MOLECULAR PHYSICS 2000 VOL 98 NO 22 1907 - 1918

Spectral moments of the rotational correlation functions for the rst- and second-rank tensors of asymmetric top molecules

Y P KALMYKOV1 and S V TITOV2

1 Centre drsquoEtudes Fondamentales Universite de Perpignan 52 avenue de Villeneuve66860 Perpignan cedex F rance

2 Institute of Radio Engineering and Electronics of the Russian Academy ofSciences Fryazino Moscow Region 141120 Russia

(Received 13 April 2000 revised version accepted 29 M ay 2000)

A method of evaluating the spectral moments M 12k of the rotational correlation functions for

the rst- and second-rank tensors of rigid asymmetric top molecules is developed It is basedon the calculation of the coe cients of a Taylor series expansion of the vector and tensororientational correlation functions about t ˆ 0 with the help of angular momentum theoryand is applicable to a pair intermolecular interaction potential with arbitrary dependence onthe angular variables Equations for the second hellipM l

2dagger fourth hellipM l4dagger and sixth hellipM l

6dagger spectralmoments are obtained as a demonstration of the ability of the method The results for M l

2 andM l

4 coincide with previously known values and the equation for M l6 is new As particular

cases the theory contains the results for classical ensembles of symmetric tops spherical topsand linear molecules

1 IntroductionInfrared absorption and Raman scattering spectra of

molecular uids contain valuable information on themolecular rotational motion and intermolecular inter-actions The usable approach in the interpretation ofthe spectra in terms of molecular dynamics is the analy-sis of spectral moments The spectral moments arede ned as [1 2]

M ln ˆ

hellipnI lhellipdaggerd hellip1dagger

where I lhellipdagger is the spectral intensity is the angularfrequency and the integral extends over all frequenciesfor which I lhellipdagger 6ˆ 0 Here the rst-order moments M 1

n

and the second-order moments M 2n correspond to the

absorption and Raman scattering spectra respectivelyIf a spectrum covers only a nite frequency range itsintensity distribution may be expanded in a complete setof polynomials in The in nite set of coe cients inthat expansion is simply a linear combination of thespectral moments In other words the spectral momentscharacterize the spectrum uniquely Any other par-ameters of the spectrum such as the frequency of themaximum intensity the half-width the intensity at anyspeci c frequency also depend on the spectral moments[3] The spectral moments allow one to analyse the

infrared and Raman spectra in terms of equilibriummolecular parameters (principal moments of inertia ofmolecules direction cosines of the dipole momentvector components of the polarizability temperature)and of the anisotropic part of the intermolecular inter-action potential In the moment analysis it is notrequired to solve the equations of molecular motionThus the determination of the spectral moments is asimpler problem than the calculation of spectral bandcontours

The spectral moments M ln are related to the coe -

cients in the Taylor series expansion of appropriateorientational correlation functions (CFs) C lhellip tdagger [1 2] viz

dn

dtn C lhellip tdaggerjtˆ0 ˆ inM ln hellip2dagger

where the correlation functions C lhellip tdagger are de ned as

C lhellip tdagger ˆhellip

I lhellipdagger exphellip itdaggerd hellip3dagger

In the theory of infrared and Raman spectra the CFsC lhellip tdagger of the rst- hellip l ˆ 1dagger and second- hellip l ˆ 2dagger rank ten-sors are given by

C1hellip tdagger ˆ hmhellip tdagger cent mhellip0daggeri0 hellip4daggerand

C2hellip tdagger ˆ hTr fahellip tdagger cent ahellip0daggergi0 hellip5daggerrespectively where m is the transition dipole momentvector a ˆ a iexcl E Tr fag=3 is the symmetric second-

Molecular Physics ISSN 0026- 8976 printISSN 1362- 3028 online 2000 Taylor amp Francis Ltdhttpwwwtandfcoukjournals

Author for correspondence e-mail kalmykovuniv-perpfr

rank tensor with a zero trace (anisotropic part of thepolarizability tensor a E is the unit tensor) and thebrackets h i0 denote an equilibrium ensemble averageThe CFs C lhellip tdagger for asymmetric top molecules havealready been evaluated in [4- 9] However these resultsare not suitable for the evaluation of the spectralmoments because they were obtained in the context ofparticular models of molecular reorientations in uidswhere either the intermolecular interactions were nottaken into account at all (as in the free rotationalmodel [4- 6]) or the theory predicted in nite values forthe moments of orders higher than the second (as in theextended rotational di usion models [7- 9] whereinstantaneous collisions of molecules are assumed)

The evaluation of the spectral moments can beaccomplished in the context of both classical andquantum theories In the context of quantum theorythe spectral moments up to the fourth order for asym-metric top molecules were obtained by Gordon [3 10-12] A further attempt to calculate higher spectralmoments was undertaken by St Pierre and Steele [2]for classical ensembles of symmetric tops sphericaltops and linear molecules They used a method basedon the evaluation of the coe cients in a Taylor seriesexpansion for the CFs C lhellip tdagger about t ˆ 0 [2 13] so thattaking into account equation (2) it is possible to eval-uate the even spectral moments M l

2k (in the classicallimit all the odd moments are zero because the Clhellip tdaggerare even functions of time) St Pierre and Steele [2]were able to calculate formal expressions for all thesecoe cients for the classical ensembles of freely rotatingmolecules (ie in the absence of intermolecular inter-actions) They also derived the sixth moments M l

6 forhindered spherical tops and linear molecules (ie ontaking account of the intermolecular torques) The pur-pose of the present work is to extend the method [2] tothe calculation of the even spectral moments of thevector (dipole moment) CF C1hellip tdagger and the tensor CFC2hellip tdagger for a classical ensemble of hindered asymmetrictop molecules As in [2] we will not take into accountthe quantization of the rotational motion so that thetheory can be used to study the infrared and Ramanrotational spectra of diluted uids under low resolution[2 4- 6] Also we will neglect rotation- vibration inter-actions and collision-induced absorption and scattering[10- 12] By making these assumptions we can considerthe transition dipole moment m and the tensor a as realvector and tensor of de nite values and orientations xed in the molecular coordinate system and rotatingwith it (In the context of the approach under considera-tion the rotation- vibration coupling e ects can betaken into account in a manner used for the evaluationof zero-rank tensor correlation functions [14] which are

related to isotropic Raman scattering of asymmetric topmolecules)

2 Time derivatives of the rotational correlationfunctions of asymmetric top molecules

The CFs C1hellip tdagger and C2hellip tdagger (equations (4) and (5)) for aclassical ensemble of rigid asymmetric top molecules canbe equivalently presented as [4]

