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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
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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.
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