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THE FREQUENCY DETUNING CORRECTION AND 

THE ASYMMETRY OF LINE SHAPES: 

THE FAR WINGS OF Efi - HjO 



Q. Ma 

Department of Applied Physics, Columbia University 

and Institute for Space Studies, Goddard Space Flight Center 

2880 Broadway 
New York, New York 10025 



and 



R. H. Tipping 

Department of Physics & Astronomy, University of Alabama 

Tuscaloosa, AL 35487-0324 



ABSTRACT 
A far-wing line shape theory which satisfies the detailed balance principle is applied 
to the HjO - HjO system. Within this formalism, two line shapes are introduced, 
corresponding to band-averages over the positive and negative resonance lines, 
respectively. Using the coordinate representation, the two line shapes can be obtained by 
evaluating ll-dimensionaJ integrations whose integrands are a product of two factors. One 
depends on the interaction between the two molecules and is easy to evaluate. The other 
contains the density matrix of the system and is expressed as a product of two 
3-dimensional distributions associated with the density matrices of the absorber and the 
perturber molecule, respectively. If most of populated states are included in the averaging 
process, to obtain these distributions requires extensive computer CPU time, but only have 
to be computed once for a given temperature. The 11-dimensional integrations are 
evaluate using the Monte Carlo method, and in order to reduce the variance, the 
integration variables are chosen such that the sensitivity of the integrands on them is 
cleariy distinguished. Numerical tests show that by taking into account about 10^ random 
selections, one is able to obtained converged results. We find that it is necessary to 
consider frequency detuning, because this makes significant and opposite contributions in 
the two band-averaging processes and causes the lines to be asymmetric. Otherwise, the 
two line shapes become symmetric, are the same, and equal to the mean of the two shapes 
obtained including the frequency effects. For the pure rotational band, we find that the 
magnitude of the line shape obtained from the positive line average is larger than that 
obtained from the negative line average for w > and vice versa for a; < 0, and their gap 
increases as the frequency displacement from the line center increases. By adopting a 
realistic potential model and optimizing its parameters, one is able to obtain these two line 
shapes and calculate the corresponding absorption coefficients which are in good agreement 
with laboratory data. Also, this same potential yields good theoretical values for other 
physical properties of the dilute HjO gas. 



I. INTRODUCTION 

In previous papers, we have presented the theory for the calculation of far— wing line 
shapes and the corresponding absorption coefficients for interacting pairs of molecules. ^'^ 
Assuming only the binary collision and quasistatic approximations, we have shown that by 
using the coordinate representation to describe the orientation of each molecule before and 
after the transition, we are able to reduce the problem to the calculation of 
multi-<iimensional integrals. The dimensionality of the integrals depends on the type of 
molecules involved; specifically for two linear molecules (e.g. COj-COj or COj-Nj) the 
dimensionality is 7, while for one asymmetric top and a linear partner (e.g. HgO-Nj) it is 
9. For the first case we were able to obtain converged results with a sophisticated 
interaction potential using conventional integration methods.^ However, for latter case, we 
had to use the Monte Carlo method.^ This system is important in atmospheric 
applications, where we have shown that the absorption (known as the "foreign continuum") 
is in good agreement with experimental results.^ Because the absorption by HjO— HjO 
pairs (the "self continuum") is more important, we would like to extend our theory to the 
case of two interacting asymmetric tops; the dimensionality in this case is 11, thus 
implying a big challenge to obtain converged results. In a previous paper, we have shown 
that by considering an interaction potential containing cyclic coordinates, the 
dimensionality is reduced to 7, and one can obtain converged results.^ 

One goal of the present paper is to remove this restriction and calculate the 
far—wing line shape for HjO— HjO using the most general interaction potential. To 
accomplish this goal, it is necessary to modify the Monte Carlo routine used previously 
such that the sensitivity of the integrand on the integration variables is clearly 
characterized. A second goal is to investigate the asymmetry of spectral lines; in order to 
do this, one has to carry out band averages in a more sophisticated way and consider the 
frequency detuning effect in the line shape calculations. Based on the present work, one 
can conclude that the band— average line shapes are asymmetric and, in addition, one would 



expect different line shapes for different bands. 

In order to reduce any unphysical effects in calculated results, we carry out all 
numerical calculations based on formulas which satisfy the detailed balance principle 
exactly and have a higher accuracy in the short-time limit. ^ As expected, this increases 
the difficulty because the formulas become more complicated. Thanks to great advances in 
computers made in recent years, we are able to overcome this obstacle successfully. 
Finally, we utilize a realistic interaction potential applicable for the dilute HjO gas that 
not only gives good agreement between theory anii experimental temperature-dependent 
second virial coefficients and differential scattering cross-sections, but also yield absorption 
coefficients that are in good agreement with the experimental self continuum data. We 
note that because the theory has a sound physical basis, we expect that theoretical results 
calculated at lower temperature, which would be extremely difficult to obtain 
experimentally because of the lower vapor pressure attainable, are valid. 

The paper is organized in the following way. In Sec. II.A, we present the 
expressions for the correlation functions and the spectral densities which satisfy detailed 
balance exactly. In Sec. II.B, we introduce two band averages and the corresponding line 
shape functions. The application of the coordinate representation for the system consisting 
of two asymmetric tops is presented in Sec. II. C. The main purpose is to introduce the 
density matrices of the absorber and the perturber molecule, which are three dimensional 
distributions, and to show how to calculate them. This is the most costly calculation in 
the present study and the results obtained are used to get the line shapes later. In Sec. 
II.D, we discuss how to calculate the frequency detuning correction. The method is similar 
to that used to calculate the line shapes without this correction, except one has to 
developed a new technique to deal with the integrand containing a derivative. The 
necessary generalization of the Monte Carlo method to 11-dimensional integrations is 
discussed in Sec. lI.E. In Sec. II.F, we discuss a realistic interaction potential which gives 
good agreement with various molecular data measured in the dilute gas phase, including 



the absorption coefficient. Then in Sec. II.G, we apply the theory to obtain two line shape 
functions numerically for several temperatures and discuss some of their general features. 
We use these results to calculate the corresponding absorption coefficients and compare 
those obtained in the 300 — 1100 cn"^ spectral region for the room temperature with 
experimental results. In Sec. Ill, we discuss the results obtained and the conclusions drawn 
from the present study. 

The present work, together with the previous calculations for simpler systems, 
constitute a general formalism in which one can obtain from first principles the far-wing 
line shapes for any colliding pair for which a realistic potential is available. Conversely, if 
experimental data for the line shape or the corresponding absorption coefficients are 
available, one can use this formalism in order to test the accuracy of the potentials. 