C1helliptdagger ˆX1

r r 0ˆiexcl1

ahellip1daggerr ahellip1daggercurren

r 0 R 1rr 0hellip tdagger

ˆ Refm2xpermilR 1

11hellip tdagger DaggerR 11 iexcl1hellip tdaggerŠ

Daggerm2ypermilR 1

1 1hellip tdagger iexcl R 11iexcl1hellip tdaggerŠ Dagger m2

zR10 0hellip tdaggerg hellip6dagger

and

C2helliptdagger ˆX2

r r 0ˆiexcl2

ahellip2daggerr ahellip2daggercurren

r 0 R 2rr 0hellip tdagger

ˆ 12 Ref4a2

xyhellipR 22 2hellip tdagger iexcl R 2

2iexcl2hellip tdaggerdagger

Dagger 4a2yzhellipR 2

1 1hellip tdagger iexcl R 21iexcl1hellip tdaggerdagger

Dagger 4a2xzhellipR 2

11hellip tdagger DaggerR 21 iexcl1hellip tdaggerdagger

iexcl hellip4axx ayy Dagger ayyazz Dagger axx azzdaggerhellipR 222hellip tdagger DaggerR 2

2 iexcl2hellip tdaggerdagger

iexcl ayyazzhellip3R 200hellip tdagger iexcl

6

ppermilR 2

2 0hellip tdagger Dagger R 202hellip tdaggerŠdagger

iexcl axx azzhellip3R 2

00hellip tdagger Dagger6

ppermilR 2

2 0hellip tdagger DaggerR 202hellip tdaggerŠdaggerg hellip7dagger

where

ahellip1dagger0 ˆ imz ahellip1dagger

sect1 ˆ iexcl 12

p hellipmx uml imydagger hellip8dagger

and

ahellip2dagger0 ˆ

32

sazz ahellip2dagger

sect1 ˆ ihellipaxz sect iayzdagger

ahellip2daggersect2 ˆ iexcl 1

2hellipaxx iexcl ayy sect 2iaxydagger hellip9dagger

are the components of two irreducible tensorial sets ofthe rst and second rank [15] in the molecular co-ordinate system Oxyz respectively mi and aij are thecomponents of the vector m and of the tensor a theR l

nmhellip tdagger are the equilibrium correlation functions de nedas

R lnmhellip tdagger ˆ

Xl

rˆiexcll

Dlcurrenr nhellip tdaggerDl

rmhellip0dagger +

0

hellip10dagger

1908 Y P Kalmykov and S V Titov

D1rmhellip tdagger ˆ D1

rmhelliprsquohellip tdagger sup3hellip tdagger Aacutehellip tdaggerdagger and D2rmhellip tdagger ˆ D2

rmhelliprsquohellip tdagger sup3hellip tdagger Aacutehellip tdaggerdagger are the elements (D functions) of the Wignerrotation matrices Dhellip1daggerhelliprsquohellip tdagger sup3hellip tdagger Aacutehellip tdaggerdagger and Dhellip2daggerhelliprsquohellip tdagger sup3hellip tdagger Aacutehellip tdaggerdagger [15] and the asterisk denotes the complexconjugate (in order to simplify equations (6) and (7)we have used symmetry properties of the D functions[15] and the fact that R l

n mhellip tdagger vanishes if n Daggerm is odd [6])The Euler angles sup3 rsquo and Aacute connecting the molecularframe Oxyz to the laboratory (space- xed) coordinatesystem OXY Z are de ned here as in [16] ie the rela-tions between the unit vectors nx ny nz of the molecularframe axes and the unit vectors nsup3 nrsquo nAacute directed alongthe angular velocities _sup3 _rsquo and _Aacute are given by

nAacute ˆ nz

nsup3 ˆ nx cos Aacute iexcl ny sin Aacute

nrsquo ˆ nx sin Aacute sin sup3 Daggerny cos Aacute sin sup3 Dagger nz cos sup3

The nine elements of the Wigner rotation matrixDhellip1daggerhelliprsquohellip tdagger sup3hellip tdagger Aacutehellip tdaggerdagger can be evaluated from the followingequations [15]

D100hellip tdagger ˆ cos sup3hellip tdagger

D110hellip tdagger ˆ iexcl 1

2p sin sup3hellip tdaggereiexclirsquohelliptdagger

D101hellip tdagger ˆ 1

2p sin sup3hellip tdaggereiexcliAacutehelliptdagger

D11 sect1hellip tdagger ˆ 1

2hellip1 sect cos sup3hellip tdaggerdaggereiexclihelliprsquohelliptdaggersectAacutehelliptdaggerdagger

and

D1curreni jhellip tdagger ˆ hellipiexcl1daggeriexcliiexcljD1

iexcliiexcljhellip tdagger

All the elements D2M Nhellip tdagger of the matrix Dhellip2dagger can be

expressed in terms of D1K Lhellip tdagger (equations relating

D2M Nhellip tdagger and D1

K Lhellip tdagger are available explicitly eg in [6])The values of the CFs C lhellip tdagger at t ˆ 0 are given by

C1hellip0dagger ˆ m2x Dagger m2

y Dagger m2z ˆ m2

C2hellip0dagger ˆ 2hellipa2xy Daggera2

yz Dagger a2xz iexcl ax x ayy iexcl ayyazz iexcl axx azzdagger

According to equations (1) and (2) the spectralmoments M l

2n are determined by the short term behav-iour of the CFs C lhellip tdagger In turn the behaviour of the CFsC lhellip tdagger de ned by equations (6) and (7) is completelydetermined by the CF R l

n mhellip tdagger from equation (10)Thus the evaluation of the spectral moments of asym-metric top molecules requires only the calculation of thetime derivatives of the 2k order at t ˆ 0 of the CFR l

n mhellip tdagger viz

d2k

dt2k R lnmhellip0dagger ˆ hellipiexcl1daggerk

Xl

rˆiexcll

dk

dtk Dlcurrenrnhellip0dagger dk

dtk Dlrmhellip0dagger

+

0

hellip11dagger

For the problem under consideration it is more con-venient to evaluate the equilibrium ensemble averagein equation (11) by using the mixed variables namelythe Euler angles sup3rsquoAacute and the components of theangular velocity vector X ˆ fOx Oy Ozg in the molecu-lar frame instead of the generalized coordinates andimpulses fsup3rsquoAacutepsup3 prsquopAacuteg (here and below we omiteverywhere the argument 0 in the Euler angles andOi) For simplicity we con ne ourselves to the evalua-tion of the second (M l

2dagger fourth (M l4dagger and sixth (M l

6daggermoments only Thus according to equation (11) weneed to evaluate the rst second and third time deriva-tives of the Wigner D functions Dl

n m at t ˆ 0 On takinginto account the Euler relations [16] viz

_sup3 ˆ Ox cos Aacute iexcl Oy sin Aacute hellip12dagger

_rsquo ˆ 1sin sup3

hellipOx sin Aacute DaggerOy cos Aacutedagger hellip13dagger

_Aacute ˆ Oz iexcl ctg sup3hellipOx sin Aacute Dagger Oy cos Aacutedagger hellip14dagger

we can write equations for these derivatives as follows

ddt

Dlnm ˆ _sup3

Dlnm

sup3Dagger _rsquo

Dlnm

rsquoDagger _Aacute

Dlnm

Aacuteˆ

X1

sˆiexcl1

OsLsDlnm hellip15dagger

d2

dt2Dl

nm ˆX1

s1ˆiexcl1

_Os1Ls1

DaggerX1

s1 s2ˆiexcl1

Os2Os1

Ls2L s1

Dl

nm hellip16dagger

d3

dt3Dl

nm ˆX1

s1ˆiexcl1

Os1Ls1

DaggerX1

s1 s2ˆiexcl1

hellip2 _Os1Os2

DaggerOs1_Os2

daggerL s2L s1

DaggerX1

s1 s2 s3ˆiexcl1

Os1Os2

Os3Ls3

L s2Ls1

Dl

nm hellip17dagger

where the new variables Osect1 O0 and the operatorsL sect1 L 0 are de ned as

Osect1 ˆ iexcl 12

p hellipOx uml iOydagger O0 ˆ iOz hellip18dagger

L sect1 ˆ i2

p eumliAacute sect ctg sup3

AacuteDagger i

sup3uml 1

sin sup3

rsquo

micro para

L 0 ˆ iexcli

Aacute hellip19dagger

The operators L sect1 L 0 have the property [15]

