WATER DIMER ATMOSPHERIC ABSORPTION

BY Tristan L'Ecuyer

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

AT

DALHOUSIE UNIVERSITY

HALIFAX, NOVA SCOTiA

JULY 1997

@ Copyright by Tristan L'Ecuyer, 1997

National Library Bibliothèque nationale du Canada

Acquisitions and Acquisitions et Bibliographie SeMces senrices bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON K1A ON4 ûttamON K1AON4 Canada Canada

The author has granted a non- exclusive licence allowing the National Libraxy of Canada to reproduce, loan, distribute or sell copies of this thesis in microform, paper or electronic formats.

The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts eom it may be printed or otherwise reproduced without the author's permission.

L'auteur a accordé une licence non exclusive permettant à la Bibliothèque nationale du Canada de reproduire, prêter, distriiuer ou vendre des copies de cette thèse sous la forme de microfiche/fih, de reproduction sur papier ou sur format électronique.

L'auteur conserve la propriété du droit d'auteur qui protege cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation.

Contents

List of Tables

List of Figures

Adcnowledgements

Abstract

List of Symbols

1 Introduction

vi

vii

ix

X

2 Interaction Potential 4

2.1 The Water Dimer, (HzO)2, Molecule . . . . . . . . . . . . . . . . . . 4

2.2 The H 2 0 - H20 interaction Potential . . . . . . . . . . . . . . . . . 5

2-2.1 The RWK2 Model , . . . . . . . . . . . . . . . . , . . . . . . 5

2.2.2 Preliminary Investigation of the Water Dimer . . . . . . . . . 9

3 Normal Mode Analysis 13

3.1 WiIson's Method . , . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Water Monomer . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . 15

3.3 WaterDimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Rotation-Vibrat ion Energy Spectrum 20

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Energy Levels 20

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Transition Frequencies 29

4.3 Nuclear Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Absorption Line Intensities 34

5.1 Absorption Cross-section . . . . . . . . . . . . . . . . . . . . . . . . . 34

. . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Initial State Probabilities 36

5.3 Dipole Moment Matrix Elements . . . . . . . . . . . . . . . . . . . . 39

5.3.1 Vibrational Matrix Elements . . . . . . . . . . . . . . . . . . . 40

5.3.2 Rotational Matrix Elements . . . . . . . . . . . . . . . . . . . 41

5.3.3 Combined Matrix Elements . . . . . . . . . . . . . . . . . . . 43

5.4 Summaryandhterpretation . . . . . . . . . . . . . . . . . . . . . . . 47

6 Results 50

6.1 Computational Techniques . . . . . . . . . . . . . . . . . . . . . . . . 50

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Monomer 51

6.3 Dimer Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Discussion 59

6.5 Water Dimer Atmospheric Concentrations . . . . . . . . . . . . . . . 62

A Direction Cosine Matrix Elernents 68

Bibliography 69

List of Tables

2.1 Experimental water dimer frequencies . . . . . . . . . . . . . . . . . . 2.2 Intramolecular potential parameters . . . . . . . . . . . . . . . . . . . 2.3 Lntermolecular potential pitlameters . . . . . . . . . . . . . . . . . . . 2.4 Equilibrium geometry of the water monomer and dimer molecules using

the RWK2 potential mode1 . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Structural properties of the water dimer . . . . . . . . . . . . . . . .

3.1 Vibrational frequencies of the water monomer molecule in the hannonic

oscillator approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Vibrational frequencies of the water dimer molecule in the harmonic

oscillator approximation . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 Moments of inertia about the principal axes of the water monomer and

&mer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Pure rotational energy levels of the water monorner

5.1 Equilibrium components of the monomer and dimer dipole moment

along the principal axes . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1 Cornparison of truncated and total rotational partition functions for

the water monomer and dimer at 300K . . . . . . . . . . . . . . . . . .

. . . . . A.l Direction cosine matrix elements in the symmetric rotor basis

List of Figures

2.1 Equilibrium structure of the water monomer . . . . . . . . . . . . . . . 6

2.2 Equilibriumstructureofthewaterdimer . . . . . . . . . . . . . . . . . 10

Normal modes of the water monomer . . . . . . . . . . . . . . . . . . . The normal modes of vibration of the water dimer molecule . . . . . .

Definition of the rotational quantum numbers . . . . . . . . . . . . . .

Absorption spectrum of the water monomer in the harmonic oscillator.

rigid rotor approximation . . . . . . . . . . . . . . . . . . . . . . . . . 2 cm-' resolution monomer absorption spectrum . . . . . . . . . . . . Pure rotationai water monorner spectrum . . . . . . . . . . . . . . . . Absorption spectrum of the water dimer in the harmonic oscillator.

rigid rotor approximation . . . . . . . . . . . . . . . . . . . . . . . . . High resolution plot of dimer bridge-hydrogen stretch mode . . . . . . High resolution plot of the dimer out-of-plane bend mode . . . . . . . High resolution plot of dimer in-plane bend mode . . . . . . . . . - . . Comparison of monomer and dimer absorption in the angle bending

band (1400 - 1600 cm-') . . . . . . . . . . . . . . . . . . . . . . . . Comparison of monomer and dimer absorption in the 3500 - 4200

cm-' region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

To Ji1 lzan,

for her wonderful support over the course of thzs work.

viii

Acknowledgement s

This work would not have been possible without the help of the following ~eople to whom 1 offer my sincerest thanks:

Dr. D.J.W. Geldart Dr. P. Chilek Dr. Q. Fu Dr. H. C. W. Tso

Abstract

A method of calculating absorption spectra which is uniformly MLid for both the water monorner, H20, and dimer, (HZ0)2 , molecules, at low temperature as well as atmo- spheric temperatures, is presented. Vibrational modes are treated in the harmonic oscillator approximation, and molecules are taken to be rigid rotors. Comparison of the absorption spectra of the water monomer and dimer molecules indicates that a significant fraction of the dimer absorption lies in regions of Little or no monomer absorption demonstrating the importance of including the effects of dimer absorption in atmospheric radiative transfer cdculations requiring accurate results. In addition, anharmonic effects may lead to dimer absorption in the û-14pm window. Further investigation of the effects of water dimers on atmospheric absorption is required.

A portion of this work is also devoted to assessing the RWK2 potential model for H20 - H20 interactions in applications involving the computation of the absorption spectra of the water dimer molecule. The intramolecular aspects of this mode1 have also been studied. Earlier cornparisons of computed dimer properties, such as the dipole moment, moments of inertia and vibrational frequencies, with amilable ex- perimental observations have demonstrated the model's ability to reproduce a wide variety of the physical attributes of the molecde [Il. In the present study, the in- tensities of the pure rotational spectral lines in the monomer absorption spectrum, obtained using the intramolecular part of the RWK2 model, are found to be in qual- itative agreement with available experimental data although the magnitude of the total absorption due to these lines is a factor of three too small. In addition, the integrated absorption coefficient over the two broad fundaaental rotation-vibration absorption bands of the monomer, is consistent with data from the Air Force Geo- physical Laboratory archive, as quoted by Goody and Yung [2].

List of Symbols

A, B, C - rotational constants

Aj - eigenvector corresponding to j th mode

ABHI AOH, AOO - intermolecular potential coefficients

â - lowering operator

ât - raising operator

a - absorption coefficient

asw, a o ~ , a00 - intermolecdar potential inverse length parameters

ai - Morse potential inverse length parameter

Ci - dispersion coefficient

Di - Morse potential depth

b - distance from oxygen atom to massless charge centre in equilibrium

Ë - electric field vector

Eo - magnitude of electric field

Eer - electronic state energy

EN - nuclear state energy

c - dielectric constant

ci - energy of initial state

E'- unit vector along electric field

1 f) - final state

fiz - intrarnolecular coupling parameter

F - "Universaln function

F - force constant matrix

F = { X , Y, 2) - space-fuced axes

f, - force constant

g = {a, b, c ) - principal axes

g - unit vector along principal a v i s g

G-' - mass matrix in cartesian coordinates

Hel - electronic Harniltonian

Hi - interaction Hamiltonian

HF,* - rotational Hamiltonian

& - nuclear Hamiltonian

H,b - vibrational Hamiltonian

li) - initial state

Ig - moment of inertia about principal axis g

I (w) - lineshape function

J - total angular momentum quantum number O

J, - angular momentum operator about principal axis g

j - total angular momentum operator

( J KM) - syrnmetric top rotational wavefunction

1 JT M) - asymrnetric top rotational wavefunct ion

k - Boltzmann's constant K - projection of angular momentum onto the figure axis

k, - absorption coefficient

L - Lagrangian

M - projection of describing angular momentum onto a space fixed axis

Ma - mass of nucleus a

mi - mass corresponding to cartesian coordinate xi i4

@ - total dipole moment operator

p;1 - equilibrium dipole moment

6jig - oscillating dipole moment

kg - dipole moment operator along principal axis g

pj - reduced mass associated with jth mode

N - number of atoms in molecule

?ZR - red part of the index of refraction

Inj) - vibrational wavefunction of harmonic oscillator j

nj - vibrational quantum number of jth state

üj - vibrational frequency of jth mode (in cm-')

vj - j th mode label

pj - moment- conjugate of qj

P,, - nuclear spin statistical weight

Plci - probability of transition from (i) to 1 f )

OFg - direction cosine between principal axis g and space-fixed axis F

- total molecular wavefunction

&1 - electronic wavefunction

îLN - total nuclear wavefunction

As - nuclear spin wavefunction

- rot ational wavefunct ion

S>& - vibrational wavefunction

Q - partial charge in RWK2 mode1

qj - normal coordinate of mode j

IR) - rotational wavefunction

& - equilibrium bond length in monomer

Rab - distance between nuclei a and b

rab - distance between electron (or nucleus) a and b

& - length of i bond

1 RV) - combined rotational-vibrational wavevector

Pi.1 - probability of finding the system in the state i,f

si - local mode coordinate

JS,,S,) - total nuclear spin state

R~ - vector position of massless charge centres

T - kinetic energy

p, - nuclear kinetic energy operator A

Te - electronic kinetic energy operator

Bo - equilibriurn monomer bond angle

Bi - ith bond angle

ej - orientation angle of j th molenile with respect to 0-0 axis

V - potential energy

- damped dispersion interaction between two oxygen atoms

Knt, - intermoleuclar interaction potential

Kntr, - intramolecular interaction potential

Vws - hydrogen-hydrogen interaction potential

VNN - interaction potential between two massless charge centres

VNH - interaction potential between hydrogen and massless charge centre A

V, - internuclear potential energy operator

V, - nuclear-electronic interaction energy operator

ce - interelectronic potential energy

1 V) - total vibrat ional wavefunction

Voo - oxygen-oxygen interaction potent ial

VRWKÎ - complete potential surface in the RWKL mode1

w - angular frequency

wfi - angular transition frequency

xi - atomic cartesian coordinate

x - column matrix with ith elernent = xi

xi - time derivative of atomic cartesian coordinate i

x - column matrix with ifh element = xi

r - pat h length

Z - total partition function

( + z), 1 - z ) - spin states of spin-+ particle

2. - atomic number of ath nucleus

2" - nuclear spin partition function

xiv

Zw - nuclear spin partition function

2"' - rot at ional partit ion funct ion ~ t r a n s - rotational partition function

ZMb - vibrational partition function

Chapter 1

Introduction

Understanding the interaction of solar radiation with the various constituents of t

atmosphere is fimdamental to our knowledge of the climate and our ability to predict

climactic changes. Despite many decades of atmospheric research, a number of uncer-

tainties remain in the magnitudes of the different components making up the earth's

radiation budget [3]. These uncertainties have direct implications on the fundamen-

t al drive of oceanic and atmospheric circulations. Any discrepancies bet ween mode1

predictions of atmospheric absorption and observations imply an inherent flaw in our

understanding of radiative transfer processes. Therefore it is important to resolve

these uncertaint ies.