II. THE GENERAL FORMALISM 

A. Symmetric Correlation Function and Spectral Density Which Satisfy the Detailed 
Balance Principle 

The absorption coefficient a{u) of a gaseous sample with a unit volume is given by 

a{u) =^n^oj tanh(^a;/2)[F((j) + F(-a;)], (1) 

where n^ is the number density of absorber molecules and the spectral density, F(a;), is the 
Fourier transform of the correlation function C(t). One separates the total Hamiltonian H 
into two parts: one commutes with the internal coordinates of the molecules while the 
second does not. We note that this distinction of H usually coincides with the division of 
the interaction into two parts: Y^^ and V^qj, the isotropic and the anisotropic interactions, 
respectively. Accordingly, the total Hamiltonian H is decomposed as 

H = H3 + Hb + Vi3„ + V,„i = Ho + V^i, (2) 

where H^^ and H^^ are the unperturbed Hamiltonians of the absorber and the bath 
molecules, respectively. In practice, for atmospheric applications where the gas pressures 



are low, one can introduce the binary collision approximation and focus on a much simpler 
system consisting of one absorber and one bath molecule. For simplicity, we don't 
introduce new symbols for the two-molecule system. 

It has been shown that one is able to express C(t) in the t -» limit, which is valid 
to order t^ as* 

C(t) = nb V Tr{(e^^-*v^ /i)^ Piso^"^^"^ [e-^^anit(e- ^^^t^^ ^)] j^ (3) 
where n^ is the number density of the bath molecules, v = Tr[e~^^*']/Tr[e~'''^] and the 
Liouville operators L^ and L^^^ corresponding to H^ and V^ni, respectively, have been 
introduced. For later convenience, we can introduce a symmetric correlation function C(t) 
{b C(t + ini3/2)) defined by 

C(t) = nbi/Tr{(e^^-t^p|pp|)t 

" Pisoe" ^ani[e-i^anit(e- i^at^ ^| ^ ^|)j ^- ^^,,y ^^^ 

From these expressions, it is easy to verify that C(-t) = C(t + i?^!?) and C(-t) = C(t) 
which guarantees both of them satisfy the detailed balance principle. ^'^ 

By expressing C(t) explicitly as summations over indices i, j, i' and j' where each 
represents all the quantum numbers necessary to specify the energy levels of the absorber 
molecule, one is able to write C(t) as 

6(t)= I Ie^"ji-^'^J'i')*nb.Tr,{<j|4Mt4^,i> 

" <ibisoe"^^*°^ e~'^anit|i.> <V\^pjtipl\y> <j'|e^^ani* e-^anij j>}. (5) 

In the above expression, u^^ = E^-E^ and the the subscript b of trace denotes the trace 
over the remaining variables including all magnetic quantum numbers. We assume that 
the interaction between two molecules does not depend on their vibrational quantum 
numbers. As a result, in Eq. (5) the vibrational quantum numbers of i and i' are identical 
and those of j and j' are also identical, but the former could differ from the latter. By 
choosing the Z axis of the space-fixed frame along the separation between the two 



molecules, the interaction V^jji(r, Q^, n^) depends on the orientations of the two molecules 
represented by tl^ and n^^, respectively, and on r, the distance between the centers of mass, 
which can be considered as a parameter since the translation^ motion is treated classically. 
In Hilbert space associated with the internal degrees, one denotes the eigenvalues and 
eigenvectors of V^i(r, fl^, fl^) by G^ and | C>, respectively; thus 

Vani(r,na.nb)lC> = G^(r)|C>. (6) 

Then, by performing the Fourier transform of the correlation function C(t) and carrying 
out an integration over r (i.e., the classical ensemble average over r which is valid within 
the quasistatic approximation), one is able to obtain the symmetric spectral density F(a;), 

^H = il Ih-i'A'^-^^ii + ^in^)]- (7) 



ij 1 J 



The explicit expression for ;tij.i,j,(a;) is given by 

^iJ;i'J'(^) = X^;^M I I I <v\h> <h\pit^^pi>fP^\ih> <^h\C> 
^^ W ib ib 

X <C|i'i^> <i%\>fp^pitipi\i%> <}%\V>, (8) 

where i^^ and i^ are indices used to specify the states of bath molecule and {m} indicates 
the summation over all magnetic quantum numbers. In the above expression, H^ (a;) is 
defined by 

H^^(a;) = n, , A^xl ^^^ e" ^'^-{ic) - ^[G;(rc)+ G^(rc)]/2^ ^g^ 



where G^^(r) denotes 3^[G^(r) - G^(r)], and r^ are roots of the equation 

G;(re) - G^(r J = u, (10) 

We note that in Eq. (7), the summation indices i, j, i' and j' exclude their magnetic 
quantum numbers since the summation over them has been carried out. The functions 
Jij.i,j,(a;) introduced above are symmetric for the exchange of indices {i j} h {i'j'}. 
Meanwhile, with respect the exchanges {i j} -» {j i} and {i'j'} -♦ {j'i'}, it is easy to verify 
that Jji.j,i,(-a;) = Jij.i,j,(it^).^ In terms of F(a;), the absorption coefficient a{u)) becomes 



a{u) = 3^ n^ cj siiih(^3?ia;/2)F(a;). (11) 

Finally, it is worth mentioning that F(a;) = e~^^/^F(a;) and F(-a;) = F(a;). The latter is 
required by the detailed balance principle. 

B. The Averaged Line Shapes with the Frequency Detuning Correction 

We consider a band consisting of transitions between states with two vibrational 
quantum numbers specified. For simplicity, we designate a pair of i and j (i.e., a line) by 
the symbol k. If necessary, we use symbols k+ and k. to distinguish a positive resonance 
line with Ej - E^ > and a negative resonance one with E: - E^ < 0. Then Eq. (7) can be 
expressed as 

k+ n k-n 

where a;i^+ (= | a;jj ) is positive. We note that in Eq. (12) we have not explicitly 
characterized the symbol n by + or — yet because the band of interest has not been 
specified. For the pure rotational band resulting from transitions without changing the 
vibrational quantum number, both the symbols n in two terms of Eq. (12) could be n+ and 
n. because there is no link between the choices of k and n. Meanwhile, for a vibrational 
band, k+ or k.is always associated with n+ or n., respectively. In this case, i and i', and 
also j and j', share the common vibrational quantum numbers and, in general, these two 
different vibrational quantum numbers determine whether k (i.e., a pair of i and j) and n 
(i.e., a pair of i' and j') belong to the positive or the negative resonance. Therefore, for the 
vibrational bands the symbol n in the first term of Eq. (12) should be understood as n^ and 
those in the second term as n.. This custom is applicable for the equations following. 