Spectral moments of CFs of asymmetric top molecules 1909

L sDln m ˆ iexcl

lhellip l Dagger1dagger

pC lmDaggers

lm1sDln mDaggers

ˆ

iexclmDln m s ˆ 0

sectlhellip l Dagger1dagger iexcl mhellipm sect 1dagger

2

sDl

nmsect1 s ˆ sect1

8gtgtltgtgt

hellip20daggerwhere CL mDaggern

l1 ml2 n are the Clebsch- Gordan coe cients [15]The time derivatives of the angular velocity componentsin equations (19) and (20) can be determined from theEuler equations [16]

I x_Ox ˆ hellipI y iexcl I zdaggerOyOz Dagger K x hellip21dagger

I y_Oy ˆ hellipI z iexcl I xdaggerOx Oz Dagger K y hellip22dagger

I z_Oz ˆ hellipI x iexcl I ydaggerOx Oy DaggerK z hellip23dagger

and their time derivatives

I xOx ˆ hellipI y iexcl I zdaggerhellip _OyOz DaggerOy

_Ozdagger DaggerX1

sˆiexcl1

OsL sK x hellip24dagger

I yOy ˆ hellipI z iexcl I xdaggerhellip _Ox Oz DaggerOx

_Ozdagger DaggerX1

sˆiexcl1

OsL sK y hellip25dagger

I zOz ˆ hellipI x iexcl I ydaggerhellip _Ox Oy DaggerOx

_Oydagger DaggerX1

sˆiexcl1

OsL sK z hellip26dagger

where

I ˆ

I x 0 0

0 I y 0

0 0 I z

0BBB

1CCCA

is the inertia tensor in the principal axis of inertia I x I y I z are the principal components of the molecular inertiatensor I and

K ˆ fK xhellipsup3rsquoAacutedagger K yhellipsup3rsquoAacutedagger K zhellipsup3rsquoAacutedaggerg

is the torque acting on the molecule The components ofK can be expressed [2 12] in terms of the anisotropicpart of the potential energy Vhellipsup3rsquoAacutedagger viz

K x ˆ L x V ˆ iexcl 12

p hellipL 1 DaggerL iexcl1daggerV hellip27dagger

K y ˆ L yV ˆ i2

p hellipL 1 iexcl L iexcl1daggerV hellip28dagger

K z ˆ L zV ˆ iL 0V hellip29dagger

Here L x L y L z are the angular momentum operators inthe molecular coordinate system [14]

On using equations (11) plus (15)- (20) we can obtainequations for the second fourth and sixth derivatives ofR l