One very significant issue currently being investigated is the magnitude of the

absorption of solar radiation in a ctoudy atmosphere. In recent years, there have been

nurnerous studies using satellite and aircraft data as well as ground measurements

[4, 5, 6, 71 which report observations of cloud absorption of 20-30 W/m2 in excess of

values predicted by current models. This amounts to 8% of the total incoming solar

radiation or up to 50% of the predicted absorption due to clouds. On the other hand,

there have been researchers who measure no excess absorption [8].

In addition to t his discrepancy, known as the "cloud absorption anomaly" , t here is

the question of the origin of observed continuum absorption in the 8 to 14 pm region

of the atmospheric spectrum. The peak in longwave radiative energy in the lower

atrnosphere resides in this region of the spectrum because absorption lines from the

major atmospheric constituents lie well outside it. Since the transfer of IR radiation

between the surface, atmosphere, and space, determines atmospheric heating rates,

accurate predictions of the absorption in the 8-14 pm region, commonly referred to as

the 'IR absorption window', are critical to the study of the earth's radiation budget.

One substance occurring in large enough concentrations under almost all atmo-

spheric conditions t hat could account for substantid discrepancies between predicted

and observed atmospheric absorption is water vapour [9]. In sufficient concentra-

tions, water molecules can form clusters such as the dimer, (H20)2, or weakly bound

complexes, H20 - X, where X represents another major atmospheric constituent such

as 02 , 03, or N2 [9] The weak hydrogen bonding in these complexes results in

strongly anharmonic intermolecular vibrational modes which can couple with the in-

t ramoledar modes of each individual molecule resulting in significant broadening of

the resulting spectrum. In a humid atmosphere, these complexes could result in a

substantial increase in the absorption of solar radiation.

The goal of this work is to obtain and compare quantitative absorption spectra

for the water monomer and dimer molecules, (Hz0)2. A procedure will be used

which is uniformly valid from low temperatures (- 10K) to 350K. This temperature

range allows comparison with data from matrix isolation studies as well as use of the

results in atmospheric applications. This problem involves a high level of complexity

due to the large number of degrees of freedom in the dimer molecule, al1 of which

must be treated precisely. As a consequence, the simplest possible conditions that

will yield a semi-quantitative description of both the monomer and the dimer is

used. Vibrational motion is treated in the harmonie oscillator approximation while

molecular rotations are considered in the rigid rotor approximation. Anharmonic

effects, tunneling transitions, and the dynamics of formation and decay of the water

dimer rnolecule will be neglected.

Complete spectra for both the monomer and dimer in the harmonic oscillator-rigid

rotor approximation will form a good basis for further study of the effects of water

dimer absorption. Since both molecules are treated in exactly the same marner,

extensive comparison between monomer and dimer properties is possible. Further-

more, comparison of the resulting monomer spectra with available experimental data

provides a means of testing the mode1 and the applicability of the mode1 to both

molecular spectroscopy and atmospheric studies.

Chapter 2

Interaction Potential

2.1 The Water Dimer, (HzO)z, Molecule

There have been a number of experimental studies of the vibrational spectrum of

water dimers using both molecular beam [IO] - [14] and matrix isolation 1151 methods.

Some of the results from these studies are summarized in Table 2.1. Due to their

Mode Acceptor angle bend Acceptor symmetric stretch Acceptor asymmetric stretch Donor angle bend Bonded H stretch Free H stretch Donor out-plane bend Donor in-plane bend Hydrogen bond stretch

Molecular Beam Ar Matrix 1593 3634 3726 1611 3574 3709

Nz Matrix 1601 3627 3725 2619 3550 3699 520 320 155

Table 2.2: Experimental water dimer vibrational frequencies (in cm-') (the modes are defined in Figure 3.2). The last three columns contain experimental data from the rnolecular beam experiment of Page et al [12] and the matrix isolation studies of Bentwood et al [15].

strong anharmonicity, the low frequency intermolecular modes are very broad and

difficult to resolve in experimental studies leading to modes for which there is no

experimental data. For example, the intermolecuiar mode presented in Figure 4 of

Bentwood et al [15] is clearly much bmader and more difficult to discern than the

intramolecular peoks (Figure 5) in the same study. For this reason, it is useful to

study intermolecular modes using a potential model.

In addition to experimental studies, there have been many recent theoretical stud-

ies of the water dimer (161 - [23]. The remainder of this chapter is devoted to describing

the H20 - HzO interaction potential to be used here and evaluating its strengths and

applicability to the problem.

2.2 The H 2 0 - H 2 0 Interaction Potential

There are numerous H 2 0 - H20 interaction potentials in use today [24] - [41]. -4

potential introduced by Reirners, Watts and Klein in 1982 [13, 11 is used in the

current study. The Reimers, Watts and Klein potential (henceforth referred to as

RWK2) is adopted because it is fitted to observational data in three bulk phases,

including the 56 band origins in the as phase spectra of H20, HDO, and D20, as

well as the bulk moduli and static lattice energies of ices Ih, VI1 and VIII. Some

success has also been realized in using this model to predict distribution funtions

and thermodynamic properties in the liquid state [l]. Al1 terms in the intermolecuiar

potential are physically based eliminating the need for excess parameters that result

in unnecessary complexity in the computations. For this reason, the RWK2 potential

is well suited to any simulation involving two or more H20 molecules.

The complete potential is separated into two parts, an intramolecular term designed

to reproduce the vibrational properties of each individual H20 molecule and an in-

termolecular terrn to characterize the interactions between the atoms on different

molecules.

Figure 2.1: Equilibrium structure of the water monomer (C2, syrnmetry). & = 0.9572A and Bo = 104.5'. a, b, and c are the principal axes.

The intramolecular potential for each molecule is comprised of three Morse oscil-

lators and a coupling term (421:

where si is the coordinate of the ith local mode [43] given by:

& = 0.957~4 and Bo = 104.5" are the equilibriurn values of the O-H bond length and

the HOH angle in the H 2 0 molecule (see Figure 2.1). The values of the parameters,

determined by fitting to 37 observed vibrational levels of H20, 9 of DzO, and 10 of

HDO using the local mode methods described in [43], are presented in Table 2.2.

Table 2.2: Parameters of the intrarnolecular potential ( h m Coker an(

Parameter Dl D2 D3 QI

a 2

a3

f 12

d Watts [42]).

Value 4.5904 x IO4 cmdL 4.5904 x IO4 cm-' 3.4369 x 104 cmd1

2.14125 A 2.14125 A

0.70600 -5.2998 x lo3 cm-' A-2

The intermolecdar potential is a simple point charge mode1 with positive partial

charges, Q, on all hydrogen atoms and a negative partial charge of twice the magnitude

on the bisector of each HOH angle, located a distance 6 from the oxygen atoms.

Q and 6 are chosen to be 0.6e and 0.26A, respectivety, to obtain good agreement

with observed monomer dipole and quadrupole moments. In addition to Coulombic

interactions between point charges on different molecules, there are also dispersion

interactions between the oxygen nuclei.

The interaction between the hydrogen atoms on different molecules consists of an

exponential repulsion term and a Coulombic repulsion term:

A Morse potential is used to represent the interaction be t~

on one molecule and a hydrogen on another:

V O H ( b H t ) = AOH (e -aOa(RO~-R") - 1)2 - AOH

The oxygeo-oxygen interaction takes the form:

VOOt ( b o t ) = ~ ~ ~ t e - ~ 0 0 ' ~ 0 0 ~ + b i s p (&O#)

{gen atom

where the damped dispersion interaction is given by:

with:

and,

g2,(Root) = [1 - exp (-0.995752&ot/n - 0.06931~~~~/J;;>]'" (2-8)

and R* = 0.948347R0~t. The dispersion coefficients, Ci, take the values of Margofiash

et al. [44] (see Table 2.3) and the damping function, F ( R ) , is the "universaln function

of Douketis et a1 [45].

There is a Coulombic attraction between the negative charge centres, N, on each

molecule and the positively charged hydrogen atoms on the other molecule:

and, finally, a Coulombic repulsion between the two negative charge centres:

In dynamic simulations, the position vector of

to a coordinate system centred on the oxygen

where fiHl and gH2 are the vector postitions

the negative charge centre with respect

atom is given by:

of the two hydrogen atoms.

The values of the parameters of the interrnolecular potential are presented in

Table 2.3. The seven adjustable parameters are fitted to the second virial coefficient

of steam and the static lattice energies and bulk moduli of ices Ih, VII, and VI11 [l].

Value 2.2101 x 105 cm-1

725.24 cm-' 1.1209x109 cm-L

3.2086 A 7.3615 A 4.9702 A

1 .1856~ 106 cm-' As 7 . 4 1 4 7 ~ 10" cmd' ALo

Table 2.3: Parameters of the intermolecular potential (frorn Coker and Watts [42]).

The complete potential is formed by cornbining the inter- and intramolecular po-

tent ids:

V = Kntra + Knter (2.12)

The empirical parameters in each potential were specified separately are not altered

when the dimer is forrned. Previous simulations of the water dimer molecule indicate

that this approach is valid [13, 46,471.

2.2.2 Preliminary Investigation of the Water Dimer

In the years following the introduction of the RWK2 potential, it has been applied

to numerous problems involving the interaction of two or more water molecules [II -

[48]. These studies were concerned with both evaluating the potential mode1 as well

as comparing three methods for detennining the vibrational frequencies; normal mode

analysis, local mode analysis (see Reimers and Watts [43, 471 for a description), and

quantum Monte Carlo procedures (see Coker and Watts [42] and Suhrn and Watts

(481 for a description).

The equilibrium energy structure of the water dimer, determined using a Monte

Carlo search procedure [47], is shown in Figure 2.2 and values of the various bond

lengths and angles are given in Table 2.4. The donor molecule resides in the plane of

the page while the acceptor molecde lies in a plane perpendicular to it. The HOH

Figure 2.2: Equilibrium structure of the water dimer. Values for each quantity are listed in Table 2.4.

RI (donor) R2 (donor)

ddonor

Ri (acceptor) Rz (accep tor)

9.cc,tor

@D

@ A

b o

Monomer m 0.9572 A 104.5"

0.9572 A 0.9572 A

104.5" n/a n/a n/a

Table 2.4: Equilibrium geometry of the water monomer and dimer molecules using the RWK2 poteotial model. The relevant quantities are defined in Figures 2.1 and 2.2.