On the other hand, by introducing two common line shape functions ^+(a;) and X_{u) 
one can express F(a;) as 

k + 



where /ij^ (= /Xy} is the reduced dipole matrix element and p^ = Vp^. In order to find 
expressions for X^{u) and ^.(o;), one performs the Fourier transform for Eqs. (12) and (13) 
and compares the results to their time-domain versions. As a result, the following 
equation has to be satisfied: 

k+n k-n 

= nyn^)]ip^\t^\'^''^'' + n^^^ (14) 

k* k- 

where for simplifying the notations we use the symbols x . (t), J . (t), ^[=-^ ^♦(^)], and 

k^^n k-iii ^ 

^[-^ ^X^)] to represent the Fourier transforms of J . (a;), J . (a;), \ ^^(cj), and -^ 

A 4') XI Ik •) XI *•' ■*' 

^.(^), respectively. Furthermore, one assumes that one can separate Eq. (14) into two 
equations, one associated with a summation over the positive resonance lines and the other 
over the negative resonance lines, 

k* n s* 

and 



k-n s + 

Then, with Eq. (15) it is easy to obtain an expression for «9^[^ ^X^)] which is valid in the 
short— time limit: 



-''Is. '"M] = N 11 KJ*) e'l«H.+''J - ■- ]', (17) 



k^n 



where N is the normalization factor defined by 

N = I/>kl/^l^ (18) 

k + 
and a; is the average positive resonance frequency defined by 

^ = ^1 Pkl/^l'^v (19) 

k + 

By performing the inverse Fourier transformation, one is able to derive the expression for 



10 



Xj{u) valid at I u;| — » 00 

},{u) = u}'^l I \^.Ju) - ^0^, + u}J + u ]. (20) 

Similarly, one can derive 

XX^) = u^^l Ix^Jt^ + ^t^.-ujJ-u]. (21) 

k- n 

In Eqs. (20) and (21) the arguments of functions depend on the summation indices 
and this results in difficulty to obtain X^{u) and JfXuj) directly. In practice, one prefers to 
derive an expressions for X^u) and Xj^u) in- which the summations are performed over 
functions whose arguments are independent of the summation indices. For this purpose, 
the frequency detuning approximation must be introduced. As an example, we consider 
Eq. (20). With the Taylor series expansion of J . [^ - 'j(£^+ + ^n) + ^ 1 ^^er u, one can 
approximate ^f^u) as 

^-,(0;) = ^,(0;) + a;2 ^ J J [^ - ^a;,, + a;J] x! . (u), (22) 

•^^ " " it+»n 

k* n 

where 

k* n 
and x! . {oj) = d? . {u)/du. In the left side of Eq. (22), the first term XJu) comes from 

simply ignoring the frequency detuning and the second term is a correction. We note that 
in our previous study, ^ a shift parameter was introduced to treat the effects from the 
frequency detuning. However, it is better to calculate the contributions from the second 
term directly because it turns out that, except for some simple cases, the previous method 
could introduce numerical errors. Similarly, one can approximate 3P-(a^) as 

}Xu}) = iXu) + oj'^l l[-u + ^i^.-u^)]x:.H, (24) 

•^"^ " " jc-jn 

k- n 

where 

k-n 



'"^^--^'^ll^.^M- (25) 



11 



Using the symmetries ? . (-a;) = ? . {u) and x . {-(^) = x(uj) (the latter is 

k-)n- k+in+ k-)n+ k+in- 

appropriate for the pure rotational bands only) mentioned above, one can show that 

h-^) = i*(^), (26) 

and 

h-^) = ^.(^). (27) 

If one does not distinguish the two shapes X^{uj) and X.{iJ) in the expression for F(a;) 
given by Eq. (13) and replaces them by only one shape -f(a;), one can pursue a similar 
derivation for X{uj) and obtain 

A^) = ^^^11 h-J" -^'^ + ^n)]/lP,M '■ (28) 

k n ' k 

A simple version X{ijj) defined by 

k^) = iJ'll\Jo:)llp^\y^\\ (29) 

k n ' k 

can also be introduced. In this case, both X{u) and X^u) become symmetric and, in 
addition, up to the first— order approximation, X{u)) is the same as X[u)) because there is no 
net contribution to X{u) from the first derivative term of the Taylor series expansion of 

In summary, we note that within this formalism no matter which functions are 
chosen for the line shape, in terms of them the symmetric spectral density F(a;) always 
satisfies the detailed balance principle. The formalism outlined above was developed 
several years ago.^ However, at that time, except for the simplest system, such as COj - 
Ar, to calculate converged line shapes for systems of interest in atmospheric applications 
was formidable. The main obstacle was the calculations involving a diagonalization 
procedure of the anisotropic potential which exhausts computer resources very quickly. 

C. The Coordinate Representation 

Recently, we have developed a formalism based on the coordinate representation in 



12 



which the eigenfunctions of the orientations of the system are chosen as the complete set in 
Hilbert space. ^"^ The advantage of introducing this representation is that the 
diagonalization of the potential becomes unnecessary and the main computational task is 
transformed to the carrying out of multidimensional integrations. For systems consisting 
of two linear molecules, or one linear and one asymmetric top molecule, or two asymmetric 
top molecules, the dimensionality is 7, 9, and 11, respectively. In addition, we have shown 
that using the Monte Carlo method, one is able to evaluate up to the 9— dimensional 
integrations required for systems such as HjO - Nj.^ ^Combined with techniques developed 
recently to handle sophisticated potential models,^ one is able to implement realistic 
potentials for these systems and derive accurate, converged results for the far-wing line 
shapes and the corresponding absorption coefficients. 

In the present study, we are interested in a system consisting of two HjO molecules. 
In order to reduce any unphysical effects, we base our study on formulas which satisfy the 
detailed balance principle exactly and have a higher accuracy in the short-time limit. 
Besides extending the Monte Carlo method to evaluate U-dimensional integrations 
required for line shape calculations, we would like to answer some questions in depth. We 
want to know whether the line shapes are asymmetric and if they are, to find out the origin 
of the asymmetry. In other words, we want to know whether it is necessary to introduce 
two line shapes instead one, and why they differ from each other. As expected, in 
comparison with our previous studies, this introduces extra difficulties. Fortunately, 
thanks to the coordinate representation and the Monte Carlo method as two powerful tools 
to perform averages, and also to great advances in computers made in recent years, we are 
able to overcome these obstacles and we make significant progresses. 

The details about the coordinate representation and the Monte Carlo method have 
been presented previously^"^ and are not repeated here. We only report new features. 
Since we want to carry out band averages in a more sophisticated way, we have to 
introduce a positive and a negative resonance dipole operator in the Hilbert space of the 



13 



absorber molecule denoted by /i„ and p^, respectively. The former is defined by 

f^m= I <i|/^mlJ> |i><Jl> (30) 



Ej>Ei 



and the latter by 

Mm= I <i|/^mlJ> liXJl- (31) 



Ej<Ei 



We note that for the vibrational bands j denotes a higher vibrational quantum number 
than i in Eq. (30) and vice versa in Eq. (31). Then, with Eqs. (8) and (23), one can rewrite 
XJ[u) as a summation over ( and r) 

^*M = '^'Nl^c,(^)G(;t,). (32) 

where G*^^, are defined by 

^Kt,) =li<C\^fP^pif^l pi\v>)*<C\>fKpit^mpi\v>, (33) 

m 

and defined by 

G(c„ =X(<ci^4Mi4i^>r<civ^4^m4^>, (34) 

m 

for the pure rotational bands and for the vibrational bands, respectively. For simplicity, 
we will only present formulas applicable for the pure rotational bands and simply mention 
differences between them and their vibrational analogs. In the expression for ^+(a;) given 
by Eq. (32), the summation terms are products of H^^(a;) and G *^^j . The former are 
functions of uj and their values depend on the interaction potential between the two 
molecules. The latter are common for all frequencies and their values are independent of 
the potential. With the coordinate representation, no matter how complicated the 
potential is, to calculate values of H^^(a;) is straightforward since the potential is a 
diagonal operator. On the other hand, to obtain the G(*a-j involves a lot of calculations, 
because they contain the density matrices which are differential operators. Fortunately, for 
a given temperature, these calculations need to be done only once since results obtained are 



14 



applicable for all potential models. Thus, we can calculate them and to store them in an 
input file. 