nmhellip tdagger at t ˆ 0

R lnmhellip0dagger ˆ iexcllhellip l Dagger1dagger

X1

s 01 s1ˆiexcl1

hOcurrens 01Os1

i0C lnDaggers 01

ln1s 01

pound C lmDaggers1lm1s1

dnDaggers 01 mDaggers1

hellip30dagger

R lhellip4daggernmhellip0dagger ˆ lhellip l Dagger 1dagger

X1

s 01 s1ˆiexcl1

h _Ocurrens 01

_Os1i0C

lnDaggers 01

ln1s 01ClmDaggers1

lm1 s1dnDaggers 0

1 mDaggers1

iexcl l3=2hellip l Dagger1dagger3=2X1

s 01 s2 s1ˆiexcl1

helliph _Ocurrens 01Os2

Os1i0C lnDaggers 0

1ln1 s 0

1

pound C lmDaggers1lm1 s1

C lmDaggers1Daggers2lmDaggers1 1 s2

dnDaggers 01 mDaggers1Daggers2

Dagger h _Os 01Ocurren

s1Ocurren

s2i0

pound C lnDaggers1ln 1s1

C lnDaggers1Daggers2lnDaggers1 1 s2

C lmDaggers 01

lm1s 01dnDaggers1 Daggers2 mDaggers 0

1dagger

Dagger l2hellip l Dagger1dagger2X1

s 02 s 0

1 s2 s1ˆiexcl1

hOcurrens 02Ocurren

s 01Os2

Os1i0

pound C lnDaggers 01

ln 1s 01C lnDaggers 0

1Daggers 02

lnDaggers 01 1 s 0

2C lmDaggers1

lm1s1

pound C lmDaggers1Daggers2lmDaggers1 1 s2

macrnDaggers 01Daggers 0

2 mDaggers1Daggers2 hellip31dagger

R lhellip6daggernmhellip0dagger ˆ iexcllhellipl Dagger1dagger

X1

s 01 s1ˆiexcl1

h Ocurrens 01

Os1i0C

lnDaggers 01

ln1s 01ClmDaggers1

lm1 s1dnDaggers 0

1 mDaggers1

Dagger l3=2hellipl Dagger1dagger3=2X1

s 01 s1 s2ˆiexcl1

pound permilh Ocurrens 01hellip2 _Os1

Os2DaggerOs1

_Os2daggeri0C

lnDaggers 01

ln1 s 01ClmDaggers1

lm1 s1

pound ClmDaggers1Daggers2lmDaggers1 1s2

dnDaggers 01 mDaggers1Daggers2

h Os 01hellip2 _Ocurren

s1Ocurren

s2DaggerOcurren

s1_Ocurren

s2daggeri0

DaggerClnDaggers1ln1 s1

ClnDaggers1Daggers2lnDaggers1 1s2

ClmDaggers 0

1lm1s 0

1dnDaggers1Daggers2 mDaggers 0

1Š

iexcl l2hellip l Dagger1dagger2X1

s 01 s1 s2 s3ˆiexcl1

permilh Ocurrens 01Os1

Os2Os3

i0

pound ClnDaggers 0

1ln1 s 0

1ClmDaggers1

lm1 s1ClmDaggers1Daggers2

lmDaggers1 1 s2ClmDaggers1 Daggers2Daggers3

lmDaggers1 Daggers2 1s3

pound dnDaggers 01 mDaggers1Daggers2Daggers3

Daggerh Os 01Ocurren

s1Ocurren

s2Ocurren

s3i0ClnDaggers1

ln1 s1

pound ClnDaggers1Daggers2lnDaggers1 1s2

ClnDaggers1Daggers2Daggers3lnDaggers1Daggers2 1s3

ClmDaggers 0

1

lm1s 01macrnDaggers1Daggers2Daggers3 mDaggers 0

1Š

iexcl l2hellip l Dagger1dagger2X1

s 01 s 0

2 s1 s2ˆiexcl1

hhellip2 _Ocurrens 01Ocurren

s 02DaggerOcurren

s 01

_Ocurrens 02dagger

1910 Y P Kalmykov and S V Titov

pound hellip2 _Os1Os2

Dagger Os1_Os2

daggeri0C lnDaggers 01

ln1s 01C lnDaggers 0

1Daggers 02

lnDaggers 01 1 s 0

2C lmDaggers1

lm1s1

pound C lmDaggers1 Daggers2lmDaggers1 1s2

dnDaggers 01Daggers 0

2 mDaggers1Daggers2

Dagger l5=2hellip l Dagger1dagger5=2X1

s 01 s 0

2 s1 s2 s3ˆiexcl1

permilhhellip2 _Ocurrens 01Ocurren

s02

DaggerOcurrens 01

_Ocurrens 02daggerOs1

Os2Os3

i0C lnDaggers 01

ln1 s 01

pound C lnDaggers 01Daggers 0

2lnDaggers 0

1 1 s 02ClmDaggers1

lm1s1C lmDaggers1 Daggers2

lmDaggers1 1s2C lmDaggers1 Daggers2 Daggers3

lmDaggers1 Daggers2 1s3

pound dnDaggers 01 Daggers 0

2 mDaggers1 Daggers2 Daggers3

Dagger hhellip2 _Os 01Os 0

2Dagger Os 0

1

_Os 02daggerOcurren

s1Ocurren

s2Ocurren

s3i0C lnDaggers1

ln 1s1C lnDaggers1 Daggers2

lnDaggers1 1 s2

pound C lnDaggers1Daggers2Daggers3lnDaggers1Daggers2 1 s3

ClmDaggers 01

lm1 s 01C lmDaggers 0

1Daggers 02

lmDaggers 01 1s 0

2dnDaggers1 Daggers2 Daggers3 mDaggers 0

1 Daggers 02Š

iexcl l3hellip l Dagger1dagger3X1

s 01 s 0

2 s 03 s1 s2 s3ˆiexcl1

hOcurrens 01Ocurren

s 02Ocurren

s 03Os1

Os2Os3

i0

pound ClnDaggers 0

1

ln1 s 01C

lnDaggers 01 Daggers 0

2

lnDaggers 01 1s 0

2C

lnDaggers 01Daggers 0

2Daggers 03

lnDaggers 01Daggers 0

2 1 s 03C lmDaggers1

lm1s1

pound C lmDaggers1 Daggers2lmDaggers1 1s2

C lmDaggers1 Daggers2 Daggers3lmDaggers1 Daggers2 1s3

dnDaggers 01Daggers 0

2 Daggers 03 mDaggers1Daggers2 Daggers3

hellip32dagger

where macrik is Kronekerrsquo s symbol Here it has been takeninto account that [15]

Xl

rˆiexcll

Dlcurrenr mhellip0daggerDl

rm 0hellip0dagger ˆ macrmm 0 hellip33dagger

Equilibrium averages from combinations of the com-ponents of the angular velocity vector in equations(30)- (32) are easily calculated on noting that

hOiOji0 ˆ kTI i

dij hO2i O2

j i0 ˆ kTI i

hellip dagger2

hellip1 Dagger2dijdagger

hO3i O2

j i0 ˆ 15kTI i

hellip dagger3

dij hellip i j ˆ x y zdagger hellip34dagger

Equations for the time derivatives of the CFs C1hellip tdagger andC2hellip tdagger at t ˆ 0 have a de nite symmetry due to the factthat the values of R lhellip2kdagger

n m hellip0dagger (appearing in those equa-tions) depend on I x I y I z K x K y K z L x L y L z onlyand in the cyclic transformation of indices

fx y zg fy z xg fz x yg

transform into each other as follows

R 1 hellip2kdagger0 0 hellip0dagger R 1 hellip2kdagger

11 hellip0dagger DaggerR 1 hellip2kdagger1 iexcl1 hellip0dagger

R 1 hellip2kdagger11 hellip0dagger iexcl R 1 hellip2kdagger

1 iexcl1 hellip0dagger hellip35dagger

and

R 2 hellip2kdagger22 hellip0dagger iexcl R2 hellip2kdagger

2 iexcl2 hellip0dagger R 2 hellip2kdagger1 1 hellip0dagger iexcl R 2 hellip2kdagger

1iexcl1 hellip0dagger

R 2 hellip2kdagger1 1 hellip0dagger DaggerR 2 hellip2kdagger

1iexcl1 hellip0dagger hellip36dagger

and

R 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger fR 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger

Dagger3R 2 hellip2kdagger00 hellip0dagger iexcl

6

ppermilR 2 hellip2kdagger

20 hellip0dagger

DaggerR 2 hellip2kdagger02 hellip0daggerŠg=4

fR 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger

Dagger3R 2 hellip2kdagger00 hellip0dagger Dagger

6

ppermilR 2 hellip2kdagger

20 hellip0dagger

DaggerR 2 hellip2kdagger02 hellip0daggerŠg=4 hellip37dagger

This circumstance allows us to evaluate in equations(30)- (32) only the time derivatives of R l

m mhellip tdagger and

R lmsect4 mhellip tdagger at t ˆ 0 Having determined R l hellip2kdagger

mm hellip0dagger and

R l hellip2kdaggermsect4 mhelliptdagger we can evaluate R 1 hellip2kdagger

0 0 hellip0dagger R 2 hellip2kdagger2 2 hellip0dagger iexcl

R 2 hellip2kdagger2iexcl2 hellip0dagger and R 2 hellip2kdagger

22 hellip0dagger DaggerR 2 hellip2kdagger2iexcl2 hellip0dagger and then the

time derivatives of all the other functions in equations(35)- (37) by changing the indices in I i K i and L i fromfx y zg to fy z xg and fz x yg respectively

Thus on using explicit equations for the Clebsch-Gordan coe cients [15] and equations (21)- (29) and(34) we can obtain from equations (30)- (32) all thequantities of interest (listed in the appendix)

3 Spectral moments for dipolar absorptionHaving determined R 1hellip2kdagger

00 hellip0dagger R 1 hellip2kdagger1 1 hellip0dagger DaggerR 1 hellip2kdagger

1 iexcl1 hellip0daggerand R 1 hellip2kdagger

1 1 hellip0dagger iexcl R 1 hellip2kdagger1 iexcl1 hellip0dagger for k ˆ 1 2 and 3 from equa-

tions (A 1) (A 3) and (A 5) of the appendix and taking

into account equations (2) and (6) we can obtain thesecond fourth and sixth spectral moments for thedipolar absorpt ion viz