1 Binding Energy 1 -6.15 kcal/mole 1 -5.4 & 0.2 kcal/rnole (

r

Dissociation Energy

D ipole Moment 1 2.24 D 1 2.6 D 1 Table 2.5: Structural properties of the water dimer. The experimentol data is that of Dyke et al [IO].

bisector of the acceptor molecule forms the intersection of these two planes and it

makes an angle Oa with the axis joining the two oxygen molecules. The presence of

the plane of inversion symmetry irnplies that the dimer molecule belongs to the Cs

point group. The principal axes (those axes which diagonalize the moment of inertia

tensor of the molecule) are also shown. The a-axis is normal to the plane of the page

while the b- and c-axes are oriented such that the c-axis makes an angle of 2-67" with

the 0-0 axis. The origin of this coordinate system is taken to be the center of mass of

the dimer molecule to facilitate the treatment of rotations in future caiculations. The

microwaveand electric resonance studies of Dyke et al [IO] indicate an 0-0 separation,

Etoo, of 2-98 f 0.0 1A and orientation angles, OD and Oa of 50 f 6 and 58 z t 6 degrees,

respectively, in good agreement with computed values. Structural propert ies of the

equilibrium dimer are compared with experimental observations in Table 2.5. Again,

good agreement is found between predicted values and those obtained experimentally

RWK2 3.83 kcal/mole

[il - The afore-ment ioned studies also evaluate three different met hods of calculât ing

vibrational frequencies of small water clusters. Normal mode and local mode proce-

dures are used as weli as a far more time consuming method combining local mode

analysis with Monte Carlo calculations and a quantum mechanical analysis of the

results. The results indicate that accuracy increases as the computational cost of

the method increases. Thus the best results were achieved using the quantum Monte

Carlo procedure while the least accurate results were obtained using the normal mode

analysis which is the simplest procedure to implement. In Light of the goal of this

study, namely to obtain a a quantitative cornparison of the absorption spectra of

the water monomer and dimer molecules, the magnitude of the computed absorption

Experiment 3-4 kcd/mole

intensities for broadband applications is more important than the exact positions of

the absorption peaks so the standard normal mode procedure will be used.

Chapter 3

Normal Mode Analysis

3.1 Wilson's Method

In order to calculate the rotation-vibration spectrum of a polyatornic molecule, it is

necessary to understand its fundamental modes of vibration. This requires knowledge

of both the frequency and the atomic motions associated with each mode. The method

used in this study of the water dimer is the classicai mechanical approach of Wilson et

al [49, 50, 51, 521. This formalism is currently used in numerous studies of molecular

vibrations and is well suited to the current investigation.

In Cartesian coordinates, the kinetic energy of an N atom molecule is given by:

where x is a column matrix consisting of the time derivatives of the Cartesian coor-

dinates of the atoms and G-' is a diagonal 3N x 3N matrix with the corresponding

atomic masses on the diagonal1.

The potential is written in Cartesian coordinates and expanded in a Taylor series

'The notation adopted here is that which commonly appears in many modern spectroscopy texts (see, for example, Barrow [50]).

about the equilibrium positions of the atoms,

where the summations run over the 3N Cartesian coordinates in the N atom molecule.

in the harmonic oscillator approximation, we tnincate this series to second order.

Noting that the first derivatives are zero in equilibrium,

where x is the column matrix formed by the deviations of the atomic Cartesian

coordinates from equilibrium, xi - zfq,* F is a square matrix whose elements are

fii = d2V/ d ~ ; a ~ ~ ( ~ ~ , and the equilibrium value of the potential is defined as zero

since the vibrational motions of the atoms are independent of the equilibrium value

of the potential energy.

Using Lagrange's equations,

where L = T - V is the Lagrangian for the system, equations of motion for each

Cartesian variable can be obtained:

The 3N coupled second order differential equations in (3.5) can be written in matrix

'Note that the matrix x in Equation (3.1) is the time derivative of the matrix x in Equation (3.3).

form,

where G-' and F have been defined earlier. The solutions are found by making the

following substitutions: 3N

X i = C Aikcos(2*~kt + p )

y ielding,

where A is a diagonal 3N x 3N matrix with X k = 4n2vE, the eigenvalues of GE', on

the diagonal. The vibrational frequencies are, therefore, determined by finding the

eigenvalues of the GF matrix. It should be noted that the six eigenvalues correspond-

ing to the three rotational and three translational degrees of freedom of the molecule

are zero.

3.2 Water Monomer

This procedure is first applied to obtain the vibrational frequencies and eigenvectors

of the normal modes of the HzO molecule. The intramolecular part of the RWK2

potential (Equation (2.1)) is used for the potential energy dong with the appropriate

equilibrium bond length, l& = 0.9572A and bond angle, 104.5'. The resulting vibra-

tional frequencies are displayed in Table 3.1. The computed vibrational frequencies

agree with experimental values as a result of the fact that the mode1 parameters were

fitted to a wide range of vibrationai band origins of &O.

The atomic motions associated with these vibrational modes can be established

via careful inspection of their eigenvectors. Diagrms of the three resulting vibrational

*

Table 3.1: Vibrational frequencies of the water monomer rnolecule in the harmonic oscillator approximation. AU fiequencies are in cm-'. The experimental data is from Goody and Yung [2].

Mode Asymmetric stretch (vl) Symmetric stretch (4 Angle bend (4

modes appear in Figure 3.1.

Figure 3.1: Normal vibrational modes of the water monomer. The modes of the monomer axe; (a) symmetric stretch, VI, (b) angle bend, v2 and (c) asymmetric stretch, u3 and their frequencies are given in Table 3.1.

Present Work 3918 38 14 1638

3.3 Water Dimer

Experiment 3755 3657 1595

Results of the application of this method to the water dimer using the RWK2 potential

(see section 2.2.1) are presented in Table 3.2. Data from the recent quantum chemical

study of Kim et al [22] are included along with the experimental results of Bentwood

et al [15] for comparison. The intramolecular frequencies are in good agreement

with experiment generdly exceeding the matrix isolation results by between two and

four percent. The disagreement between the harmonic oscillator and experimental

frequencies for the intermolecular modes is far more pronounced. This confirms the

expectation that, due to weak hydrogen bonding, these modes are very anharmonic

and are, therefore, very poorly represented by a parabolic potential.

Mode Free-hydrogen st ret ch (vl )

1 Acceptor asymmetric stretch (u9) Acceptor symmetric stretch (vz) Donor bridge-H stretch (4

1 Donor angle bend (4 Acceptor angle bend (us) Out-plane bend (via) In-plane bend (24) 0-0 stretch (u7)

1 Acceptor twist (yl) Acceptor bend ( u ~ ) Torsion ( u 4

Present Work 3887 3884 3780 3520 1682 1621 780.6 479.2 263.2 219.0 174.4 102.9

QC Study 3967 3950 3846 3765 1667 1639 637 362 182 146 130 137

Experiment 3726 3709 3634 3574 161 1 1593 520 290 147

Table 3.2: Vibrational frequencies of the water dimer molecule in the harmonic 1 os- cillator approximation. The mode assignments follow that of Reimers et ai (311 (see Figure 3.2). The results of the present work are presented in the second column, the third column provides results from a recent quantum chernical study [22], and some experimental results [1 O] are given in the final column for comparison. Al1 frequencies are in cm-'.

The eigenvectors of the dimer vibrational modes reveal the atomic motions asso-

ciated with each mode, which can be attributed to the twisting or stretching of the

various bonds in the molecule. These are depicted in Figure 3.2.

The three acceptor intramolecular modes correspond very closely to the monorner

vibrational modes, as expected since dirnerization has little effect on the geometry of

the acceptor molecule. The frequencies of these modes, however, are slightly shifted

in the acceptor molecule due to the slight increase in the OH bond length and the tiny

DQnm Acceotor Donor Acceotor

Figure 3.2: The normal modes of vibration of the water dimer molecule (adapted from Coker et al [31]. Column (a) illustrates the intramolecular modes while column (b) provides the low frequency intermolecular modes. The frequencies of al1 modes and t heir names are given in Table 3.2.

reduction in HOH angle. In contrast, the intramolecular modes of the donor molecule

are significantly altered due to the presence of the additional hydrogen bond in the

dimer.

The results obtained here will be employed, in conjunction with an expression for

the absorption cross-section, to produce quantitative absorption spectra for the water

monomer and dimer in the harmonic oscillotor-rigid rotor approximation.

Chapter 4

Rot ation-Vibrat ion Energy

Spectrum

4.1 Energy Levels

The non-relativistic Hamiltonian for a molecule can be written as a s u m of the nuclear

kinetic energy, T,, the electronic kinet ic energy, Te, the repulsion energy between A *

nuclei, V,, the attraction energy between electrons and nuclei, V,, and the repulsion A

energy between electrons, V., ,

w here,

with a, p referring to nuclei and i, j speciSing electrons in the molecule. Ma is

the mass of the a nucleus, me is the mass of an electron, 2, is the atomic oumber

of nucleus ath, and rab is the distance between nucleus or electron a and nucleus or

electron b (see [52]). To find the energy levels of the molecule we must solve the

Schrodinger equation,

Due to the enormous complexity of the Hamiltonian, it is useful to expoit a number

of approximations in solving Equation (4.3).

First, it is necessary to separate nuclear and atomic motion making use of the

Born-Oppenheimer approximation [53]. Since typical electronic energies are much

larger than typical nuclear energies', the nuclei appear, to the electrons, to be sta-

tionary. This allows the nuclei to be 'frozen in' during the calculation of the electronic

where fî.1 = Te + Y, + e,. The repulsion energy between nuclei is now just a oumber

that can be added to give the total energy of a given electronic state,

The Schrodinger equation has now been separated into electronic and nuclear

'The kinetic energy of a typical electron in the molede is on the order of several electron volts while typical nuclear kinetic energies (resulting from vibrational and rotational motions) are on the order of 1ob4 to loW3 electron volts [54].

parts which can be solved separately,

[ fn + ~ ( g a ) ] $ ~ ( r a ) = E N ~ N ( L )

and the total statevector of the molecule is,

@ = ilC Ci^

The eigenfunctions and energy O f electronic states depend on the positions of the

ouclei, Fa, in a parametric way owing to the fact that both V, as well as are

functions of nuclear position. The nuclei, on the other hand, experience a potential

corresponding to an 'average' distribution of the extremely rapid electrons and, thus,

the potential U(Q derived in Equation (4.6 a) serves as the potential energy for

the nuclear motion in Equation (4.6 b). The effective potential energy surface can

be found by either numerical solution of the appropriate Schr6dinger equation or by

constmcting a semi-empirical model.

The RWK2 potential is constructed by empirically fitting to experimental data,

which is comprised of measurements made on s&ciently long time scales that the

resulting potential inctudes the 'average' over electronic positions automatically. To

determine the energy levels of the water dimer rnolect.de, described by the RWK2

potential, the nuclear Schrodinger equation (4.6 b) needs to be solved.

If we neglect coupling between rotational and vibrational motions and nuclear

spins, the nuclear statevector can be written,

and the Hamiltonian c m be separated into a rotational and a vibrational p u t

There is no nuclear spin term in the Hamiltonian but the nuclear spin statevector,

$nr determines the statistical weights of rotation-vibration states.

In the harmonic approximation, the vibrational Hamiltonian is,

where qj is the normal coordinate and wj = S T C V ~ with üj the vibrational frequency

of the jth mode (see Table 3.2). The sum runs over au 3N - 6 degrees of freedom of

the molecule. A set of raising and lowering operators,

are defined [55] such that acting upon a statevector corresponding to the jth mode,

Inj), results in,

In terms of these operators the Hamiltonian takes the simple form,

The most convenient coordinate system to use in describing rotations is that

Table 4.1: Moments of inertia about the principal axes of the water monomer and dimer. The principal axes are defined in Figures 2.1 and 2.2.

formed by the principal axes of the molecule. In terms of the angular momentum

operators and moments of inertia about these axes, the rotational Hamiltonian is

given by [56],

where the required moments of inertia are listed in Table 4.1.