In the coordinate representation, the basis functions | (> are nothing but the direct 
product 1 6{n^ - n^^)> ® I tf(n^j - nb;)> w^^^^ t^^ notations of | 6{n^ - "&;)> ^^ I *("b ■ 
tt^()> are used to represent specified orientations of the absorber and the perturber 
molecules, respectively. Accordingly, one can separate the dependence of G(*^^) on the 
absorber and on the perturber molecules and express it as the product of G^^ ^„j and 
^b{;T])> " 

GKTi)=I(<*(«a-nac)l4/^S4K(na-na,)>)*<*^^^ 
m 

= GactTi) "" Gi,(^^). (35) 

For the vibrational bands, the dipole moment operator fi^ in Eq. (35) is replaced by /x^. 

In comparison with G *( ^^) , the expressions for G^^^ ^^j are simpler because they do 

not contain the dipole operator. For the linear, symmetric top, and asymmetric top 

molecules, the explicit expressions for G^^^ ^ j have been presented and the corresponding 

profiles have been discussed in our previous papers ^"^ and we do not repeat them. With 

respect to G*(^^j , one has to derive the corresponding expressions valid for the linear, 

symmetric top, and asymmetric top molecules, respectively. We do not present all of 

them, rather only the last and the most complicated one applicable for HjO. It is well 

known^'^ that the wavefunctions of HjO, | J7'm>, can be expressed in terms of an expansion 

of symmetric— top wavef unctions |jkm>, 

|jnn>=5;uijjkm> 
k 

k 
where DJ, , (a, 0, 7) (= e"^™"di, M^'^^^^) is the rotational matrix. With Eq. (36), one 

XXI , Jv XXX , iw 

is able to express G^(^^) as. 



15 



°a( Ct,) = n ^KK' ^K,K'(^( Cti) ' /'c Ct)) > 7( C^i, ), (37) 

L EK' 

where a^ ^^j , P^ ^^j , and 7^ ^^j are the three Euler angles used to represent a rotation 
resulting from two successive rotations, i.e., 

R(^( Cti) . Pi Cn) , 7( Ct,) ) = R''(^o ^C T';) ^(S' ^T)> 7j, (38) 

and the summation index L = 0, 1, 2, • • • ; both indices K and K' run from - L to L. In 
the above expression, the coefficients A^ , are given by 



KK' 



1 tt 



Jl^l J2^2 



{jl^i/* J2^2 

1 1 « 



X 

k 



k' ^ ^ ^ 

{I ^nlr, "nln ^0 1 Jl. -> n,)) . Q U^' U^j C(j, 1 jj, n, n,)}, (39) 



^ .,. -r. . -r, .. -W-.l, -l-~iyj lA-.^^,^ ^^^^ 

°1 °2 



where E(j,r) are the energies of the state labeled by the quantum numbers j and r, g is its 
nuclear spin degeneracy factor, C(ji jj L, k K-k K) is a Clebsch-Gordan coefficient, W(jJ 
h Ji Ji>l ^) ^s a Racah coefficient, and the summation over j*^ and t\ indicated by a symbol 
{Jir'J^ is limited to a range with E(j'^ri) > E(jjrj). We note that for the vibrational bands, 
the expression for A , is similar to that given above except that the states labeled by }\t\ 
and jjrj and the states labeled by j^Tj and JjTj belong to a higher and a lower vibrational 
level, respectively. In addition, another limitation of E(jjr J) > E(J2r2) is also enforced in 
the summations. 

With Eqs. (37) and (38), one can conclude that G*^^^) are three-dimensional 
distributions over three Euler angles used to represent a rotation of the molecule from the 
initial orientation to the final one, labeled by ( and r/, respectively. With respect to /?.(a;). 



16 



one can introduce G '( ^^j and write down similar equations to Eqs. (32) and (35). 
Meanwhile, G^-( ^^j can be given in terms of aJ^, and dJ; ^^.(a, ;^) , ^( <;t,) , 7( ^t,) ) in the 
same way as shown by Eq. (37). The expression for A^ , is almost identical to A^l, shown 

A.K KK 

by Eq. (39), except that the summation over i\ and t\ is limited to a range with E{}\t[) < 

Although the calculations of aJ^,, and A^^,, are straightforward, there are many 
summation loops involved. Usually one introduces a cut-off j^^^^^ to exclude less populated 
states. It turns out that as j^^^^^ increases, not only more CPU time is required to calculate 
each Aj^g, and A^^y because the ranges of loops become larger, but also the number to be 
evaluated increases quickly. Fortunately, one does not need to calculate all of them since 
some are identical and others are zero. For HjO, due to the synmietry of U^ , all the 
coefficients A^^,, and A^^^, are zero unless their indices K and K' have the same evenness or 
oddness. In addition, for the non-zero coefficients there are symmetries A^ , = A^ ,, 
Aj.g, = A_j,_^„ and A^^, = A^,^,. As a result, if one introduces a cut-off j^^^ = 23 (which 
is the highest angular quantum number of the initial states listed in the pure rotational 
band of the HITRAN 92 database^), there are 18424 values of A^ * and 18424 of A^ ", 
needed to be evaluated. If one uses an even higher cut-off jj^^^^^ = 26, these numbers 
become 26235. By utilizing a dozen workstations, we are able to manage the latter in less 
than two days. We note that to obtain these coefficients is the most costly calculational 
part in the present study. 

After all A^^^^, and A^.^^, are available, we can easily calculate G *( ^^j and G^( ^^i) 
which are three— dimensional distributions over the Euler angles c^( (-ti) > ^( Cn) > ^^^ '^t Cti) • 
In cases where no confusion results, the subscripts of these Euler angles are omitted. 
However, it is better to express them as distributions over their two sensitive variables /? 
and u (= (a + j)/2) and one insensitive one v (= (a - 7)/2). The explicit expression for 
^a( C^i) ^^^^ ^^ *^^ numerical calculations is given by 



17 



L KK' 

X {cos[(k+k')u] cos[(k-"K')v] - sin[(K+K')u] sin[(K- k')v]}, (40) 

where the ranges of the indices L, K, and K' are from to 2j„^, from to L, and from - L 
to L, respectively; e^^, = 1 for K = 0, and €^^, = 2 otherwise. The expression for G^( (^^) 
is similar to Eq. (40), except a replacement of aJ *, by A^^^,. However, due to the 
symmetry A^.^^, = A^,^ mentioned above, G *( ^^j {0, u, v) does not differ from G^^ ^^, (/?, u, 
v) significantly. In fact, it is easy to show that 

G^(CTi)(^>^>v) = G-t^^,(^,u,-v). (41) 

This means that with respect to the sensitive variables p and u, they have same 
distribution patterns. Meanwhile, with respect to the insensitive v, one is the others' 
mirror image. 

