational spectroscopy has improved, particularly with the development of submillimeter wave technology, so that microwave spectroscopy of free radicals and ions,6 often with Land S not equal to zero, is now an important area of research. However, it is beyond the scope of this book.

Vibrationally-excited linear polyatomic molecules can display the effects of vibrational angular momentum. A molecule like H-~C:==N or H ·C=C CI has doubly de

s in transition generate bending modes, since the molecule could bend in plane or out of plane (Figure,, 6.20). For example, HCN has 3N - 6 = 4 vibrational modes with

•


x

f yE (.,H-C/ N-z

y end on view

Figure 6.20: The doubly degenerate bending mode of a linear molecule.

VI (17+) 3311 cm- I for the H--C stretching mode,

V2(7r) 713cm- 1 for the H C:=N bending mode,

IV3((j+) 2 097 cm- for the C:=N stretching mode,

with V2 being a doubly degenerate bending mode. The degenerate bending mode V2 is modeled by a two-dimensional harmonic

oscillator7 with a Hamiltonian operator given by

2 2 A -!i

2 ( 8 8 ) 1 2 2

H = 2Jl 8x2 + 8y2 + 2k (x + y ), (6.56)

in which Jl and k are the effective mass and force constant, respectively. The x and y parts are separable, so the Schrodinger equation is solved by writing the wavefunctions as

'l/J(x, y) = 'l/JHo(X)'l/Jno(Y) (6.57)

and splitting the total energy into two parts as

E hv(vx + 1) + hv(vy + 1) hv(v+1), v=0,1,2, ... , (6.58)

with v = V x + vy , and each level v has a degeneracy of v + 1. In general for the d-dimensional harmonic oscillator

d E=hv(v+ 2), v=0,1,2, ... , (6.59)

with d = 1,2,3, ... , depending on the number of degenerate oscillators, each contributing hv/2 of zero-point energy.

The two-dimensional harmonic oscillator Hamiltonian operator7 can be converted to plane polar coordinates in which p = (x2 +y2 )1/2 and ¢ = tan-1(y/x) (Figure 6.21). The Hamiltonian operator becomes

6.2 Diatomic a

In this co'';:,:, 1

in which: ,of ±Illfi. T:,

since

The possil '" ':, Follo\\'iL:: r

although = clockwise '.If

As before. r :,'

in the x Rn,: ." small amp;:r.'l only ±/f1 ',:.i'~

to designCl~' '.: forth, to r, ; ;- ~

about th0 ie-, as a super-- ;:;:

Altho,,:::, . harmonic, ,- :. cules arc '1..".'~;

few cm-: T :" molecule f i,Z •

lIla1 Spectroscopy

view

mokocule.

I:Sioonal harmonic

(6.56)

!\'ely. The x and Ig the wavefunc

(6.57)

(6.58)

g('neral for the

(6.59)

f'ach contribut

11 be converted 'Figure 6.21).


y

'" , X

Figure 6.21: Plane polar coordinates.

2 2)" ~ ti (1 a a 1 a 1 2 (6.60)H = 2f-t papPap + p2 acp + '/p . In this coordinate system the problem is also separable and results in a wavefunction

'l/Jvl = Rvl (p )ei1 ¢, (6.61)

in which I is a new quantum number associated with vibrational angular momentum of ±lllti. The operator for vibrational angular momentum about z is

a pz = -iti arb (6.62)

since

PzWvl = lti'l/Jvi' (6.63)

The possible values7 of III are v, v - 2, ... , 0 or 1. Following the usual custom in spectroscopy, a single positive value of III is used

although ±III are possible. The double degeneracy for each value of I is associated with clockwise or counterclockwise motion of the nuclei in a linear molecule (Figure 6.22). As before, the total degeneracy for the level v is v + 1. Classically, the two oscillators in the x and y directions can be phased such that the nuclei execute circular motion of small amplitude about the z-axis. In quantum mechanics this motion is quantized and only ±Iti units of angular momentum are possible. Sometimes Greek letters are used to designate vibrational angular momentum (in analogy to the use of 2:, II, ~, and so forth, to represent A = 0,1,2, ... for the component of the orbital angular momentum about the internuclear axis of a diatomic molecule, see Chapter 9) and I is often written as a superscript, v& (Figure 6.23).

Although the different I values for a given v are degenerate for the two-dimensional harmonic oscillator, they become split if the oscillator is anharmonic. Since real molecules are always anharmonic oscillators, the different III values are split by typically a few cm~l. The twofold degeneracy for each I value (±III) remains in the nonrotating molecule (Figure 6.23).


Figure 6.22: Classical picture of vibrational angular momentum.

degeneracy g=v+1

, 44 r /

424 AI , } 5 1:+4°

, 33 C!»

3 '" "' 31 }4"' II

, 22 A 2 '" "' }3"' 2° 1:+

111 II }2

1:+o o } 1

v2 v2'

Figure 6.23: Vibrational energy-level pattern for the bending mode of a linear molecule.

When only vibrational and rotational angular momentum (Figure 6.24) are present, we have

J=R+l. (6.64)

The possible values of the quantum number J are Ill, III + 1 ... , since a vector cannot be shorter than its projection on the z-axis (Figure 6.25).