M 12 ˆ kTpermilm2

xhellipI iexcl1y Dagger I iexcl1

z dagger Dagger m2yhellipI iexcl1

x Dagger I iexcl1z dagger

Daggerm2zhellipI iexcl1

x Dagger I iexcl1y daggerŠ hellip38dagger

Spectral moments of CFs of asymmetric top molecules 1911

M 14 ˆ m2

x fhellipkTdagger2permil6I iexcl1y I iexcl1

z iexcl I iexcl1x hellipI iexcl1

y Dagger I iexcl1z dagger

Dagger I iexcl1x hellipI zI

iexcl2y Dagger I yI

iexcl2z dagger

Dagger I x I iexcl1z I iexcl1

y hellipI iexcl1y Dagger I iexcl1

z dagger Dagger I iexcl2y Dagger I iexcl2

z Š

Dagger hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z g

Daggerm2y fhellipkTdagger2permil6I iexcl1

x I iexcl1z iexcl I iexcl1

y hellipI iexcl1x Dagger I iexcl1

z dagger

Dagger I iexcl1y hellipI zI

iexcl2x Dagger I x I iexcl2

z dagger

Dagger I y I iexcl1z I iexcl1

x hellipI iexcl1x Dagger I iexcl1

z dagger Dagger I iexcl2x Dagger I iexcl2

z Š

Dagger hK 2x i0I

iexcl2x Dagger hK 2

z i0I iexcl2z g

Daggerm2z fhellipkTdagger2permil6I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y Dagger I iexcl1

x dagger

Dagger I iexcl1z hellipI yI

iexcl2x Dagger I x I iexcl2

y dagger Dagger I zIiexcl1x I iexcl1

y hellipI iexcl1x Dagger I iexcl1

y dagger

Dagger I iexcl2x Dagger I iexcl2

y Š Dagger hK 2x i0I

iexcl2x Dagger hK 2

yi0I iexcl2y g hellip39dagger

M 16 ˆ kT

I 3x I3

yI3zhellipm2

x M x Dagger m2yM y Daggerm2

zM zdagger hellip40dagger

where

M x ˆ I x I yIzhellipI2xhellipIy DaggerI zdagger DaggerhellipI y DaggerI z iexcl2IxdaggerhellipIy iexcl I zdagger2daggerhK 2

x i0

DaggerI2x I zhellipI3

y iexcl2I2yhellipI x Dagger2Izdagger DaggerIyhellipIx Dagger2I zdagger2 Dagger9I x I2

zdaggerhK 2yi0

DaggerI2x I yhellipI3

z iexcl2I2zhellipIx Dagger2Iydagger DaggerI zhellipI x Dagger2Iydagger2 Dagger9Ix I2

ydaggerhK 2zi0

DaggerI2x I2

zhellipI x IzhhellipL yK ydagger2i0 DaggerIy IzhhellipL x K ydagger2i0

DaggerI x I yhhellipL zK ydagger2i0dagger DaggerI2x I2

yhellipIx IzhhellipL yK zdagger2i0

DaggerI yIzhhellipLx K zdagger2i0 DaggerIx I yhhellipLzK zdagger2i0dagger

Dagger2I2x I yI zpermilhK x L yK zi0IyhellipIx iexclI y DaggerI zdagger

iexclhK xL zK yi0I zhellipI x iexcl I z DaggerI ydagger

DaggerhK yL xK zi0I yhellipI x iexcl Iy Dagger2Izdagger

iexclhK zL xK yi0I zhellipI x iexcl I z Dagger2I ydaggerŠiexcl2hellipkTdaggerI2x

poundpermilhLyK yi0I2zhellipI2

x Dagger3Ix Iy iexcl I2y DaggerI2

z DaggerI x I zdagger

DaggerhLzK zi0I2yhellipI2

x DaggerI x Iy DaggerI2y iexcl I2

z Dagger3I x I zdaggerŠ

DaggerhellipkTdagger2I xpermil3I4xhellipIy DaggerIzdagger DaggerI3

xhellip7I2y Dagger2I yI z Dagger7I2

zdagger

iexclI2xhellipIy DaggerIzdaggerhellip5I2

y iexcl34I yIz Dagger5I2zdagger

DaggerI xhellipIy iexcl Izdagger2hellip7I2y Dagger22I yIz Dagger7I2

zdagger

Dagger3hellipIy iexcl Izdagger2hellipIy DaggerI zdagger3Š hellip41dagger

Equations for M y and M z are obtained by changing allthe indices in equation (41) by cyclic permutation of x y z

4 Moments of Raman spectraUsing the symmetry propert ies of the functions in

equations (36) and (37) we can also obtain from equa-tions (A 1)- (A 6) all the R 2hellip2kdagger