Because G*( ^^j {p, u, v) and G^( ^^, {0, u, v) are three-dimensional, it is impossible 
to plot their profile in one figure. We calculate several two-dimensional distributions of 
G*( (jf^){a, j9q, 7) over the Euler angles a and 7 at 296 K obtained with the fixed 0^ = 5, 22, 
38, and 50 degrees, respectively, and present their corresponding three-dimensional plots in 
Fig. 1. From the figure, one can easily see that the magnitudes of G *( ^^j decreases very 
fast as Pq increases. More specifically, for ^q = 22, 38, and 50 degrees the magnitudes 
decrease by about one order each. In addition, these G *( a^j exhibit symmetry with respect 
to the axes (a + 7)/2 and (a - 7)/2. In order to show the profile of G *( ^^j at 296 K over 
the two sensitive variables p and u, we calculate its average over v and present the 
resulting two— dimensional distributions in Fig. 2. One has to imagine that the profile 
shown in Fig. 2 extends along another dimension, i.e., the v axis which is perpendicular to 
the p—u plane and is missing in the figure. Furthermore, the range of extension along the v 
axis varies from the minimum at u = and 2w to the maximum 27r at u = tt. As shown 
in Fig. 2, there are five sharp peaks located along the u axis at u = 0, 7r/2, tt, 37r/2, and 27r, 
respectively, and they are symmetric with respect to the plane u = tt. The magnitudes of 



18 



these peaks decrease very fast as fi increases. We note that in contrast with Fig. 1, a 
logarithmic coordinate is used to plot the magnitudes. From Figs. 1 and 2, one can 
conclude that the peak at u = tt is dominant. We do not present the profiles of G *( a„j for 
other temperatures, but simply mention that they have similar patterns, but the peaks 
become lower and wider as the temperature decreases. On the other hand, it is 
unnecessary to present similar figures for G^^^ ^-^j because one can easily obtain them from 
Figs. 1 and 2. In fact, Figs. 1 and 2 are also applicable for G^^ ^^j except that one has to 
switch the labels a and 7 in Fig. 2. Finally, we note that because G *( ^^j (^, u, v) and 
G^( <;t]) (^> ^j v) ^^^ independent of the potential, it is wise to calculate them first and store 
them in files. Then, when one carries out repeated calculations for XJ^u) and X,{u)) to 
optimize the potential models, one does not need to evaluate the values of G^j a , (^, u, v) 
and Gl^ ^^j (^, u, v) again. In addition, using the interpolation method one can easily 
obtain their values for a random selection of ^, u, and v from these input files. Otherwise, 
one has to independently evaluate them about 10^ times in the Monte Carlo calculations. 

D. Contributions from Frequency Detuning 

So far, our discussion has been focused on how to apply the coordinate 
representation for calculating Xj^uj) and X,{u)). In order to calculate /+(a;) and ^.(o;), one 
has to go further by adding contributions from the frequency detuning. We briefly explain 
a method used to obtain the second term of Eq. (22) associated with Xj^uj). In comparison 
with evaluating the first term -^'♦(a;), the only additional obstacle is that the integrand 
contains a derivative J' . (a;). In practice, except for special cases in which analytic 

expressions for this derivative are available, to evaluate values of J' . (cj) is much more 

difficult than x . (^) because a numerical subroutine is required to obtain the former from 

the latter. Given the fact that there are about 10^ random selections in the Monte Carlo 
calculations, this means that this subroutine must be called 10^ times. Fortunately, except 



19 



for this part, the other parts of the second term do not depend on the frequency. 

Therefore, it is better to reverse the order of the derivative and the summation (i.e., 

integral) operations. In other words, instead ofj^Y l[u)-^(^*+ ^n)! X' . (^)» we can 

k+ n 

calculate a new term given by jq^ ^ 5| [^ " i(^+ + ^n)) X . (^)- The results obtained are 

k* n 

a function of the frequency represented by a set of values of the integrations and a set of 
corresponding frequencies. Then, with the numerical subroutine, one is able to obtain the 
derivatives which are just the second term we want to calculate. As explained above, the 
subroutine is called only once. With this technique, the costs to calculate the second term 
of Eq. (22) is comparable to the first. 

The same method used for -^+(0;) explained in Sec. II. C is also applicable for 
evaluating this new term. We do not repeat a detailed discussion, but simply mention 
things which are different. In this case, a new set of coefficients B , can be introduced 

Bk.Dk, 

whose expression is almost the same as A , given by Eq. (39) except a factor of {u — 
i[E(Jir'j) — E(jiri) + E(j^rJ) — E(J2r2)]} is inserted into the inside of summation loops over 
the indices j^, r^, jj, Tj, Ji, t[^ J2, and rj. Similarly, one can introduce a set of coefficients 
B , associated with the second term of Eq. (24) for ^.(o;) whose expression is the same as 
B ,, except the summation over j^ and r\ is limited to a range with E(j'^T*^) < E(JjTj). We 
note that in comparison with A , and A ,, B , and B , have similar symmetries B , 
= B^j,_^, and B^^,, = B^^_^,y except B^^^, = - Bj,,^^. Then, one can introduce two 
three— dimensional distributions associated with B , and B ,, respectively, and store 
their values in two input files the same way as G *( ^^j (^9, u, v) and G^^ ^^j (^9, u, v). 
Finally, by comparing these two distributions, their patterns are closely related each other 
as shown by Eq. (41) except one has to add a minus sign on the left side since their values 
become opposite. 

As expected, to evaluate B , and B , requires considerable CPU time, and the 
costs are approximately the same as for A , and A ,. In fact, for specified L , K and K', 



20 



one does not need to calculate the four coefficients separately and one can evaluate all of 
them simultaneously. 

E. A Monte Carlo Calculation of 11— Dimensional Integrations 

As an example, we explain how to calculate X^uj) from Eq. (23) in detail. In the 
coordinate representation, the summation of H^^(a;)G(*^^j over ( and tj becomes a 
11-dimensional integration of H^^(a;)G(*^^) over the Euler angles P^^, 7^^, a^;, /?b;) 7bo 
^at]) ^at]) 7aTi» Q^bTi> /'bri' ^^^ %r\ ^^ ^hich the first five (including a^^ = 0) specify the initial 
orientations of the system and the last six specify the final ones. We note that due to the 
rotational symmetry of the whole system, one can always assume a^^- = 0. For such high 
dimensionality, the Monte Carlo method is the only way to evaluate the integrals. 