By virtue of the fact that two of its moments of inertia are very nearly equal, the

dimer can be approximated as a symmetric top. Taking 1, = 4, then, the rotational

Hamiltonian can be rewritten in terms of the total angular momentum operator, 12 J = 3: + j' + jz, and the operator for the angular rnornentum dong the secalled

figure axis, &,

12 It is easy to show that J and jC commute and, therefore, share a cornmon set of

eigenvectors. If J is the total angular momentum quantum number, K is the projection

of the angular momentum on the figure axis, and M is the projection of the angular

momentum onto an arbitrary space-fixed axis2 (see Figure 4.1), t hen statevectors,

1 J K M ) can be defined which satisfy,

2The rotational energy of a rnolecule does cot depend on the value of M. For this reason many authors use only the J and K quantum numbers to describe a rotating molecule, When discussing the interaction of the molecu1e with external radiation, however, care needs to be taken to ensure that the direction of the electric field is adequately taken into account. The M dependence of the statevectors is explicitly included at this time to maintain consistency with the remainder of this work.

Figure 4.1: Definition of the rotational quantum numbers.

The energy of a given rotation-vibration state is found by solving,

Using Equations (4.12) and (4.14) along with the relations in Equations (4.1 1 a),

(4.1 1 b), (4.15 a) and (4.15 b) we find the energy levels of a rigid symrnetric top in

the harmonic oscillator approximation to be,

where A = h2/21, and C = fi2/21,.

The monomer possesses three distinctly different moments of inertia and is, there-

fore, an asymmetric top. In contrast to the dimer, which is an "accidentdy symmet-

ricn top, no closed forrn exists for its energy levels for arbitrary J [56]. The energy

levels of the asymmetric top are most easily obtained by diagonalizing the general

rigid rotor Hamiltonian (Equation (4.13)) in the symmetric top basis.

The asymmetric top statevector will be denoted by IJTM) where r = 1,2,. -, 25 + 1. J and M have the same meaning as in the symmetric top case. T , however,

is a pseud-quantum number, inserted to label the states. It a c tudy represents a

combination of the projections of the angular momentum about au three axes. Each

statevector is a linear combination of the symmetric top statevectors,

where K runs from -J to J. These are the vectors which diagonalize the rotational

Hamiltonian matrix in the symmetric top basis, a matrix of rank 25 + 1 for a given

J . Thus the coefficients, CEM, are found by computing the 25 + 1 eigenvectors of

the Hamiltonian.

Recall that the Hamiltonian for an asymmetric top rigid rotor is given by

As opposed to the symmetric top, this Hamiltonian can no longer be reduced to a

fonn involving o d y two commuting operators since the moments of inertia are al1

different. As usual, the energy levels are found by solving

For given J and M, kS is expanded in the symrnetric top basis using the properties of

the angular momentum operators. It is easy to show that the elements of the rnatrix

representation of the asymmetric top Hamiltonian in the symmetric top basis, H p K ,

are given by [56],

B-A - 4 J ( J - K ) ( J - K - I ) ( J + K + ~ ) ( J + K + z ) ; A K = l t 2

= [ J ( J + ~ ) 4 - K * ] + C K ~ ; AK=O (4.21)

0; for all other AK

The fact that the only non-vanishing matrix elements have AK = O, I 2 is easily

explained. Just as was the case with the normal coordinate operators, t j j , the angdar

momentum operators can be writteo in terms of raising and lowering operators. Since

the Hamiltonian consists of the squares of these operators, the initial state must either

be raised twice, AK = 2, lowered twice, AK = -2, or raised once and lowered once,

MY = O. Hence fj., takes the form,

(4.22)

The eigenvolues of this 25 + 1 x 25 + 1 matrix are the non-degenerate energy levels of

the asymmetric top and the eigenvectors form the asymmetric top statevectors. Note

that asyrnmetric top energy levels are also independent of the M quantum number.

Table 4.2 provides a List of selected pure rotational energy levels of the water

monomer. The computed values agree well with the experirnental data given in

Dennison [57] deviating by less than one percent for the low J values and by about

three percent for J = 9. Due to the fact that the rigid rotor approximation does

not take centrifuga1 rotation-vibration coupling into account, the errors continue to

Present Work Experiment

Table 4.2: Pure rotational energy levels of the water monomer (al1 energies are in cm-'). The experimental data are from Dennison (571.

increase for higher J. The occupation of higher J states is so srnail, however, that

they have little impact on the absorption s p e c t m at atmospherk temperatures.

4.2 Transition Frequencies

It is now possible to determine the energies corresponding to d monomer ond dimer

rotation-vibration transitions. Using the result in Equation (4.17) it is found that to . -

undergo a transition from the state (i) = Inin;. . .n& J'K'M') to the state 1 f ) =

~n{n{ . . . ni,-,; Jf K f ~ f ) , a rigid symmetric top rnolecule must absorb a photon of

energy,

which results in a peak in the absorption spectrum of the molecule at,

The expression in Equation (4.24) with N=6 and appropriate values for A, C, and

the uj, provides the locations of all peaks in the absorption spectrum of the water

dimer. The approximations used to arrive at this conclusion are summarized here:

i. The electronic and nuclear motions are separated by the Boni-Oppenheimer

approximation.

ii. AU vibrational modes are harmonie.

iii. The dimer is a rigid symmetric top.

iv. There is no coupling between rotations, vibrations or nuclear spins in the Harnil-

tonian.

The consequences of these approximations wilI be discussed further in section 6.4.

Since no closed form expression for the energy levels of an asymmetric top is avail-

able for arbitrary J, an expression equivalent to Equation (4.24) for the monomer

doesn't exist. The frequencies of each monomer transition must be evaluated indi-

vidually using,

with the rotational energies ob tained in the diagonalization of the asymmetric top

Hamiltonian matrix using values of A, B, and C for the monomer. The same ap-

proximations that are used in the dimer case apply here except, of course, that the

monomer is treated as a rigid asymmetric top rotor.

4.3 Nuclear Spin

Until now the spins of the nuclei making up the molecule have not been considered.

The nuclear statevector (Equation (4.8 a)) has a third factor, the nuclear spin wave

function, which does not affect the energy of rotation-vibration levels but plays a

crucial role in determining their statistical weights.

The complete nuclear statevector is

If a symmetry operation (eg. rotation of 360/n, n = 2,3, ..., degrees about a symmetry

axis) results in a molecular configuration that can dso be obtained by the permutation

of of two or more identical nuclei in a molecule, care must be taken to ensure that

the Pauli Exclusion Principle is not violated.

In the case of the water monomer, a rotation of 180' about the c principal axis is

equivalent to interchanging the two hydrogen nuclei. Permutation of two hydrogen

nuclei must result in a change of sign of the total nuclear statevector since they are

fermions (spin-$). The product of statevectors making up +N rnust, therefore, be

antisymmetric under the interchange of these two nuclei.

The vibrational statevectors of the symmetric stretch (v l ) and angle bending (vz)

modes are both symmetric while that of the asymmetric stretching mode is antisym-

metric as can be seen by simply switching the two nuclei in the corresponding eigenvec-

tors. Determining the symrnetry of the rotational statevectors is also straightforward.

Due to the form of the matrix representation of the asymmetric top Harniltonian (see

equation (4.22)), the eigenvectors consist of a sum of symmetric top statevectors with

K either even or odd, never a combination of the two. Since the symmetric top stat-

evectors are symmetric under a rotation of 180" for even K and antisymmetric for

odd K,3 the asymmetric top statevectors can be classified according to which CE,% are non-zero. For simplicity, those statevectors for which the even K's contribute will

be labeled r(even) while the others will be labeled r(odd).

The total nuclear spin statevector of the monomer is denoted by (S,, Sz2) = IS,, ) 8

IS,) where S, = +z or -2 is the z-component of the spin on the ith hydrogen

nucleus4. There are four such total spin states (see Townsend [55]) three of which are

symmetric under the exchange of nuclei; this is the spin-l triplet state,

and one which is antisymmetric; the spin-0 singlet state,

Suppose the monomer is in a symmetric vibrational state (i.e. ul or 4. If it is

also in a ~ ( o d d ) rotational state, the nuclear spin state must be symmetric to yield

3The rotational statevector depends on K thmugh e-iK6 [52]. For 0 = 180' this factor is +l when K is even and -1 with odd K.

4Note that the oxygen nucleus has spin zero and, therefore, must be in the same spin state in every case.

a total statevector which is antisymmetric. Since there are three possible nuclear

spin statevectors which satis% this requirement, states with a symmetric vibrational

state and a r(odd) rotational state have a statistical weight of three. Conversely, if

the rotational state has ~ (even) , the nuclear spin statevector is the antisymmetric

singlet state. In this case the statistical weight is one. When in the antisymmetric,

us, vibrational state, on the other hand, the statistical weights are reversed yielding

a statistical weight of three for ~ (even) rotational states and a statistical weight of

one for the r(odd) case.

Statistical weights become very important when considering the intensity of the

individual absorption peaks in the spectrum. They can lead to modulations in the

intensity of absorption bands and, in some cases, even prevent certain transitions from

occurring at all. The monomer spectrum provides a good example of a case where

both effects can be obsewed. It is clear that the statistical weight factors result in

peaks whose intensity alternates depending on whether the rotational statevector is

made up of even or odd K states. To discover the origin of this effect, note that

the rotational and nuclear spin dependence of the strength of a rotation-vibration

transition takes the form, ( J ' ~ ' M ' ; n s ' l ô l ~ r ~ ; ns). The operator, Ô cannot induce

changes in the nuclear spin state5, sol

(J'T'M'; ns'1Ô I J T M ; ns) = (J'T'M'IÔ(JT M ) (ns'lns) (4.28)

This result indiates an immediate selection rule on the r labels, narnely that no

transitions can occur between r(even) and r(odd) states.

The water dimer molecule has no rotational syrnmetry but does possess a reflection

plane (this is the plane of the donor in Figure 2.2). Experimental studies of the

water dimer rnolea.de indicate that tunneling modes exist in which the two acceptor

hydrogen atoms are effectively interchanged [IO]. This is a very different situation

from that of the monomer and two different approaches can be taken. There are two

modes, the acceptor asymrnetric stretch (u9) and the acceptor twist (y ,), which are

'This is discussed in more detail in chapter 5.

antisymrnetric under the interchange of the acceptor hydrogen atorns. If one couples

nuclear spin statevectors to these modes, they must have a statistical weight of three.

AU other dimer vibrational states have statistical weights of unity. On the other

hand, since there is no non-trivial rotation which is equivalent to reflection over the

symmetry plane, one rnight choose to ignore spins altogether. Both possibilities are

considered in chapter 6. It should be ernphasized that this choice of approaches need

not imply a contradiction. Selection ' des" are always defined with respect to a

specific choice of a complete set of approximate eigenstates so the use of different

approximations wiU result in somewhat different selection d e s .

Chapter 5

Absorption Line Intensities

The following few sections outline the details of the quantum mechanical theory of the

absorption of electromagnetic radiation by a system of molecules. Section 5.1 provides

a brief d e r i ~ t i o n of the absorption cross-section, cr(w), and the following sections

describe the computation of the dipole moment matrix elernents in the vibrational

and rotational bases defhed in chapters 3 and 4. These analyses are summarized in

the final section along wit h

5.1 Absorption

sorne interpretation of the resdt.

Cross-section

The Hamiltonian for the interaction between a rnoiecule and an external electric field

is given by,

&(t) = -ji(t) Ë ( t )

provided the wavelength of the radiation field is large compared with molecular di-

mensions (i.e. the field is uniform over al1 parts of the molecule). $ ( t ) is the total

electric dipole moment operator of the molecule and Ë( t ) = EoEcoswt. Eo is the

amplitude of the field and Z is the unit polarization vector of the electric field. Only

single photon absorption or emission processes are considered in this study.