6.2 Diatomic aJ

Figure G~~

V2 =

V2 =

v = 2

Figure 6.2:) j-,

rotational CI1(f~·

There i- -- : arily exprc~c' .:

with di the:· 2

the rotatior.',

II _'.'BVIV2V3

This Be \'i'd;value of O~·;c: j

[13} Spectroscopy

"urn.

ICY 1

n€ar molecule.

~4 I are present,

(6.64)

w'Ctor cannot

,


... ... ... J = R +l

Figure 6.24: The rotational R and vibrational i angular momenta couple to give j.

J = 3 J = 1V2 =2

J = 2 J = 0l=2,A. l = 0,1;+

J = 3 J = 2

V2 = 1 J = 1l = l,n

___~J=2

V2 =0 _-:---_~J = 1 l = 0,1;+ J = 0

Figure 6.25: Rotational structure (not to scale) of the first few bending vibrational and rotational energy levels of a linear triatomic molecule.

. There is a different rotational constant for each vibrational level, which is customarily expressed as

B v = Be - l: at (Vi + ~t ) , (6.65)

with di the degeneracy of the ith mode. For example, the vibrational dependence of the rotational constant for BeF2 isS

BVl112V3 = 0.235356 - 0.000 794(VI + ~) + 0.001 254(V2 + 1) - 0.002 446(V3 -r- ~ )Cll-1.

This Be value gives an T e = 1.374971 A for the Be--F bond length while the Booo value of 0.234990 cm- 1 gives an TO value of 1.374042 A (see section 6.6).


6.3 Line Intensities for Diatomic and Linear Molecules

In quantitive applications of spectroscopy for astronomy, remote sensing, or analytical chemistry, one seeks to determine the amount of material by application of the equations governing absorption or emission of radiation presented in Chapters 1 and 5. For the case of pure rotational emission of diatomic and linear molecules, the value of IL10 in equation (1.53) for the Einstein A coefficient needs to be found. For the transition IJ'M ' ) ....... IJ"Mil) between two quantum states, the transition dipole moment (6.35) is

M = (JI M'IJLIJ" Mil) (6.66)

or equation (6.38) in spherical polar coordinates. The integrals in (6.38) for the x, y, and z (el' e2, and e3) directions will be evaluated separately to obtain an explicit expression for M.

For the z component, the selection rule on M is 11M = 0 and the ¢ part of the integral is 271", leaving only the epart to be evaluated as

M z = JLo 1"" 8 J'M COSe8J"M sinede. (6.67)

From the definition of the spherical harmonics, YJM, and the associated Legendre functions, pt, in Table 5.1, one finds

8 = (_l)M ((2J + l)(J - M)!) 1/2 pM( e) (6.68)JM 2(J + M)! J cos

and the recursion relationship (6.39) becomes

(2J + 1) cos ept (cos e) = (J + M)P!t-l + (J - M + 1)P.n-1 (6.69)

or

j2 _ M2 ) 1/2 ( (J + 1)2 _ M2 ) 1/2 ~OO = 8 8.1M ( (2J + 1)(21 _ 1) J-1,M + (2J + 1)(2J + 3) J+1,M·

(6.70) Substituting equation (6.70) into equation (6.67) yields

(JI)2 _ M2 ) 1/2 71"

M z JLo (( (2JI + 1)(21' _ 1) 18J'-I,M8J",M sinede

(JI + 1)2 _ M2 ) 1/2 71"

(6.71)+ ( (2JI+1)(2J +3) 18J'+I,M8J",Msinede)'

For the case of IJ + 1, M) <- IJ, M), I1J = +1 or J' - 1 = J" the integral becomes

(JI)2_M2 )1/2 ((JII +1)-M2 )1/2 (6.72)

Mz = JLo ( (21' + 1)(2J' _ 1) = JLo (2JII + 3)(2J" + 1)

and for the IJ - 1, M) <- IJ, M), I1J = -1 transition,

6.3 Line Intel

M. c= ._,

Similarly lIe.:;

for the iJ - ~

for the I.l ~ ~

Startill':: :~(

IJ+ 1, AI - ~

dipole mo::., :.~

However. r :,.~.

expressioL ,

For emissj, <. :1

but now r L'~'

is the sam, -~

The ll!J;'" ~

per lvI' st.:~· ,

state) bee' ":."

and

so

~ Spectroscopy

.inear Mole

nsing, or analytical ion of the equations r.; 1 and 5. For the the value of 1110 in For the transition

IE- moment (6.35) is

(6.66)

:6.3~1 for the x, y, obtain an explicit

thf: 0 part of the

(6.67)

lDCiated Legendre

(6.68)

p~( '/-1 (6.69)

2

) 8 J + I ,M.