It is well known that in the Monte Carlo computation, it is important to distinguish 
the sensitive and insensitive variables of the integrand, and to incorporate this into the 
integration variables since this enables one to tailor the important sampling and to reduce 
the variance dramatically. In the present case, the integrand is a product of H^ (a;) and 
G(*Q^j . With respect to their variables, the former is a smooth function as shown by Eq. 
(9), but the latter 's values vary wildly and could differ from each other by many orders of 
magnitude. This means that the sensitivity of integrand is mainly determined by G(*a„j , 
or more specifically, by G *( ^^j and Gb( ^t]) • We note that G *( ^^, and G^^^^^ depend on 
relative orientations between the initial and final positions of the absorber and disturber 
molecules, respectively. Therefore, it is proper to represent the final orientations of the 
system labeled by 7/ in terms of the body— fixed frames instead of the space— fixed firame. 
The body— fixed frames introduced here are those attached to the two molecules at their 
initial orientational positions. For the asymmetric— top molecule, one chooses a^, j9a, 7^, 
^(Ct|) » ^(;ti) ) ^^^ 7((;ti) ii^stead of a^, /3^, 7^, o^^, /?^, and 7^ as variables. However, similar 
to the behavior of G^^^ ^^j explained in our previous study, ^ the sensitivity of G*( a»j on its 
two variables Qf(^^) and 7((-^j is interwoven such that neither is a sensitive or insensitive 



21 



variable, but their combinations u^ ^^) , and v^ ^^j are. Therefore, in order to well 
characterize the sensitivity, a replacement of a^ q^, , )9( ^^j , and 7^ ^^j by i9( q^, , U( ^^) , and 
V( ^^j as variables in G *^ ^^) is appropriate. The same conclusion is also true for G^^^ ^^j . 
This is a further step necessary to evaluate 11-dimensional integrations because, in 
comparison with 7 or 9— dimensional ones, not only the dimensionality becomes higher, but 
also the distributions of the integrand become more nonuniform. However, in order to 
incorporate these new choices for the integration variables, one has to pay extra attention 
to their ranges. More specifically, since U( ^^) varies from to 27r and V( a j varies from — 
U( Q^j to U( ^^j when < u < tt and from u^ ^^) — 27r to 27r - u^ ^^j when tt < u < 27r, the 
integration volume becomes a lozenge-shaped area. Because the algorithm VEGAS ^^ is 
designed for carrying out integrations over rectangular volumes, one cannot incorporate the 
integration variables directly. Therefore, we have modified VEGAS such that the new 
version enables one to evaluate integrations over a volume containing a lozenge-shape 
area. Then, with respect to the sensitivity, the dependence of integrand on all integration 
variables is well characterized. This enables one to fully exploit the power of the Monte 
Carlo method. As a result, numerical tests show that with a few more random selections 
than before, one is able to evaluate the 11-dimensional integrations successfully. 

The above discussion is also applicable for evaluating other 11-dimensional 
integrations, i.e., those for ^.{uj), and for the frequency detuning correction terms of J(^{u) 
and ^.(cj) because their integrands have similar features. Numerical tests show that with 
the modified version of VEGAS, we are also able to evaluate these as well. 

F. Potential Models 

Based on the progress mentioned so far, we are able to calculate converged line 
shapes for HjO — HjO without or with the frequency detuning correction, i.e., ^+(0;) and 
/.(a;), or X^{uj) and -J'-(a;), from potential models provided unless they are too complicated 
(e.g. those consisting of several decades terms and parameters). There are several potential 



22 



models available in literatures, such as HF^\ CI^^ Watts^^ RWKl^^ and RWK2 model ^1 
We have tested some of them to calculate the line shapes and the corresponding absorption 
coefficients. Unfortunately, it turns out that the results obtained from these models 
predict too much absorptions in comparison with experimental data. It has been shown in 
our previous study ^'^ that the far-wing shape is very sensitive to the angular gradients of 
the potential. The reason is that contributions to the line shape come mainly from energy 
contour areas at which large angular gradients of the potential are exhibited while the 
potential values themselves are relatively small or even negative. We expect that such 
features would not fully manifest their effects on other physical measurements on which 
these models were developed. Therefore, the failure of these models is not surprising. 

As an alternative, we assume that the potential for HjO - HgO consists of a 
Coulomb interaction represented by a site-site model, a short range repulsive interaction 
represented by another site-^ite model, and an isotropic attractive dispersion interaction 
proportional to -1/r^ For each HgO, there are two positive point charges + q located at 
the H atoms and one negative point charge - 2q at a position along its symmetry axis a 
distance d from the atom. (A positive value of d means the charge - 2q is located on the 
same side as the center of mass.) In addition, we assume that there are three repulsive 
force centers: two located at the two H atoms and one at the atom, and the repulsive 
interactions have a form Ay e" ^^j^^ij, where r^j are distances between force centers, and 
Ay and py are adjustable parameters. Accordingly, the potential V(r, fi^, n^^) considered 
here is given by 

^('. n.. "b) = X I ^ + I I A„ e- 'ijMi - f, , (42) 

i€a j6b iea jeb 

and it contains 7 adjustable parameters: A^^, p^^, A^^, p^^, A^^, p^^, and B. The values 

of q = 0.60 (a.u.) and d = 0.4991 (a.u.) are determined such that they match the well 

known dipole moment value 1.8546 D and yield quadrupole moments 0^^ = - 2.319 D A, 

0^ = 2.635 D A, and 6^^^ = — 0.316 D A, which are reasonable in comparison with 



23 



experimental values 0^^ = - 2.50 D A, 6^^ = 2.63 D A, and 0^^ = - 0.13 D A.^^ ^^ ^^^^^ 
the potential form given by (42) and search for a set of potential parameters that enables 
us to obtain satisfactory results for several properties of a dilute H^O gas; e.g. the 
absorption coefficient, the second virial coefficient, and the differential cross section. 

G. General Features and Numerical Results for the Line Shapes 

We have presented the line shape formulas applicable for both the pure rotational 
band and vibrational bands. But, in the present study, the numerical calculations are 
carried out for the former because not only is this the strongest band of HjO, but also most 
of the continuum absorption measurements are performed in its high— frequency wing. 

It is worthwhile to report general features of line shapes found from numerous test 
calculations before we present more specified results. First of all, we find that differences 
between ^^{u) and ^.(cj) calculated from the same potential are always less than numerical 
errors. The formulas used to get X^{uj) and /.(a;) are the same, but the input files 
representing the two distributions G^(^^,()9, u, v) and G^(;^)(A u, v), respectively, are 
different. However, these two distributions differ from each other only slightly. In fact, as 
explained above, they exhibit the same profiles over two sensitive variables and u, and 
are mirror images over the insensitive v. In cases that the potential contains cyclic 
coordinates, it is easy to show analytically that these two distributions must yield the same 
line shapes. For more general cases, given the fact that G *( ^^j (/?, u, v) and G^( ^ , (^, u, 
v) differ from each other as discussed above, we suspect that effects resulting from these 
differences could cancel out in the averaging processes when the integrations are evaluated 
by about 10^ random selections. In the present study, we do not seek a general and 
rigorous proof of this finding, rather we assume it. Then, we can draw the important 
conclusion that by not considering the frequency detuning correction, the line shape 
obtained from the formalism satisfying the detailed balance is symmetric. In other words, 
if one does not consider the frequency detuning correction, one only needs to introduce one 



24 



line shape function. 