nm hellip0dagger which are needed forthe calculation of the spectral moments M 2

2M24 and M 2

6Thus on taking into account equations (2) (7) and(A 1)- (A 6) we can obtain

M 22 ˆ 2kTpermilhellipa2

xy iexcl axx ayydaggerhellipI iexcl1x Dagger I iexcl1

y Dagger 4I iexcl1z dagger

Daggerhellipa2yz iexcl ayyazzdaggerhellip4I iexcl1

x Dagger I iexcl1y Dagger I iexcl1

z dagger

Daggerhellipa2xz iexcl axx azzdaggerhellipI iexcl1

x Dagger 4I iexcl1y Dagger I iexcl1

z daggerŠ hellip42dagger

M 24 ˆ 2hellipkTdagger2 fhellipa2

xy iexcl axx ayydaggerpermil23I iexcl1z hellipI iexcl1

y Dagger I iexcl1x dagger

Dagger4I iexcl2z hellipI y I iexcl1

x Dagger I x I iexcl1y dagger Dagger I iexcl1

z hellipI yIiexcl2x Dagger I x I iexcl2

y dagger

Dagger I zIiexcl1x I iexcl1

y hellipI iexcl1x Dagger I iexcl1

y dagger DaggerhellipI iexcl1x Dagger I iexcl1

y dagger2

Dagger40I iexcl2z DaggerhellipkTdaggeriexcl2helliphK 2

x i0Iiexcl2x Dagger hK 2

yi0I iexcl2y

Dagger4hK 2z i0I iexcl2

z daggerŠ Daggerhellipa2yz iexcl ayyazzdaggerpermil23I iexcl1

x hellipI iexcl1y Dagger I iexcl1

z dagger

Dagger4I iexcl2x hellipI y I iexcl1

z Dagger I zIiexcl1y dagger Dagger I iexcl1

x hellipI y I iexcl2z Dagger I zI

iexcl2y dagger

Dagger I x I iexcl1y I iexcl1

z hellipI iexcl1y Dagger I iexcl1

z dagger DaggerhellipI iexcl1y Dagger I iexcl1

z dagger2 Dagger40I iexcl2x

DaggerhellipkTdaggeriexcl2hellip4hK 2x i0I iexcl2

x Dagger hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z daggerŠ

Daggerhellipa2xz iexcl axx azzdaggerpermil23I iexcl1

y hellipI iexcl1x Dagger I iexcl1

z dagger

Dagger4I iexcl2y hellipI x I iexcl1

z Dagger I zIiexcl1x dagger Dagger I iexcl1

y hellipI x I iexcl2z Dagger I zI

iexcl2x dagger

Dagger I y I iexcl1x I iexcl1

z hellipI iexcl1x Dagger I iexcl1

z dagger DaggerhellipI iexcl1x Dagger I iexcl1

z dagger2 Dagger40I iexcl2y

DaggerhellipkTdaggeriexcl2helliphK 2x i0I iexcl2

x Dagger 4hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z daggerŠ

iexcl 9axx ayyhellipI iexcl1x iexcl I iexcl1

y dagger2 iexcl 9ayyazzhellipI iexcl1y iexcl I iexcl1

z dagger2

iexcl 9axx azzhellipI iexcl1x iexcl I iexcl1

z dagger2g hellip43dagger

M 26 ˆ 2kT

I3x I3

y I3z

fhellipa2xy iexcl axx ayydaggerM xy Daggerhellipa2

yz iexcl ayyazzdaggerM yz

Daggerhellipa2xz iexcl axx azzdaggerM zx iexcl axx ayyN xy iexcl ayyazzN yz

iexcl axx azzN zxg hellip44dagger

where

1912 Y P Kalmykov and S V Titov

M xy ˆ Ix IzhellipI2x I2

zhhellipLyK ydagger2i0 DaggerI2yI

2zhhellipL yK xdagger2i0

Dagger4I2x I2

yhhellipL yK zdagger2i0dagger DaggerI yIzhellipI2x I2

zhhellipL xK ydagger2i0

DaggerI2yI

2zhhellipL xK xdagger2i0 Dagger4I2

x I2yhhellipL x K zdagger2i0dagger

DaggerI x I yhellipI2x I2

z hhellipL zK ydagger2i0 DaggerI2yI

2zhhellipLzK xdagger2i0

Dagger4I2x I2

yhhellipL zK zdagger2i0dagger DaggerIyI zhellipI3xhellip4I y DaggerI zdagger

iexcl2I2xhellip4I2

y iexcl2I yIz DaggerI2zdagger DaggerI xhellip4I3

y Dagger20I2yIz

Dagger20I yI2z DaggerI3

zdagger Dagger9I2yI

2zdaggerhK 2

xi0 DaggerIx IzhellipI3yhellip4Ix DaggerIzdagger

iexcl2I2yhellip4I2

x iexcl2I x Iz DaggerI2zdagger

DaggerI yhellip4I3x Dagger20I2

x Iz Dagger20Ix I2z DaggerI3

zdagger Dagger9I2x I2

zdaggerhK 2yi0

DaggerI x I yhellipI3zhellipIx DaggerI zdagger iexcl2I2

zhellipI2x Dagger10Ix Iy DaggerI2

ydagger

DaggerI zhellipI x DaggerIydaggerhellipI2x Dagger34Ix Iy DaggerI2

ydagger Dagger144I2x I2

ydaggerhK 2zi0

Dagger2I x I yIzpermil2I x I yhelliphK x LyK zi0hellip2I x iexcl2I y iexcl Izdagger

DaggerhK yLx K zi0hellip2I x iexcl2I y DaggerI zdaggerdagger

DaggerI yIzhelliphK yL zK xi0hellip4Ix DaggerIy iexcl Izdagger

DaggerhK zLyK x i0hellip5I x DaggerIy iexcl I zdaggerdagger

iexclI x I zhelliphK xL zK yi0hellipIx Dagger4Iy iexcl I zdagger

DaggerhK zLx K yi0hellipIx Dagger5Iy iexcl I zdaggerdaggerŠ

iexcl2kTpermilhL yK yi0I2x I2

zhellipI2z Dagger3I yIz iexcl I2

y DaggerI xhellipI x Dagger12I y DaggerIzdaggerdagger

DaggerhLx K xi0I2yI

2zhellipI2

z Dagger3I x I z iexcl I2x DaggerI yhellipIy Dagger12Ix DaggerIzdaggerdagger

Dagger4hLzK zi0I2x I2

yhellipI2x Dagger10Ix Iy DaggerI2

y DaggerIzhellip3Ix Dagger3I y iexcl IzdaggerdaggerŠ

DaggerhellipkTdagger2permil3I5zhellipI x DaggerI ydagger Dagger7I4

zhellipI2x Dagger10I x Iy DaggerI2

ydagger

iexcl5I3zhellipI x DaggerI ydaggerhellipI x iexcl Iydagger2

DaggerI2zhellip7I4

x Dagger68I3x Iy Dagger90I2

x I2y Dagger68Ix I3

y Dagger7I4ydagger

DaggerI zhellipI x DaggerIydagger

poundhellip3I4x Dagger64I3

x Iy Dagger586I2x I2

y Dagger64I x I3y Dagger3I4

ydagger

Dagger4I x I yhellip3I4x Dagger52I3

x Iy Dagger130I2x I2

y Dagger52Ix I3y Dagger3I4

ydaggerŠ hellip45dagger

N xy ˆ 9I2zhellipI x iexcl Iydaggerf3I zhellipI2

xhK 2yi0 iexcl I2

yhK 2x i0dagger

iexcl2kT IzhellipI2xhL yK yi0 iexcl I2

yhLx K xi0dagger

DaggerhellipkTdagger2permil5hellipIx iexcl IydaggerhellipI2x Dagger10Ix Iy DaggerI2

y

Dagger3I zhellipIx DaggerI ydagger DaggerI2zdaggerŠg hellip46dagger

Equations for M yz M zx and N yz N zx in equation (44)can be obtained readily by changing all the indices inequations (45) and (46) from fx y zg to fy z xg andfz x yg respectively

5 Linear spherical and symmetric top moleculesThe results we have obtained contain the particular

cases of classical ensembles of symmetric tops sphericaltops and linear molecules In these cases the theory isconsiderably simpli ed as the calculation of all the spec-tral moments M l

2k needs only that of R l hellip2kdaggermm hellip0dagger [2] Thus

for symmetric top molecules (I x ˆ I y ˆ I 6ˆ I zdagger equa-tions (A 1) (A 3) and (A 5) from the appendix yield

R lmmhellip0daggerˆiexclkT

Ipermilx Daggersup2m2Š hellip47dagger

R l hellip4daggermm hellip0daggerˆ kT

Ihellip dagger2

xhellip3x iexcl1daggerDaggersup2 6xm2 Daggerm2 iexclx1Daggersup2hellip daggerDagger3sup22m4

Dagger 12I2permilhellipx iexclm2daggerhK 2

i0 Dagger2hellip1Daggersup2dagger2m2hK 2zi0Š hellip48dagger

R l hellip6daggermm hellip0daggerˆiexcl kT

Ihellip dagger3 11Daggersup2

copy5xhellip3x2 iexcl3x Dagger1dagger

Dagger3sup2hellipxhellip5x2 iexcl10x Dagger8daggerDagger3m2hellip5x2 iexcl4daggerdaggerDagger15sup22m2hellipxhellip3x iexcl2daggerDaggerm2hellip3x Dagger1daggerdagger

Dagger15sup23m4hellip3x Daggerm2daggerDagger15sup24m6 iexcl3sup2hellipx iexclm2dagger1Daggersup2