The probability per unit time that the system will undergo a transition from the

initial state, li), to a final state, 1 f) , is given by Fermi's Golden Rule (see, for example,

Davydov [58] ) ,

where wfi = w j - wi (see section 4.2). The rate at which the radiation field loses

energy to the system is obtained by multiplying Equation (5.2) by the energy of the

transition, summingover au final states, and, findy, multiplying by pi , the probability

that the molecule is in the initial state, and summing over aU initial states, i.e.

The summations i and f both run over all allowed states of the system so these

"dummyn indices may be interchanged in the summation over the second S function

aLlowing Equation (5.3) to be reduced to,

If it is assumed that the system is initially in equilibrium, then the probability of

finding the system in the state I f ) is related to that of the initial state li) through,

where f l = l / k T . When Equation (5.5) is substituted into Equation (5.4) and the

result is divided by the magnitude of the total energy flux in the radiation field,

191 = veEi/8r = mRg/8a, the following expression for the absorption cross-section

is obtained [59],

v is the speed of light in the medium, e is the dielectric constant of the surrounding

medium, and ?%R is the real part of its index of refraction. The subscript is dropped

from the w's since the delta function requires w = w/i .

The goal of the present chapter is to compute a(w) for the fundamental absorp-

tion bands of both the water monomer and dimer molecules. The problem requires

knowledge of both the probability that the system will be in a given initial state, pi, 2

as well as the modulus çquared dipole moment matrix elements, 1 (f 12 - qi) 1 . The

remaining sections of this chapter wiU examine t hese terms in detail. All approxima-

tions employed will be discussed as they are introduced and the significance of the

final results wilI be clarified in the summary.

5.2 Initial State Probabilities

The probability of finding a system in the initial state, (i), is gi r the Boltzmann

factor divided by the rotation-vibration partition function for the system,

where ci is the energy of the state li), and P = l / k T . The vibrational partition

function of any molecule is given by the surn over ail allowed vibrational states of 3N-6

e - ~ c : t b where e y b = ((n + -)kiwi, 1 with nj = 0 ,1 ,2 ,..., SO, j=i 2

If nuclear spin statistics are included an additional factor, Xj, must be included in the

dimer case to account for the stat istical weights of the antisymmetric dirner vibrations.

This factor will be three for the antisymmetric modes and one for the symmetric

modes-

From section 4.2, the energy of a rotational state is given by,

for the dirner, a rigid symmetric top rotor. Agaio, the partition function is obtained

by sumrning ë P L : O t over all allowed rot at ional st ates,

where the factor 2 J + 1 is inserted to account for the degeneracy of the rotational

energy levels resulting from the M quantum nurnber l .

In the case of the monomer, an asyrnmetric top. the rotational partition function

must be left in the fom, w 2J+i

Zrot = C C e - / 3 c ~ 7

J=O r=I

Nuclear spins require that for symmetricvibrational states (y and v2) the T consisting

'The M quantum number represents the projection of the angular momentum along a space-fixed axis and, therefore, can tabe on a11 values from - J to J inclusive resulting in the 2J + 1 degeneracy noted above (see section 4.2). This is a consequeace of the isotropy of space irrespective of molecular symrnetry.

I

of a sum over odd K have a statistical weight of three (see section 4.3).

Asymmetric vibrational states (in this case 4, however, must couple to a symmetric

product of rotational and nuclear spin statevectors and, therefore, have the statistical

weights reversed.

The rotation-vibration partition function, 2, is the product of the rotational and

vibrational partition funct ions, = p p b

Substituting the expression for the energy of a symmetric top rotation-vibration state

(see section 4.2) Equation (5.7) yields the probability of finding the dimer ini t idy in

the state li),

- pidimer -

where the +Aj are a11 one if nuclear spins are ignored. Similady,

+ 00 2J+l e - P h ~ 3 / 2

J=0 r (odd) J=O ~ ( e v e n ) 1 - e - P h

yields the probability of hd ing the monomer in the initial state.

5.3 Dipole Moment Matrix Elements 2

The rnodulus squared dipole moment matrix elements, 1 (f 15 - qi) 1 , now need to

be addressed. There are two sets of axes which need to be defined before these

matrix elements can be calculated. Again, the most convenient set of molecule-fuced

coordinates are the principal axes of the rnolecule, g = {a, b, c) . These axes rotate

with the molecule and, thus, the equilibriurn components of its dipole moment along

these axes as well as its moments of inertia about these axes remain constant as it

rotates. In addition, a set of laboratory-fixed axes, F = (X, Y, Z), is defined by the

polarization of the incident electric field. The amplitude of the electric field along

these axes will remain constant regardless of the motion of the molecule.

The dipole moment can be decomposed into components along the moleculefixed

axes, = Mgg, where is a unit vector along the g principal axis, and the modulus 9

squared matrix elements become,

where the aF, are the direction cosines relating the principal axes, g, to the space-

fixed axes, F. The dipole moment operators, &, are expanded in a Taylor series

in the normal coordinates. If the effects of electrical anhamonicity are neglected,

these expansions can be truncated at the first order term, resulting in dipole moment

operators that are linear in the normal coordinates. This gives,

where 6bg = 4 . Since vibrational transitions require an ascillating dipole aqj ,

moment, the matrix element , (f 1 (&q + rotational and a rotation-vibration part,

(f 1 (P? + &fi,) ~ F L T I ~ ) = P ~ ( E I ~ F ~ I

6kg) OFg li), can be separated into a pure

R)&Pv + (v '~~&Iv)(R I ~ + ~ I R ) (5.19)

where V, V', R, and R' are used to denote initial and final vibrational and rotational

statevectors2. The vi brational and rot at ional dipole moment matrix element s can

t herefore be considered separately.

5.3.1 Vibrational Matrix Elements

The necessary vibrational rnatrix elements are given, in the linear dipole moment

The normal mode displacement operators, pj7 can be written in t e m s of the raising

and lowering operators (see Equation (4.10 a))

and, writing Dgj = ,/= 21 , the vibrational matrix elements becorne, ~ P , w , =q

These matrix elements are now easily e d u a t e d using the properties of the raising and

lowering operators (Equations (4.11 a) and (4.11 b)). Immediately, the vibrational

2Note that the nuclear spin statevectors are implicit on the right hand side of Equation (5.19). Al- though they are invariant under application of the dipole moment operator, they affect the statistical weights of transitions and impose additional selection rules.

selection d e s in the harmonic oscillator approximation are found to be Anj = H.

The absorption of a photon corresponds to an innease in the energy of the system so

the Anj = 1 matrix elements are needed.

The complete spectmm will include emission t e m s which might appear to have been

neglected here. These terms, however, have already been accounted for in the deriva-

tion of the absorption cross section. R e c d that the summations in Equation (5.3) nin

over all initial and final states (including those final states which are lower in energy

thon the initial states). In interchanging the indices in the second delta function to

reduce this equation to a single sum, emission terms, which correspond to the original

initial state losing energy, are replaced by absorption terms, which correspond to the

state that was origindy the final state, gaining energy.

The vibrational matrix elements, then, are given by,

The constants Dgj are found using the eigenvalues and eigenvectors of the GF matrix

from the vibrational frequency calculation of section 3.3. In the case of the dimer,

the two antisyrnmetric vibrational modes, ug and y l , must be multiplied by their

stat ist ical weight of t hree.

5.3.2 Rotational Matrix Elements

The rotational matrix elements of the dipole moment for the water monomer and

dimer are different since the dimer is an "accidentaiiy symmetric" top while the

monomer is an asymmetric top. It has been shown, however, that the rotational stat-

evectors of an asymmetric top can be written as a linear combination of the symmetric

top statevectors (see section 4.1). As a result, the rotational dipole moment matrix

elements of the asymmetric top can be written in terms of those for a symmetric top,

where (J' K' M' (&, ( J K M ) are the symmetric top rotational dipole moment matrix

elements. Consequently, it is sficient to compute only these matrix elements to

obtain a complete description of rotations for both the monomer and the dimer.

Recall that the &Fg are direction cosines between the space-fixed coordinate sys-

tem F and the molecule-fixed coordinate system g.3 These operators belong to a class

of vector operators called T-operators and satisfy the following selection d e s ,

These results are very useful in evaluating the rotational matrix elements (5.25). The

Wigner-Eckart theorem can be used separate the K and M dependence from the full

matrix element resulting in the following relation (see Kroto [60] or Wollrab [61]),

Each term on the right hand side can be evaluated individuaily using the properties

of the angular momentum operators (Equations (4.15 a), (4.15 b) and (4.15 c)) along

wit h the commutation relations of the direction cosine matrix elements.

The applicable selection rules on J, K, and M are also easily derived from the

commutation relations (see Schutte [54]). For a rigid rotor, one finds that the only

3For a lucid discussion of the rotational matrix elements, see Kroto [6O].

I

non-vanishing matrix elements occur for,

and,

With these in mind, the direction cosine matrix elements can be evaluated for any

ailowed transition using Equation (5.27) and Table A.1 in Appendix A which provides

values for each term. As an example, the matrix element for the transition from

I J K M ) to IJ + 1 KM) is given by,

A11 other necessary rotational matrix elements con be evaluated in a similar fashion.

5.3.3 Combined Matrix Elements

It is now necessary to combine the results from the two previous sections to obtain

an expression for the complete rotation-vibration matrix elements. Recall that the

square of the modulus of the dipole moment matrix elernents is needed to compute the

absorption cross section. Returning to Equation (5.19), the square of the combined

rotation-vibrat ion matrix elernents is given by,

A doser inspection of Table A.1 shows that there are no cross terms when the

sum over g is squared, so Equation (5.30) can be written,

It is, therefore, only the modulus squared vibrational and rotational matrix elements

that are required to compute the absorption coefficients. The modulus squared vi-

brational matrix elements are given by,

Squaring the real matrix element in Equation (5.29) gives,

Since the incident radiation in this study is unpolarized sunlight, Equation (5.33) can

be surnmed over all allowed values of M. Noting that,

the desired modulus squared matrix elernent becomes,

Assuming the atmosphere to be isotropic, the molecules are randomly distributed

and randomly oriented with respect to the incident radiation. Then, if the result

is summed over all rnolecular orientations, it is possible to fix the direction of the

incoming radiation. Here, the direction of incoming radiation is taken to be the Z

space-fixed axis. This eliminates the need to average over the incident directions of the

radiation field with no loss of generality of the final results. AU of the modulus squared

the Z space-fixed a i s ) for a rigid symrnetric

a)-(5.36 i),

Al1 the necessary information to compute all rotation-vibration matrix elements of a

rigid symmetric top rotor in the harmonic oscillator approximation, is embodied by

Equations (5.31), (5.32) and (5.36 a)-(5.36 i).

Ln the case of the monomer, the modulus squared rotational matrix elements in

Selection rules for these matrix elernents must now be determined. Since T is merely a

label, not a "real" quantum number related to molecular symmetry, ail T transitions

are ailowed. However, the symmetric top selection rules still apply to the basis state

matrix elements so, for a given J and T , only 4 K = O, f l transitions will be non-zero.