Second, we find that the calculated contributions from the frequency detuning 
associated with X^{(J) are opposite to those associated with ^Xui), and the differences 
between their magnitudes are always less than numerical errors. This indicates that these 
correction contributions have the same magnitudes, but with different signs; this is not 
surprising since we have already noted our finding for Jf^{u) and ^X^)- I^ general, the 
values of the correction contribution for Jt^u) are positive and those for X,{uj) are negative 
for a; > and vice versa for u <0. In addition, we find that the correction contributions 
are significant, especially at high frequencies, but they tend to become negligible near the 
line center. 

Now, we are ready to present some results. We find that by adopting A /k = 1.05 
"" 10' K' ^00 = °'245 A, A^jj/k = 2.0 X 103 K, p^^ = 0.36 A, A^^^^ = 4.0 x 10^ K, p^^ = 
0.46 A, and B/k = 9.0 x 10^ K, one is able to obtain Jt^u) and ^.(^) such that the 
calculated absorption at 296 K can fit the experimental results in the spectral region 300 - 
1100 cm"^ very well. We note that because this window region is located at the 
high— frequency side of the pure rotational band, the calculated absorption are mainly from 
the contributions ofJ^^u). The two line shapes X^u) and ^.(a;) at 296 K, together with 
their mean X{uj), are plotted in Fig. 3. As shown in the figure, the magnitudes of Jf^u) are 
larger than ZJ[iJ) and the relative gaps between them increase significantly as a; increases. 
The corresponding self-broadened absorption coefficients in the spectral region 300 - 1100 
cm'^ based on HITRAN 92 data are plotted in Fig. 4, together with the experimental 
results of Burch et al.^^ and some recent measurements of Cormier et al.^^ Using this 
potential, we calculate the second virial coefficients at several temperatures^^ and compare 
them with experimental data^® in Fig. 5. In addition, the calculated differential cross 
sections together with experimental data^*^ are plotted in Fig. 6. We note that all these 
physical quantities are associated with dilute water vapor, and as shown by these figures, 
the agreements between the theoretical predictions and the laboratory measurements are 



25 



good. 

Based on the same potential model, we calculated the two line shapes Z^{u) and 
/^{j) for several temperatures ranging from 220 K to 330 K which are of interest in the 
atmospheric applications and the corresponding absorption coefficients. Some of J(^{uj) 
obtained for 220, 240, 260, 280, 300, and 320 K with frequencies raging from - 1600 cm"* to 
1600 cm"* are presented in Fig. 7 and the corresponding ^.(a;) are their reflections about 
the a; = axis. As shown in the figure, these ^^{uj) and X^u)) are asymmetric and their 
magnitudes increase as the temperature decreases. We note that these line shapes J^^{u) 
and X.{^) presented here do not include a factor l/iJ^ as shown in Eq. (13). If one wants to 
compare them to other line shapes (e.g. a Lorentzian), one has to multiply them by the 
factor 1/uP. Also, the magnitude of /+(a;) or X,{u) as a;-* should approach the Lorentzian 
halfwidth, although the present theory is not valid in this limit. It is clear, however, that 
the theoretical shapes are super— Lorentzian for the displacements up to around 400 cm"*, 
and then become sub— Lorentzian for larger displacements. In fact, the line shape must 
approach zero at least as fast as an exponential. This can be shown from the analyticity of 
the correlation function; the successive derivatives of C(t) in the t = limit correspond to 
moments of the line shape in frequency space. Because the derivative are all finite, this 
implies that all the moments of the line shape are also finite and, therefore the line shape 
must approach zero faster than any inverse power of cj. In Fig. 8, we present all the 
calculated absorption coefficients in the window region 600 — 1250 cm"*. We note that for 
a specified frequency cj, we exclude completely any contribution from lines that are within 
[a; — 25 cm"*, a; + 25 cm"*] in the calculations. As shown by Fig. 8, the negative 
temperature dependence of self— continuum is clearly demonstrated. 

III. DISCUSSIONS AND CONCLUSIONS 

In comparison with our previous studies on the far— wing line shape, there are 
several important advances which have been made in the present study. First of all, by 



26 



clearly distinguishing the sensitive and insensitive variables and by modifying the Monte 
Carlo subroutine used previously to handle integrations whose volume is not rectangular, 
the effectiveness of the important sampling is enhanced significantly. As a result, by 
accounting for random selections the order of 10^ (which is comparable to or slightly more 
than that required for lower dimensionality cases), one is able to evaluate ll--dimensional 
integrations. 

Second, we have carried out numerical calculations based on the formalism which 
satisfies the detailed balance principle exactly and has aiiigher accuracy in the short-time 
limit. As expected, this increases the difficulty because the evaluation of the integrands 
requires more calculations. As shown by Eqs. (3) and (4), within this formalism the dipole 
moment operator appears in formulas in such a way that it is always sandwiched by the 
density matrix. However, these two operators have different characters: one depends on 
the coordinates only while the other contains the differential operators. No matter what 
kind of representation is chosen, the sandwiched operators require more loops to evaluate 
their values. As in our previous studies, we use the coordinate representation because it 
enable us to include more populated states; we also introduce the distribution functions and 
store them in files because it enable us to obtain values of the integrand with less CPU 
time. But, to calculate these distributions with a high cut-off jj^^^^ requires long CUP 
times. We note that for temperatures of interest in atmospheric applications, jj^^^^ = 23 is 
enough, but for higher temperatures, a higher j^^^ would be necessary. With these input 
files, to obtain values of the integrand becomes relatively easy. Combing the technique 
mentioned above to reduce the number of random selection to the order of 10^, one is able 
to complete the evaluation of 11— dimensional integrations and to obtain a line shape in one 
day with one workstation. 

Finally, by carrying out band averages in a more sophisticated way, we can 
calculate the two line shape functions. In addition, the effects resulting from the frequency 
detuning have been taken into account in the averaging processes. We note that except for 



27 



the simplest system, e.g. CO^ - Ar,^ the latter has not been done previously. In fact, most 
of calculations are similar to those without including the frequency detuning correction, 
except the integrand contains a derivative. As explained above, we have developed a 
technique which enables one to overcome this obstacle and calculate the frequency detuning 
corrections. 

Based on numerous test calculations of line shapes during the course of study, we 
can draw several important conclusions. It is necessary to consider the frequency detuning 
correction because the effect on the line shape is significant, especially in the 
high— frequency region. However, this effect shows up only when one distinguishes the 
positive and the negative resonance lines and carries out the band average over them 
separately. In other words, we have demonstrated that the line shapes obtained from the 
two band averages is asymmetric and we have found that this asymmetry results from the 
frequency detuning effect, or more specifically, from the distribution of lines within the 
band of interest. In addition, we find that for the pure rotational band the magnitudes of 
Z^{u) are significantly larger than X^ui) for a; > and vice versa for a; < 0, and these gaps 
become larger as a; increases. We note that the conclusion concerning the asymmetry 
claimed here is applicable for the band average line shapes and has nothing to do with 
individual lines. 