(6.70)

n iJdO

9dO) (6.71)

~a1 becomes

2

(6.72)

6.3 Line Intensities for Diatomic and Linear Molecules 183

(J'+1)2-M2 ((JII)2_ M 2 )1/2)1/2 (6.73)

M z = flo ( (2.J' + 1)(21' + 3) = flo (2J" - 1)(2J" + 1)

Similarly using equation (5.108) yields

M =-iM = -flo ((.1+M+2)(J+M+l))1/2 (6.74) x y 2 (2J+l)(2J+3)

for the IJ + 1, M + 1) <- IJ, M) transition and

. flo ((J-M+l)(J-M+2))1/2 ,-M x = tMy = 2 (2J + 1)(2J + 3) (6.7,,)

for the 1J + 1, M - 1) <- 1.1, M) transition. Starting from IJ M) there are thus three possible transitions to IJ + 1), i.e., to

IJ + 1, M + 1), IJ + 1, M) and IJ + 1, M -I)-so the square magnitude of the transition dipole moment from IJ, M) to IJ + 1) is

IMI}+l~J,M = L M; + M; + M 2 = fl6(J + 1) (6.76)M' z 2J + 1 .

However, there are 2J + 1 values of M to be counted for the lower state so the final expression is

IMI}+l~J = fl5(J + 1). (6.77)

For emission from state IJ + 1, M) to state IJ), a similar calculation gives

IMI}+I,M~J = fl6(J + 1) (6.78)nT, .. ~

but now there are 2J + 3 values of M in the upper state so the final expression (6.77) is the same as for emission or absorption.

The upper state has a total population NJ+l and a population density N J+d(2.1+3) per !vI' state so the rate of emission, equation (1.17) (which applies to a single J'M')1

state) becomes

N J +1dNJ+l = - L LAJ+l,M'~J,M" 2.1 + 3 (6.79)

dt M'M"

and

dNJ+l _ 161l"3v3 NJ+l 2 3cohc3 2J + 3 L L 1(1 + 1, M'IILIJM")1dt

M'M"

_ 161l"3V 3fl6 (J + 1) (6.80)3cohc3(2J + 3) NJ+l

so

161l"3V 3fl6 (J + 1) (6.81)AJ+I~J = 3cohc3(2J + 3) .

I


If a lineshape function g(v - VIO) is included, then the Einstein A coefficient becomes

I61r3 v3 J.L6(J + 1)(AJ+I->J)// = 3c hc3 (2J + 3) g(v - VIO). (6.82)

o As noted earlier, the same symbols are often used with (6.82) and without (6.81) the lineshape function so care with units is required. All of the equations for A, D, a, and f derived for atoms in section 5.7 apply with 2J' + 1 = 2J + 3 and S J' J" replaced by J.L5(J +1). The (J+ I)-part of the square of the transition dipole moment is an example of a rotational line strength factor, commonly called a Honl-London factor (see Chapter 9).

The absorption cross section for the transition J + 1 t--- J for a linear molecule is thus

21r2vJ.L5(J + 1) a = 3c hc(2J + 1) g(v - VIO) (6.83)

oand Beer's law, equation (1.62), including the stimulated emission correction, is

I = Ioe-<t(No-N, ~~t~)1 = Ioe- exl (6.84)

with the absorption coefficient Q (units of m- t) given as

2J + 1)Q = a No - Nt -J-- . (6.85)( 2 + 3

In equation (6.85) it is convenient to replace No and Nt by the total concentration, N. For a system at temperature T, the absorption coefficient Q becomes

2 2 E /kT _ 21r VJ.Lo(J + I)Ne- J (I -h///kT) ( )

Q - h . - e g v - VIO (6.86)3co cq

with q = qelqvibqrot as the partition function and assuming that

N(2J + I)e- EJ / kT

No = PJN = , (6.87) q

i.e., that the state J has (2J + I)-fold rotational degeneracy, but no additional vibrational or electronic degeneracy.

At low frequencies, Doppler broadening is generally negligible relative to pressure broadening so the molecular line shape g(v - VLO) is typically given by the Lorentzian function (1.78). Interestingly, for high precision work at low frequency, the "antiresonant" term containing W + WIO neglected in going from (1.71) to (1.72) needs to be included and the lineshape function is then approximately

v (!:1V/2 !:1V/2)g(v - VIO) = -- + ---:-:-----:--:-::,-----'-:---__:_;:_ (6.88)1rVIO (!:1v/2)2 + (v - VIO)2 (!:1v/2)2 + (v + VIO)2 '

which is called the Van Vleck--Weisskopf lineshape function. 9 The !:1v parameter is given as (1rT2 )-I, equation (1.81), with T2 the average time between collisions. For large VIO (in the infrared and optical regions) the second term on the right-hand side of equation (6.88) can be neglected and the usual Lorentzian line shape is recovered with !:1v = !:1vI/2' the full width at half maximum. The Van Vleck Weisskopf line shape agrees well with experimental observations of pure rotational transitions in the microwave and millimeter wave spectral region.

6.4 Symmetr

6.4 Syn

The clas"i, r,.

or

For simpli, .. :: to an obLF'

or

then

The corr·<' ':

Thesolur; :. quantulIl ... ,,

Molecule

The synlll.":laboraton e-: both wit L ,,:-:1

system [(I " ;',

defined 11. :_~

6.27. Th' o:,~

molecllbr >" molecuk c' - -.\

describce ':'-ThC'k:

matrix S ':.1

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