Furthemore it has been shown that the Exclusion Principle requires that transitions

between T (even) and T (odd) transitions vanish. This immediately requires t hat only

A K = O transitions are nonzero. This results in the dramatic consequence t hat, in the

harmonic oscillator-rigid rotor approximation, the asymmetric stretch mode does not

give rise to any allowed transitions! Inspection of the atomic motion associated with

this mode (see Figure 3.1) indicates that an osciUating dipole moment perpendicular

to the figure axis, c, is required for a vibrational transition to occur. Table A.l

shows that al1 rotational transitions applicable to this mode must have Ah' = rtl

which would violate the selection rule imposed by the nuclear spin statevector. Al1

transitions in the spectral bands produced by the other two vibrations with odd K

will have statistical weights of three, while even K transitions have statistical weights

of one. The only surviving matrix element is

A simple extension of the arguments used in deriving the symmetric top matrix

elements yields the necessary matrix elements

5.4 Summary and Interpretation

The previous results can now be combined to yield complete expressions for the

spectra of the water monomer and dimer in the harmonic oscillator, rigid rotor ap-

proximat ion.

The absorption spectrum of the water monomer consists of a large number of delta

function peaks located at,

while the peaks in the dimer spectrum occur at

Table 5.1: Equilibnum components of the monomer and dimer dipole moment along the principal axes (e is the electronic charge).

where wj = 27tcüjj. The strength of each of these peaks is given by,

4n2 2 a = PnS-w(I - e-B'W)pi ((f 1; - qi)(

hcn

where the initial state probabilities, pi, of the dimer and monomer, are given by

Equations (5.15) and (5.16), respectively. The factor P . . is the statistical weight of

the transition determined by considering the overall symmetryof the total statevector.

It has been shown that the dipole moment matrix elements cau be separated into

rotational and vibrational parts

where the modulus squared vibrational matrix elements are given by,

and the required modulus squared rotational matrix elements are listed in Equations

(5.39 a)-(5.39 c) for the monomer and in Equations (5.36 a)-(5.36 i) for the dimer . In al1 cases the appropriate statistical weights must be inserted.

Finally, the principal axis dipole moment components of each molecule in equilib-

rium that are needed in the calculation of the pure rotational part of the spectrum

are listed in Table 5.1.

The results of this chapter indicate that the spectrum of each molecule will consist

of a pure rotational band and 3N-6 rotation-vibration bands. Each of these bands

will be made up of numerous, densely packed, rotational peaks. The details of the

method used to plot the spectrum will be described in chapter 6 and representative

spectra wiLi be andyzed.

Chapter 6

Results

6.1 Computational Techniques

The formalism discussed in the previous two chapters was implemented using a math-

ematical computational program, Maple V, and a plotting routine written in C.

Maple V was used to compute the vibrational frequencies and reduced masses of

the dimer using the RWK2 potential. The potential was entered as it appears in

section 2.2.1 and the interatomic distances, Rij, and bond angles, Bk, were written in

terms of the Cartesian coordinates of each atom using the distance formula and law

of cosines:

where i, j nui over all atoms in the water dimer and k = donor or accepter. The force

constant matrix, F, is computed by taking all second derivatives of the resulting

~otential , evaluated at the equilibrium geometry of the dimer. The matrix G-' is

simply a diagonal rnatrix with the masses of each atom along the diagonal (three

times each since there are three Cartesian coordinates for each atom). This matrix is

then inverted, GF is computed, and the eigenvalues and eigenvectors of the resulting

matrix are found.

The 12 eigenvectors that correspond to the non-zero frequencies form the so-called

'normal coordinates' of the molecule. When transformed into this basis, G and F are

both diagonal matrices (see [62]). The vibrational frequencies of the molecule appear

along the diagonal of F and the corresponding reduced masses lie on the diagonal of

the inverse of G.

To compute absorption intensities, a C program was constructed. The spectral

region of interest is broken down into intervals (bins) whose width is determined

by the prescribed resolution. The absorption intensities of all possible rotat ional-

vibrational transitions, within the harmonic oscillator, rigid-rotor approximation, are

computed and added to the appropriate bin. In addition, the number of transitions

occurring in each interval is recorded to aid in the analysis of the resulting spectrum.

After each of the allowed transitions has been included, the contents of each bin

as well as the central frequency of the bin are printed to a data file to be plotted

using any standard graphing package. The program allows the user to speci- the

spectral region of interest, the desired resolution, and the atmospheric temperature.

Special care has been taken to facilitate modification of the code to allow for a vertical

temperature profile as well as the inclusion of absorption bands which occur as a result

of anharmonic correct ions.

Representative spectra will be presented and discussed in the remaining sections.

In al1 graphs the absorption coefficient is in and wavenumber is in cm-'

unless otherwise specified.

6.2 Monomer

The water monomer spectrum in the harmonic oscillator-rigid rotor approximation

is presented in Figure 6.1. Values of J higher than nine are not included since the

probability of finding the system in such a state is a factor of 1000 less than that of the

Figure 6.1: Water monomer absorption spectnun in the harrnonic oscillator, rigid rotor approximation. The temperature is taken to be 300K and the resolution is 10 cm-'.

low J states a t 300K and much less at lower temperatures A resolution of 10 cm-'

is chosen to be consistent with representative broadband atmospheric experiments.

If the resolution is increased, the peaks become sharper and more intense as can be

seen in Figure 6.2, the monomer spectrum at 2 cm-' resolution. Thus the resolution

provides a built in method of introducing varying amounts of line broadening to the

spectra. The spectrum consists of three broad peaks which correspond to the pure

rotations (O - 500 cm-'), the angle bend mode (1500 - 2050 cm-') and the symmetric

stretch mode (3700 - 4250 cm-'). The integrated absorption cross section for the

pure rotational band is found to be 17.634 x 10-l8 cm while values of 7.9686 x IO-''

cm and 5.5994 x 10-l8 cm are obtained for the angle bending (- 1640 cm-') and bond

stretching (- 3820 cm-') modes, respectively. One of the most up to date archives of

atmospheric molecuiar data is that of the Air Force Geophysical Laboratory (AFGL).

AFGL data, based on a Hamihonian with 25 adjustable parameters, yields integrated

absorption cross sections of 52.7 x IO-'' cm for the pure rotational band, 10.4 x 10-18

cm for the angle bending mode and 7.456 x 10-l8 cm for the stretching modes [2].

Figure 6.2: High resolution water monomer absorption spectrum in the harmonic oscillator, rigid rotor approximation. The temperature is taken to be 300K and the resolution is 2 cm-'.

The source of the disagreement in integrated absorption over the pure rotational

bands is not known at this time.

The peaks in the pure rotational spectnun of the monomer, (see Figure 6.3) , arise

from transitions between the rotational energy levels computed in section 4.2. For

example, the peak at 17.894 cm-' is a result of the transition J = 1; r = 3 t

J = 1; T = 1. Another example is the transition J = 3; T = 1 t J = 2; T =

2 which Lies at 58.82 cm-'. Randall et al. [63] provide experimental spectra of

the monomer pure rotational bands. Although quantitative cornparison of absolute

intensities is impossible due to the lack of a vertical scale, the relative intensities of

these peaks generally agree qualitatively, as do similar theoretical predictions made

by the aut hors.

Wavenumber - - -- - - - - . -- - - -- -

Figure 6.3: Pure rotational water monomer spectrum at 300 K (Note that the y-axis is offset slightly so that x-axis tick marks aren't confused with rotational peaks). The resolution is 10 cm-'.

6.3 Dimer Spectrum

Figure 6.4 is the dimer spectrum at 300K and 10cm-' resolution. In this case values of

J up to 70 are needed to achieve accuracy comparable to that of the monomer spectra.

The spectrum has three main features, the pure rotational band, centered at about

20 cm-', the low frequency intermolecular bands between 70 and 800 cm-', and the

high frequency modes resulting from intramolecular vibrations. The monomer lines

appear much more intense than those of the dimer. This is a direct result of the fact

that there are far fewer excited monomer states at 300K. The total intensity of the

dimer absorption is divided among many more individual peaks which are weaker

than those in the monomer spectrum. It is interesting that this spectrum results

independent of whether or not nuclear spin modification of selection mles is included.

This is a result of a cancellation of the statistical weight of each state with an identical

factor in the vibrational partition function.

Unlike the monomer spectrum which has the appearance of a 'randorn' scatter of

Figure 6.4: Water dimer absorption spectnim in the harmonic oscillator-rigid rotor approximation. The temperature is taken to be 300K and the resolution is 10 cm-'.

lines, the peaks in the spectrum of the symmetric top water dimer have recognizable

shapes. Vibrational modes which involve a change in the component of the dipole

moment along the figure axis are referred to as parallel bands. An example of such a

band is the bridge-hydrogen stretch mode (4 which is shown in Figure 6.5. Parallel

bands consist solely of AK = O transitions so the peaks are separated by

as is seen in by the plot.

Modes which involve an oscillating dipole moment along an axis perpendicular to

the figure axis are referred to as perpendicular bands. The out-of-plane bend mode

(y,), shown in Figure 6.6, is an example of a perpendicular band. Perpendicular

bands consist of a number of equidistant lines corresponding to A J = O transitions as

well as many other much weaker lines resulting from A J = I l transitions. This pro-

nounced difference in intensity is due to the fact that only the AK = f l transitions

I Wavenumber

Figure 6.5: High resolution plot of dimer bridgehydrogen stretch mode, v3. The resolution is 0.02 cm-' and the temperature is 300K.

760 780 800 820

Wavenumber

Figure 6.6: High resolution plot of the dimer out-of-plane bend mode, v10. The resolution is 0.1 cm-' and the temperature is 300K.

Wavsnumber

Figure 6.7: High resolution plot of dimer in-plane bend mode. The resolution is 0.05 cm-' and the temperature is 300K.

are observed. Transitions with the A J = f 1 have different origins whereas AJ = O

transitions (wit h a given value of K and A K) coincide for all values of J, surnming

to give the resulting intense peak.

There are some modes which involve a component of the oscillating dipole moment

along the figure axis as well as a component perpendicular to it. This results in a

superposition of a pardlel and a perpendicular band. Figure 6.7 illustrates one such

band, the dimer in-plane bend mode, us.

Figures 6.8 and 6.9 compare monomer and dimer absorption over the two main

monomer absorption bands at high resolution. The former shows the region around

the monomer angle bending mode 1400 - 1600 cm-' and the latter encompasses both

of the monomer stretching vibrations 3500 - 4200 cm-'. Both spectra illustrate the

disordered appearance of the monomer lines compared to Figures 6.5 - 6.7. The rela-

tively small moments of inertia of the monomer result in its peaks being spread out.

Dimer peaks are densely packed in cornparison and, as a result, have a tendency to fil1

in many of the 'gaps' in the monomer spectrum. In addition, the second graph shows

Figure 6.8: Cornparison of monomer (solid lines) and dimer (dashed Lines) absorption in the vicinity of the 1640 cm-', angle bending band. The resolution is 0.2 cm-' and the temperature is 300K.

Wavenumbsr

Figure 6.9: Comparison of monomer (solid lines) and dimer (dashed lines) absomtion 8 L

in the 3500 - 4200 cm-' region. The resolution is 0.2 cm-' m d the temperature is 300K.

that dimerkation causes slight shifts in the locations of some of the intramolecular vi-

brational peaks, as a result, the entire 3520 cm-' (bridge-H stretch) vibrational band

of the dimer lies in a region of minimal monomer absorption. These results suggest

that dimer absorption may be important, even in broad spectral ranges where there

is significant water monomer absorption, depending on the dimer concentration and

the shape of monomer lines'.