Finally, we would like to make a few of comments on the vibrational bands. We 
expect the main conclusions about the line shapes for the pure rotational band would 
remain true, but some different features could show up because the contributions to ^+(a;) 
and /.(cj) from the frequency detuning terms depend strongly on the band structure. We 
expect that the more unevenly and the more widely the lines are distributed within the 
band of interest, the contributions come from the frequency detuning corrections will 
increase and the more /+(a;) differs from -^.(a;). It is well known that the line distributions 
of the vibrational bands are quite different from that for the pure rotational one. The 
former's lines are, more or less evenly, located on both sides of the band centers, but the 



28 



latter 's are always on the high-frequency side because the band center is zero. We note 
that according to the definition, the average positive resonance frequency u introduced here 
is not the band center. But these two are very close for the vibrational bands and are quite 
different for the pure rotational band. We expect that for the vibrational band the 
frequency detuning corrections could become smaller. This means that Jf^{uj) and -f.(a;) 
differ from each other by smaller amounts. Meanwhile, we could not draw any conclusion 
about which magnitude is larger because this is related to the special structure of the pure 
rotational band. It has been known for years that the line shapes for CO 2 derived from 
experimental data are asymmetric and they are not the same for different bands. ^^'^^ So 
far, there has been no theoretical explanation why the different bands have different shapes 
without assuming that the interaction depends sensitively on the vibrational quantum 
numbers which seems unlikely to be true. We think that both from the theoretical and 
practical points of view, to investigate the lines shapes for different bands is an interesting 
subject to pursue. 

ACKNOWLEDGEMENT: 

This work was supported in part by the Department of Energy Interagency 
Agreement under the Atmospheric Radiation Measurement Program, by NASA through 
grants NAG5-6314, NAG5^269, and NAGW-4693. The authors would like to thank the 
National Energy Research Supercomputer (Livermore, CA) for computer time and facilities 
provided. Also, we would like to thank J. G. Cormier for providing his results prior to 
publication. 



29 



LIST OF FIGURES 

Fig. 1. The two-dimensional distribution of G^^ ^^j (a, ^, 7) of HjO over the Euler angles 
a and 7 obtained at T = 296 K for j^^^^^ = 26. The values of the Euler angle ^ is 
fixed and the four plots presented here correspond to ^ = 5, 22, 38, and 50 degrees, 
respectively. 

Fig. 2. The two-dimensional distribution of G^( ^^j (^, u, v) of HjO over the two sensitive 
variables ^ and u obtained at T = 296 K for j^^^^ = 26. This distribution results 
from the averaging G *( ^^j (^, u, v) over the one insensitive variable v. In contrast 
with Fig. 1, a logarithmic coordinate is used for the G *( a j (^, u, v) axis. 

Fig. 3. The self— broadened far— wing line shape of HjO (in units of cm"^ atm"^) as a 
function of frequency u (in units of cm'^) for T = 296 K. The dashed curve 
represents Xj^tJ) calculated from the positive resonance line average and the dotted 
curve represents X_{u)) calculated from the negative resonance line average. The 
solid curve is X{iJ) which is the mean of ^+(0;) and X,{uj). 

Fig. 4. The calculated self— broadened absorption coefficient (in units of cm^ molecule"^ 

atm"^) at T = 296 K in the 300 — 1100 cm"^ spectral region is represented by A. For 
comparison, the experimental values of Burch et. al are denoted by + and those 
from Cormier et al. are denoted by d. 

Fig. 5. The calculated second virial coefficients (in units of cm^ mol'^) as a function of 

temperature is represented by a solid line. The experimental data are denoted by d. 

Fig. 6. The calculated differential cross section (in arbitrary units) as a function of the 

laboratory scattering angle 6 is represented by a solid line. The experimental values 
of Duquette are denoted by d. 

Fig, 7, The self— broadened far— wing line shapes ^+(0;) of HjO (in units of cm"^ atm"^) as a 
function of frequency u) (in units of cm"^) obtained for T = 220, 240, 260, 280, 300, 
and 320 K; these are represented by 6 curves in order from top to bottom. The 
frequency uj varies from — 1600 cm'^ to 1600 cm'^ The corresponding X_{u) are 



/ 

_L 



30 



reflections of /^^(w) about the a; = axis. 
Fig. 8. The self-broadened absorption coefficient (in units of cm^ molecule*' atm"') in the 
window region 600 - 1250 cm'* calculated for T = 220, 230, 240, 250, 260, 270, 280, 
290, 300, 310, 320, and 330 K in order from top to bottom. A cut-off 25cm*' is used 
to exclude completely any contribution of lines that are closer than this limit. 



31 



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32 



1984. 

17) J. G. Cormier, R. Ciurylo, and J. R. Drummond, J. Chem. Phys. to be published. 

18) A. J. Stone, A. Dullweber, M. P. Hodges, P. L. A. Popelier, and D. J. Wales, Orient: a 
Program for Studying Interactions between Molecules, Version 3.2, University of 
Cambridge (1995). 

19) J. R. Reimers, R. O. Watts, and M. L. Klein, Chem. Phys. 64, 95 (1982). 

20) G. Duquette, Ph. D. Thesis, University of Waterloo, Waterioo, Canada (1977). 

21) D. E. Burch, D. A. Gryvnak, R. R. Patty, and C. E. Bartky, J. Opt. Soc. Am. 59, 267 
(1969). 

22) M. V. Tonkov, N. N. Filippov, V. V. Bertsev, J. P. Bouanich, Nguyen Van-Thanh, C. 
Brodbeck, J. M. Hartmann, C. Boulet, F. Thibault, and R. Le Doucen, Appl. Optics 35, 
4863 (1996). 



p = 5 



P = 22 




360 



360 




360 



360 




100 



u (= [a+Y]-2) 




600 800 1000 1200 1400 1600 
FREQUENCY (cm"^) 



10 



-19 



I 

D 
O 

o 

E 

CM 

E 
o 



10 



-21 



UJ 
O 



LxJ 
O 

o 
g 

^10 

o 

CO 

m 
< 



10 



-23 




-22 



300 400 500 



600 700 800 900 
FREQUENCY (cm"^) 



1000 1100 



-50 - 



o 

E 



-100 - 



E 
o 

m 

C 

0) 

o 

a> 
o 
O 



c 
o 
o 

0> 
CO 



-150 - 



-200 - 



-250 - 



-300 - 



-350 




300 400 500 600 700 800 900 1000 1100 1200 

Temperature (K) 



-I — ,_, — p. 



I ' ' ' ' I 



I I — r 



-, — , — , — P 



0) 



b 10' 

o 

c 



© 



c 
o 

o 

6 



c 



HjO - H2O 




■ ■ ■ ' 



■ ■ ■ 



-I — I I I I ■ ■ ' 



.J L. 



10 15 

(in degrees) 



20 



25 



-1 



LINE SHAPE FUNCTION (cm"' atm"') 



m 
o 

c 
m 

z 

Q 

3 




-I 1 I I I 1 1 il I I ■ ■ I 1 1 il 



I I I 1 1 il t ' ■ i ■ « 1 il 



/ 



.-V 



ABSORPTION COEFFICIENT (cm'' molecule"^ atm"^