6.4 Discussion

For practical purposes, the calculations of al1 spectra were tnincated at a finite value

of J . A convenient way of estimating the resulting error is to compare the rotational

partition function, Zrot, tnincated at the fixed value of J with the value obtained

by summing over aLI J. At high temperature, rotational levels are sufficiently close

together that we con replace the summations in Equation (5.10) with integrals over

J and K which can be integrated to give [60],

In the case of the dimer A = B. Table 6.1 compares the partition function obtained

by truncating the sum over over ePCroc at the values of J used in the plotting code to

those resulting from Equation (6.4). The lineshape function, defined by

provides another measure of the effect of neglecting higher values of J. The Lineshape

function can be summed over al1 pure rotational levels (terms for which = 0) - .

lExperimenta1 spectral lines have a finite width and shape resulting from the thermal energy of the molecules (temperature broadening) , and collisions between molecules (pressure broadening) . In most atmospheric applications the line-widths arising from these broadening effects are on the order of 1 cm-' or l e s .

'The system under consideration is a t typical atmospheric temperatures, T - 300K. At these temperatures, kT - 200 cm-' > rotational level spacing which is on the order of 0.5 cm-'.

1 Dimer (J,, = 70) 1 8284.86 1 8261.76 1 Table 6.1: Cornparison of tmca t ed and total rotational partition functions for the water monomer and dimer at 300K.

to give 13. In the case of the monomer, truncating at J = 9 leads to an error

of just 0.3 percent while truncating at J = 70 in the dimer case results in an error

of one percent. The uncertainty introduced in the total rotational partition function

when J is truncated a t 9 in the monomer calcdation and 70 in the dimer cdculation

is less than one percent, much less than the errors incurred as a result of neglecting

anharmonic effect s and rot at ional-vibrat ional coupling.

Anharmonic correct ions make up the most significant omission from t his st udy.

It is well known that hydrogen bonded systems exhibit strong anharmonic effects

and that the anharmonic effects are most severe in the intermolecular modes. As an

example, due to neglecting anharmonic corrections, the low frequency intermolecular

bands of the dimer appear very intense. This is an artifact of the harmonic oscillator

approximation in which the transitions between excited states, so called 'hot bands',

Lie at exactly the same frequency as the fundament al transit ions. Anharmonic correc-

tions lead to slight shifts in the hot band frequencies resulting in broader, less intense

peaks. Rough estimates based on higher order derivàtives of the RWK2 potential

energy surface have been made indicating that the anharmonicity is too strong to be

treated as a simple perturbation. More sophisticated (self- consistent) methods of

including such effects are necessary in future studies. Initial estimates, obtained by

treating the intermolecular modes as Morse oscillators, suggest that the intensity of

ailowed overtones is strong.

Combination bands involving two or more transitions and overtone bands, Anj =

2,3 , . . ., are also a direct result of anharmonicity. Again, both effects have the greatest

impact on the low frequency intermolecular modes of the dimer. Overtones and

combinations of these modes wil1 result in further broadening of the low frequency

part of the spectrum. These efEects rnay result in significant spreading of the spectnun

possibly accounting for at least a fraction of the observed continuum absorption in the

8 - 14 pm window. It is also likely that combinations of inter- and intramolecular

modes occur when anharmonic effects are included. If appreciable mixing of the

inter- and intrarnolecular modes transpires, the higher frequency end of the spectrum

rnay be extended suggesting a possible source of the observed shortwave anomalous

absorption.

In addition to the 'mechanical' anhmonici ty described above, there exists 'elec-

trical' anhannonicity stemming from the use of the linear dipole moment approxima-

tion. Higher order terrns in the Taylor series expansion of the dipole moment operator

also allow overtone and combination bands increasing the likelihood of extending the

dimer spectnun to higher frequency.

For a complete treatment of the dimer, rotational-vibrational coupling must also

be included. As a molecule rotates, bonds are stretched resulting in shifts in the

vibrational frequencies of the molecule. This effect is known as centrifuga1 distortion.

In excited vibrational states the effective equilibrium positions of the nuclei change

causing shifts in the rotational energy levels, called C0n01is coupling. In light of

evidence that the dimer is not rigid [64] these effects both should be taken into

account if very accurate results are desired.

Uncertainties in the strengths of individual spectral Lines arise from errors associ-

ated with the molecular dipole moments. The equilibrium structure of both the dimer

and monomer show some dependence on mode1 and as a result, different equilibrium

values of the dipole moment exist . Variance in the equilibrium dipole moments lead

to uncertainties in the intensities of pure rotational lines. Little can be done to avoid

such uncertainties unless the results of a number of different models are compared.

6.5 Water Dimer Atmospheric Concentrations

Finally, to assess the importance of small water clusters, their concentrations (relative

to that of the monomer) must be considered. This information wiU not only aid in

evaluating the significance of dimer absorption, but will also indicate whether or not

clusters larger than the dimer need to be examined.

The concentrations of small water clusters are estimated following a procedure

similar to that of Suck et al [65, 661. As radiation passes through an absorbing

medium its attenuated intensity is given, as a function of path length, z, by the

Beer-Bouguer-Lambert Law:

where 1: is the intensity of incident radiation at wavenumber ü and kI = Ca, is the

absorption coefficient. cr, is the absorption cross-sect ion and C is the concentrat ion

of absorbers in the atmosphere. To estimate the total atmospheric absorption of a

particular species of water cluster their concentration must be known.

The concentrations of small water clusters and water complexes have been consid-

ered in a number of recent studies [65] - [ 6 ï ] . Here water vapor is treated as a system

of small water clusters, (HzO)i, i = 1,2,3, . . . in equilibrium, obeying the following

react ion:

(H*O)i-i+ H z 0 + (H2O)i (6.7)

Using the grand canonical ensemble representation for the above reaction, the follow-

ing law of mass action can be obtained (651:

where Ni is the number of (HzO)i clusters and Zi is the single particle partition

function of the cluster. Equation (6.8) is subject to the constraint that the total

number of HzO molecules in the system remains constant,

The single particle partition function c m be written as a product of translational,

electronic, rotational, intemolecular, and int ramolecular partit ion functions:

provided it is assumed that the respective motions can be uncoupled. The individual

partition functions in Equation (6.10) are given by:

6(i-1) exp (- 2 3

j=i 1 - exp ( - f lhvy tcr)

t a = exp (-+y") j=i 1 - exp (-phv?'")

where V is the volume of systern, ,d = llkT, ci is total binding energy of the cluster,

Ro is the degree of degeneracy, r ) is the symmetry number of the point group of the

cluster [68], and I, is the moment of inertial about the a principal axis.

Combining Equations (6.8) and (6.10), the concentration of an i molecule cluster

is:

(6.12)

where it has been assumed that the intramolecular vibrational frequencies of the

clusters are equd to the those of the monomer3. The electronic partition function of

the monomer is taken to be unity.

Since the hydrogen bonding in clusters is much weaker than the chernical bonding

in the monomer, it can be assumed that there will be many more monomers than

clusters in the system and, therefore, the total water wpor pressure, P, is largely due

to the monomer,

PV x NIRT (6.13)

which, upon rearranging, yields an expression for the monomer concentration, Ci:

Finally, inserting (6.14) and (6.11 a)-(6.11 e) in (6.12) yields an expression for the

concentration of any small water cluster,

If the values of the binding energy, moments of inertia and vibrational frequencies,

computed using the RWK2 model, are substituted into Equation (6.15), it is found

that the relative concentration of dimers to that of monomers, C2/C1, ranges from 4 to

7 x IO-* over the range of typical atmospheric temperatures. This suggests that the

3Their have been numerous studies which indicate that cluster formation results in only negligible deviations of the intramolecular frequencies from those of the monomer (see, for example, 1461).

average relative concentration of dimers is on the order of 1 0 - ~ although these results

are very sensitive to the choice of vibrational frequencies and binding energy used in

the calculation. In regions of Little or no monomer absorption the total effect due to

the presence of dimers may be significant and certainly warrants further investigation.

Furthermore, Equation (6.15) also indicates that the concentration of water dimers

is approximately 200 times that of trimers, (H20)3, under similar conditions. The

concentrations of larger clusters decrease rapidly with increasing cluster size [66]. As

a result, it is likely that ody the effects of water dimers need to be considered in most

applications involving the atmospheric absorption of s m d water clusters.

Chapter 7

Conclusion

An extensive study of the equilibrium and absorption properties of the water monomer

and dimer molecules has been conducted using the potential energy surface of Reimers,

Watts and Klein (11. These authors have shown that the computed equilibrium struc-

tures and many structural properties of both molecules agree with experiment. There

is agreement with experimental data in the solid, Liquid and gas phases illustrating

the versatility of the model.

Using normal mode analysis, the vibrational modes and their frequencies were

determined for each molecule. AU of the monorner frequencies and the intramolecular

dimer frequencies agree weU with experimental data. The intermolecular frequencies

of the dimer molecule, however, deviate by as much as 75 % from the limited exper-

imental data available. The most likely explanation for t his is strong anharmonicity

associated with the weak hydrogen bonding in the molecule.

Rotation-vibration spectra of the water monomer and dimer have also been ob-

tained in the harmonic oscillator-rigid rotor approximation. The monorner spectrum

is comprised of three bands. Pure rotational bands are obsenred from O - 450 cm-':

the angle bending mode lies between 1400 - 1950 cm-' and the 3550 - 4200 cm-'

band is a result of the bond stretching modes. The positions of the rotational bands

agree with experiment and qualitative agreement is obtained for the intensities of

t hese lines.

Dimer absorption bands can be assigned to four broad categories, the pure rota-

tional bands, 20 - 65 cm-', the intermolecular vibrational modes, 75 - 900 cm-', and two regions dominated by intramolecular vibrations, 1500 - 1800 and 3500 -

4050 cm".

The present results indicate that with anharmonic corrections and rotation-vibration

coupling the interrnolecular contribution to the water dimer spectrum may be spread

out and extended to higher wavenumber. It is, therefore, important to extend this

work to include both mechanical and electrical anharmonicity as well as rotation-

vibration coupling in order to determine absorption in the 8-14pm window.

To obtain accurate quantitative estimates of the absorption coefficient of molecular

complexes as large as the dimer is a very complicated problem. As a result, in addition

to the further refinements of the current method, it would be instructive to compare

the results of other procedures such as quantum rnolecular dynamics calculations and

semi-classical methods. Due to the absence of experimental results, such cornparisons

form a d u a b l e way of assessing the overall accuracy of the computation.

In conclusion, results from this study form a solid foundation for future studies of

water dimer absorption. The current method is valid over a temperature range which

include those used in matrix isolation studies and typical atmospheric temperatures

and the RWK2 potential energy surface has been found to be quite suitable for studies

of this nature. In the future it will be necessary to include a comprehensive treatrnent

of anharmonic corrections.

Appendix A

Direction Cosine Matrix Elernents

Table A.1 provides the direction cosine matrix elements in the symmetric rotor basis.

The values include an arbitrary choice of phase which is chosen to be consistent with

that used by Wohab [61].

Table A.1: Direction cosine matrix elements in the symmetric rotor basis.

Bibliography

[l] J. R. Reimers, R. O. Watts, and M. L. Klein, Intermolecular potential functions

and the properties of water, Chem. Phys. 64, 95-1 14 (1982).

[2] R. M. Goody and Y. L. Yung, Atmospherie Radiation Theoretical Basis: 2nd

Ed, Oxford University Press, New York, 1989.

[3] G. Stephens and S.-C. Tsay, On the cloud absorption anomaly, Q. J. R. Meteor.

SOC. 116, 671-704 (1990).

[4] R. D. Cess, M. H. Zhang, P. Mirmis, L. Corsetti, E. G. Dutton, B. W. F. D. P.

Garber, W. L. Gates, J.-J. Morcrette, G. L. Potter, V. Ramanathan, B. Subasi-

lar, C. H. Whitlock, D. F. Young, , and Y. Zhou, Absorption of solar radiation

by clouds: Observations vs. models, Science 267, 496-499 (1995).

[5] 2. Li, H. W. Barker, and L. Moreau, The variable effects of clouds on atrnospheric

absorption of solar radiation, Nature 376, 486-490 (1995).

[6] P. Pilewskie and F. Valero, Direct observations of excess solar absorption by

clouds, Science 267, 1626-1629 (1995).

[7] V. Ramanathan, B. Subasilar, G. Zhang, W. Conant, R. D. Cess, J. Kiehl,

H. Grassl, and L. Shi, Warm pool heat budget and short wave cloud forcing: A

missing physics, Science 267, 499-503 (l995).

[8] T. Hayasaka, N. Kikuchi, and M. Tanaka, Absorption of solar radiation by

stratocumulus clouds: Aircraft measurements and theoretical caldations, J.

Appl. Meteor. 34, 1047-1055 (1995).

[9] P. Chylek and D. J. W. Geldart, Water vapor dimers and atmospheric absorption

of solar radiation, to be published in Geophys. Rev. Lett., 1997.

[IO] T. R. Dyke, K. M. Mack, and J. S. Muenter, The structure of water dimer

dimer from molecular beam electric resonance spectroscopy, J. Chem. Phys. 66,

498-510 (1977).

[Il] M. F. Vernon, D. J. Krajnovich, K. S. Kwok, J. M. Lisy, Y.-R. Shen, and Y. T.

Lee, lnfrared vibrational predissociation spectroscopy of water clusters by the

crossed laser-molecular beam technique, J. Chem. Phys. 77, 47-57 (1982).

[12] R. P. Page, J. G. Frey, Y.-R. Shen, and Y. T. Lee, Infrared vibrational predis-

sociation spectra of water dimer in a supersonic molecular beam, Chem. Phys.

Letters 106, 373-376 (1984).

[13] D. F. Coker, J. R. Reimers, and R. O. Watts, The infrared predissociations

spectra of water clusters, J. Chem. Phys. 82, 3554-3562 (1985).

[14] F. Huisken, M. Kaloudis, and A. Kulcke, Infrared spectroscopy of small size-

selected water clusters, J. Chem. Phys. 104, 17-25 (1996).

[15] R. M. Bentwood, A. J. Barnes, and W. J. Orville-Thomas, Studies of intermolec-

ular interactions by matrix isolation vibrational spectroscopy, J. Mol. Spectry.

84, 391-404 (1980).

1161 L. A. Curtiss and J. A. Pople, Ab initia calculation of the vibrational force field

of the water dimer, J. Mol. Spectry. 55, 1-14 (1975).

[17] 0. Matsuoka, E. Clementi, and M. Yoshirnine, CI study of the water dimer

potential surface, J. Chem. Phys. 64, 1351-1361 (1976).

[18] 2. Slaninô, Isomerism, energetics and thermodynarnics of (H20)2 and its role in

real gas phase of water, Cou. Czech. Chem. Commun. 45, 3417-3435 (1980).

[19] D. J. Swanton, G. B. Bacskay, and N. S. Hush, The infrared absorption intensities

of the water molecule: A quantum chemical study, J. Chem. Phys. 84,5715-5727

(1986).

[20] E. Honegger and S. LeutwyIer, htramolecular vibrations of small water clusters,

J. Chem. Phys. 88, 2582-2595 (1988).

(211 K. Laasonen, F. Csajka, and M. Parrinello, Water dimer properties in the

gradient-corrected density functional theory, Chem. Phys. Letters 194, 172-1 74

(1992).

[22] K. S. Kim, B. J. Mhin, U.-S. Choi, and K. Lee, Ab initio studies of the water

dimer using large basis sets: The structure and thermodynamic energies, J.

Chem. Phys. 97, 6649-6662 (1992).

[23] K. Laasonen, M. Parrinello, R. Car, C. Lee, and D. Vanderbilt, Structures of

small water clusters using gradient-corrected density functional theory, Chem.

Phys. Letters 207, 208-213 (1993).

[24] F. H. Stillinger and A. Rahman, Improved simulation of Lquid water by molecular

dynamics, J. Chem. Phys. 60, 1545-1557 (1974).

[25] A. Rahman, F. H. StiUinger, and H. L. Lemberg, Study of central force model

for liquid water by molecular dynamics, J. Chem. Phys. 63, 5223-5230 (1975).

[26] Hg L. Lemberg and R. H. StiUnger, Central-force model for liquid water, J.

Chem. Phys- 62, 1677-1690 (1975).

(27) G. D. Carney, L. A. Curtiss, and S. R. Langhoff, Improved potential functins

for bent AB2 molecules: water and ozone, J. Mol. Spectry. 61, 371-381 (1976).

[28] R. O. Watts, An accurate potential for deformable water moledes, Chem. P hys.

26, 367-377 (1977).

[29] F. H. StiUinger and A. Rahman, Revised central force potentials for water, J.

Chem. Phys. 68, 666-670 (1978).

[30] W. L. Jorgensen, Transferable intermolecular potential functions for water, al-

cohols and ethers. Application to liquid water, J. Am. Chem. Soc. 103, 335-340

(1981).

[31] D. F. Coker, J. R. Reimers, and R. O. Watts, The infrared absorption spectrum

of water, Aust. J. Phys. 35, 623-638 (1982).

1321 W. L. Jorgensen, Revised TIPS for simulations of liquid water and aqueous

solutions, J. Chem. Phys. 77, 4156-4163 (1982).

[33] W. L. Jorgensen, J. Chandrasekhar, J. P. Madura, R. W. Impey, and M. L.

Klein, Cornparison of simple potential functions for simulating liquid water, J.

Chem. Phys. 79, 926-935 (1983).

[34] G. C. Lie and E. Clementi, Molecular-dynamics simulation of liquid water with

an a b initio flexible water-water interaction potential, Phys. Rev. A 33, 2679-

2693 (1986).

[35] L. X. Dang and B. M. Pettitt, Simple intramolecular mode1 potentials for water,

J. Phys. Chem. 91, 33493354 (1987).

[36] H. J. C. Berendsen, J. R. Grigera, and T. P. Straatsma, The miçsing term in

effective pair potentials, J. Phys. Chem. 91, 6269-6271 (1987).

(371 U. Niesar, G. Corongiu, E. Clementi, G. R. Kneller, and D. K. Bhattacharya,

Molecular dynarnics simulations of liquid water using the NCC ab initio poten-

tial, J. Phys. Chem. 94, 7949-7956 (1990).

[38] U. Dinur, "Flexiblen water molecules in external electrostatic potentials, J.

Phys. Chem. 94, 5669-5671 (1990).

[39] M. Sprik, Hydrogen bonding and the static dielectric constant in Liquid water,

J. Chem. Phys. 95, 6762-6769 (1991).

[40] S.-B. Zhu, S. Yao, J.-B. Zhu, S. Singh, and G. W. Robinson, A flexi-

ble/polarizable simple point charge water model, J. Phys. Chem. 95, 6211-6217

(1991).

(411 S.-B. Zhu, S. Singh, and G. W. Robinson, A new flexible/polarizabIe water

model, J. Chem. Phys. 95, 2791-2799 (1991).

[42] D. F. Coker and R. O. Watts, Structure and vibrational spectroscopy of the

water dimer using quantum simultion, J. P hys. Chem. 91, 2513-2518 (1987).

[43] J. R. Reimers and R. O. Watts, A local mode potential function for the water

molecule, Mol. Phys. 52, 357-381 (1984).

[44] D. J. Margofiash, T. R. Proctor, G. D. Zeiss, and W. J. Meath, Tripkdipole

energies for H, He, Li, N, O, H2, Nz, 02, NO, NzO, H20, NH3, and CH4 evaluated

using pseudo-spectral dipole oscillator strength distributions., Mol. Phys. 35,

747-757 (1978).

[45] C. Douketis, G. Scoles, S. Marchette, M. Zen, and A. J. Thakkar, Intemolecular

forces via hybrid Hartree-Fock-SCF plus damped dispersion (HFD) energy cal-

culations. An improved spherical model., J. Chem. Phys. 76, 3057-3063 (1982).

[46] J. R. Reimers and R. O. Watts, Analysis of the OH bend and stretch region in

the vibrational spectrum of water, Chem. Phys. Letters 94, 222-226 (1983).

[47] J. R. Reimers and R. O. Watts, The structure and vibrational spectra of small

clusters of water moledes, Chem. Phys. 85, 83-1 12 (1984).

[48] M. A. Suhm and R. O. Watts, Quantum Monte Carlo studies of vibrational

states in molecules and clusters, Physics Reports (Review section of Physics

Letters) 204, 293-329 (1991).

[49] E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations, McGraw-

Hill, Toronto, 1955.

[50] G. M. Barrow, introduction to Molecular Spectroscopy, McGraw-Hill, New York,

1962.

[51] P. Gans, Vibrating Molecules, Chapman and Hall Ltd., London, 1971.

[52] P. F. Bernath, Spectra of Atoms and Molecules, Oxford University Press, New

York, 1995.

[53] M. Born and J. R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann. Physik

84, 457-484 (1927).

[54] C. J. H. Schutte, The Theory of Molecular Spectroscopy, North-Holland Pub-

lishing Company, Oxford, 1976.

(551 J. S. Townsend, A Modem Approach to Quantum Mechanics, McGraw-Hill, New

York, 1992.

[56] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory,

Permagon Press, New York, 1965.

[57] D. M. Dennison, The infrared spectra of polyatomic molecules. Part II, Rev.

Mod. Phys. 12, 175-194 (1940).

[58] A. S. Davydov, Quantum Mechanics: 2nd Edition, Permagon Press, New York,

1976.

[59] D. A. MacQuarrie, Statistical Mechanics, Harper and Row Publishers, New

York, 1976.

H. W. Kroto, Molecular Rotation Spectra, Dover Publications, Inc., New York,

1992.

J. E. Wollrab, Rotutional Spectra and Molecular Structure, Academic Press, New

York, 1967.

B. Friedman, Principles and Techniques of Applied Mathematics, John Wiley

and Sons, Inc., New York, 1956.

H. M. Ibndail, D. M. Dennison, and N. G. L. R. Weber, The far iafrared

spectrum of water vapor, Phys. Rev. 52, 160-174 (1937).

C. Braun and H. Leidecker, Rotation and vibration spectra for the H20 dimer:

Theory and cornparison with experirnental data, J. Chem. Phys. 61, 3104-31 13

(1974).

S. H. Suck, J. J. L. Kassner, and Y. Yamaguchi, Water cluster interpretation

of IR absorption spectra in the û-14 pm wavelength region, Appl. Opt. 18,

2609-2617 (1979).

S. H. Suck, A. E. Wetmore, T. S. Chen, and J. J. L. Kassner, Role of various

water clusters in IR absorption in the 8-14 pm window region, Appl. Opt. 21,

1610-1614 (1982).

G. J. Frost and V. Vaida, Atmospheric implications of the photolysis of the

ozone-water weakly bound cornplex, J. Geophys. Res. 100, 18803-18809 (1995).

G. Herzberg, Molecufar Spect~a and Mo~ecdar Structure, Vol II, Krieger Pub-

Lishing Company, Malabar, FL, 1950.

APPLlEi IR/LAGE. lnc S 1653 East Main Street - -. - Rochester. NY 14609 USA -- -- - - Phone: 71 W82-0300 -- -- - - FaX: 71 6t208-5989

O 1993. Applied Image, Inc.. Aii Rlgtits Resennrd