Dagger 1

hellipkTdagger2permil12hK 2

i0hellipxhellip9x iexcl5daggeriexclm2hellip9x iexcl7dagger

Daggersup2permilxhellip9x iexcl5daggeriexclm2hellip9m2 iexcl1daggerŠDaggersup22permilx Daggerm2hellip9x iexcl9m2 iexcl7daggerŠdaggerDaggerhK 2

zi0hellip2x Daggerm2hellip9x iexcl11daggerDaggersup2permil2x Daggerm2hellip27x Dagger9m2 iexcl20daggerŠDagger9sup22m2hellip3x iexcl1Dagger3m2daggerDagger9sup23m2hellipx Dagger3m2daggerDagger9sup24m4daggerDagger1

2helliphhellipLx K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipLyK xdagger2

DaggerhellipL yK ydagger2i0 Daggerhellip1Daggersup2daggerhhellipLzK xdagger2

DaggerhellipL zK ydagger2i0daggerhellip1Daggersup2daggerhellipx iexclm2daggerDaggerhelliphhellipL xK zdagger2

DaggerhellipL yK zdagger2i0 Daggerhellip1Daggersup2daggerhhellipL zK zdagger2i0daggerhellip1Daggersup2dagger3m2

DaggerhK yLx K z iexclK x LyK zi0m2hellip1Daggersup2dagger2

DaggerhK yLzK x iexclK x LzK yi0permilm2hellip5Dagger4sup2daggeriexclxhellip2Daggersup2daggerŠpoundhellip1Daggersup2daggerŠ

Dagger 1kT

permilhLxK x DaggerL yK yi0hellipx iexclm2daggerhellip1iexcl3hellip1Daggersup2dagger

poundhellipx Daggersup2m2daggerDagger2sup2daggerDagger2hL zK zi0hellip1Daggersup2dagger2m2

poundhellip1iexcl3hellip1Daggersup2daggerhellipx Daggersup2m2daggerdaggerŠordf

hellip49dagger

Spectral moments of CFs of asymmetric top molecules 1913

where

x ˆ lhellip l Dagger1dagger sup2 ˆ I=I z iexcl 1 K 2 ˆ K 2

x DaggerK 2y

Here we have taken into account that [14]

L x L y iexcl L yL x ˆ L z

Equations (47) and (48) for the second and the fourthmoments coincide with the results of [2] which wereobtained from the consideration of an ensemble of hin-dered symmetric tops Equation (49) for freely hellipV ˆ 0daggerrotating symmetric tops was obtained in [2] However itwas presented there with several misprints

For spherical tops hellipI x ˆ I y ˆ I z ˆ Idagger we simplyput sup2 ˆ 0 in equations (47)- (49) Thus for m ˆ 0 wehave

R l00hellip0dagger ˆ iexcl

kTI

x hellip50dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 1dagger Dagger 12I2

hK 2i0 hellip51dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 15x Dagger 5 iexcl 3x iexcl 1kT

pound hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 5dagger Dagger4hK 2

z i0

Dagger4hK x L zK y iexcl K yL zK x i0

Dagger hhellipL x K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipL yK xdagger2

DaggerhellipL yK ydagger2 DaggerhellipL zK xdagger2 DaggerhellipL zK ydagger2i0Šordf

hellip52dagger

Equations (50)- (52) are in accordance with the results of[2]

For linear molecules hellipI x ˆ I y ˆ I I z ˆ 0dagger one has

R l00hellip0dagger ˆ iexcl kT

Ix hellip53dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 2dagger Dagger 12I2

hK 2i0 hellip54dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 30x Dagger 24

iexcl 3x iexcl 2kT

hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 1dagger Dagger hhellipL x K xdagger2

DaggerhellipL x K ydagger2 DaggerhellipL yK x dagger2 DaggerhellipL yK ydagger2i0Šordf

hellip55dagger

Equations (53)- (55) are also in agreement with theresults of [2]

6 Spectral moments and coe cients in the Taylorseries expansion of the memory functions

As has been shown on many occasions (eg [4- 6 8])the calculation of the CFs C lhellip tdagger for asymmetric topmolecules can be simpli ed considerably by makinguse of the memory function approach [13] Thisapproach allows one to express the absorpt ion andRaman spectra in terms of the memory functionsK l

nhellip tdagger of arbitrary order n [2 13] The coe cients inthe Taylor series expansions of the memory functionsK l

nhellip tdagger are also related to the spectral moments M ln as

the CFs C lhellip tdagger are connected to the correspondingmemory functions K l

nhellip tdagger by equations (see eg [13])

ddt

C lhellip tdagger ˆ iexclhellip t

0K l

1hellip t iexcl t 0daggerChellip t 0daggerdt 0 hellip56dagger

ddt

K lnhellip tdagger ˆ iexcl

hellip t

0K l

nDagger1hellip t iexcl t 0daggerK lnhellip t 0daggerdt 0

hellip57dagger

In the classical limit the CF C lhellip tdagger and the memory func-tions K l

nhellip tdagger which are both even functions of time t canbe expanded in Taylor series as [13]

C lhellip tdagger ˆX1

kˆ0

Chellip2kdaggerl hellip0dagger t2k

hellip2kdaggerˆ

X1

kˆ0

M l2k

hellip itdagger2k

hellip2kdagger hellip58dagger

K lnhellip tdagger ˆ

X1

kˆ0

K lhellip2kdaggern hellip0dagger t2k

hellip2kdagger hellip59dagger

Here we have taken into account equation (2)Substituting equations (58) and (59) in (56) and (57)one can obtain after some algebraic transformations

C lhellip0daggerK lhellip2ndagger1 hellip0dagger ˆ iexclChellip2nDagger2dagger

l hellip0dagger

iexclXn

sˆ1

Chellip2sdaggerl hellip0daggerK lhellip2hellipniexclsdaggerdagger

1 hellip0dagger hellip60dagger

K lihellip0daggerK lhellip2ndagger

iDagger1 hellip0dagger ˆ iexclK lhellip2nDagger2daggeri hellip0dagger

iexclXn

sˆ1

K lhellip2sdaggeri hellip0daggerK lhellip2hellipniexclsdaggerdagger

iDagger1 hellip0dagger

hellip61dagger

It is convenient to present equations (60) and (61) interms of determinants

1914 Y P Kalmykov and S V Titov

K lhellip2ndagger1 hellip0dagger ˆdet

M l2=Clhellip0dagger 1 0 centcent cent 0

M l4=Clhellip0dagger M l

2=Clhellip0dagger 1

0

M l6=Clhellip0dagger M l

4=Clhellip0dagger M l2=Clhellip0dagger

1

M l2hellipnDagger1dagger=Clhellip0dagger M l

2n=Clhellip0dagger cent centcent M l4=Clhellip0dagger M l

2=Clhellip0dagger

7777777777777777777

7777777777777777777

hellip62dagger

K lihellip0daggerK l hellip2ndagger

iDagger1 hellip0dagger ˆ i2n det

K lhellip2daggeri hellip0dagger 1 0 cent cent cent 0

K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger 1

1

K lhellip2hellipnDagger1daggerdaggeri hellip0dagger K lhellip2ndagger

i hellip0dagger cent cent cent K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger

777777777777777

777777777777777

hellip63dagger

where n ˆ 0 1 2 and

M l2n ˆClhellip0daggerdet

K l1hellip0dagger 1 0 centcentcent 0

K lhellip2dagger1 hellip0dagger K l

1hellip0dagger 1

1

K lhellip2niexcl2dagger1 hellip0dagger K lhellip2hellipniexcl3daggerdagger

1 hellip0dagger centcent cent K lhellip2dagger1 hellip0dagger K l

1hellip0dagger

777777777777777

777777777777777

hellip64dagger

Thus if the moments M l2n are known it is possible to

calculate all the memory functions and their derivativesat t ˆ 0 Conversely if the derivatives of the memoryfunctions K lhellip2ndagger

i hellip0dagger are known it is possible to calculateall the spectral moments M l

2n

7 DiscussionAs we have already mentioned in the analysis of

absorption and scattering spectra of molecular uidsin terms of the moments of the CFs the key problemis the evaluation of the short term behaviour of thecorrelation functions R l

nmhellip tdagger This problem is relatedin turn to the calculation of the coe cients in aTaylor series expansion of R l

n mhellip tdagger The results obtainedin the present work allow one to calculate these coe -cients and hence the spectral moments in a systematicmanner Equations for M l

n obtained in the present paperpermit one also to calculate from equation (64) the co-e cients in Taylor series expansions of the appropriatememory functions Here for simplicity we have con- ned ourselves to the evaluation of the second (M l

2)fourth (M l

4) and sixth (M l6) moments only However

the method used is quite general and it may be extendedreadily for evaluating higher spectral moments as thehigher time derivatives of R l

n mhellip tdagger at t ˆ 0 can be calcu-lated by a similar way For example according to equa-tion (11) on evaluating the fourth derivative of theWigner D function Dl

n mhellipsup3rsquoAacutedagger at t ˆ 0 one can calcu-late the eighth derivative of R lhellip8dagger

n mhellip0dagger etcThe second moment M 1

2 for the dipole absorption ofasymmetric top molecules has been obtained in many

works as within the framework of classical andquantum theory (eg [10 17]) The derivation of thefourth moment M 1

4 for molecules of C2v symmetry wasgiven in [17] The second and fourth moments for anarbitrary asymmetric top molecule have been calculatedby Gordon [11] on using quantum-mechanical methodswith the subsequent transition to the classical limithellip -h 0dagger Our results (equations (38) and (39)) are incomplete agreement with those of Gordon [11] Inmatrix notations of [11] equations (38) and (39) canbe written as

M 12 ˆ kTpermilm2Tr fBg iexcl m cent B cent mTŠ hellip65dagger

M 14 ˆ hellipkTdagger2 fm2permil2hellipTr fBgdagger2 DaggerTr fB2ghellipTr fBgTr fBiexcl1g iexcl 3dagger

iexcl Tr fB3gTr fBiexcl1gŠ

Daggerm cent B2 cent mThellip7 iexcl Tr fBgTr fBiexcl1gdagger

iexcl 6 m cent B cent mTTr fBg Daggerm cent B3 cent mTTr fBiexcl1gg

Daggerm2hK cent B2 cent KTi0 iexcl hhellipm cent B cent KTdagger2i0 hellip66dagger

where B ˆ Iiexcl1 and the upper index T denotes the trans-position

Equations (42) and (43) for the second and fourthRaman spectral moments M 2

2 and M 24 coincide with

the results of Gordon [12] and can be rewritten in thematrix form as follows

M 22 ˆ 2kTpermil2Tr fBgTr fa2g iexcl 3Tr fa cent B cent agŠ hellip67dagger

M 24 ˆ hellipkTdagger2permil27hellipTr fBgdagger2Tr fa2g iexcl 72Tr fa cent B cent agTr fBg

Dagger18Tr fa cent B cent a cent Bg

Dagger6Tr fa cent B2 cent aghellip1 iexcl Tr fBgTr fBiexcl1gdagger

DaggerTr fa2gTr fB2ghellip5 Dagger4Tr fBgTr fBiexcl1gdagger

iexcl 4Tr fa2gTr fB3gTr fBiexcl1g

Dagger6Tr fa cent B3 cent agTr fBiexcl1gŠ

Dagger4hK cent B2 cent KTTr fa2gi0 iexcl 6hK cent B cent a2 cent B cent KT i0

hellip68dagger

It should be noted that there was a misprint in theequation for the fourth moment M 2

4 in [12] equation(58) To our knowledge the sixth moments forthe dipole absorpt ion (equation (40) and Ramanscattering (equation (44)) spectra of an arbitrary asym-metric top molecule have been calculated for the rsttime

Spectral moments of CFs of asymmetric top molecules 1915

The spectral moments allow one to carry out thequantitative analysis of experimental data on IR absorp-tion and Raman spectra of uids with asymmetrictop molecules As is well known the spectral momentM l

2 does not depend on intermolecular interactions Thespectral moment M l

4 depends on the mean-squaretorque acting on the molecule The spectral momentM l

6 includes also averages of angular derivatives ofthe torque Thus the spectral moments allow one toobtain information both about molecular motions andintermolecular interactions Another important prop-erty of spectral moments is the possibility of evaluatingasymptotic properties of spectra For example ifspectral moments M l

2 M l4 and M l

6 exist the integral inthe right hand side of equation (1) exists for n ˆ 2 4and 6 As a result the spectral function I lhellipdagger mustdecrease faster than iexcl6 Moreover the spectralmoments play a rather important role in the theoreticalcalculation of the absorption and scattering spectraand in the modelling of molecular rotation in uidsAs the second spectral moment M l

2 does not dependon the torques acting on molecules (it is exclusivelydetermined by molecular parameters) this momentcalculated in the context of any particular modelshould not depend on parameters describing inter-actions of molecules In other words the secondmoment can be used as a criterion of the correctnessof the model assumptions As the higher momentsdepend on angular derivatives of the intermolecularpotential energy they can be used for the evaluationof model parameters in systems of interacting moleculesFor example the mean-square torque acting on themolecule can be determined from the fourth spectralmoment M l

4 [18] With the help of the spectral momentsM l

2 and M l4 one can evaluate the mean time of molecular

collisions [19 20] Examples of such evaluations forasymmetric top molecules like H 2O and D 2O weregiven in [9]

We thank Professor J L Dejardin for usefulcomments and suggestions The support of this workby the International Association for the Promotionof Co-operation with Scientists from the NewIndependent States of the Former Soviet Union(Project INTAS 96-1411) is gratefully acknowledged

Appendix

Equat ions for R l hellip2kdaggermm hellip0dagger and R l hellip2kdagger

mmsect4hellip0daggerEquations (21)- (32) and (34) yield

R l hellip2daggermm hellip0dagger ˆ iexcl

kT2

permilhellipx iexcl m2daggerhellipI iexcl1x Dagger I iexcl1

y dagger Dagger 2m2I iexcl1z Š hellipA 1dagger

R l hellip2daggermmsect4hellip0dagger ˆ 0 hellipA 2dagger

R l hellip4daggermmhellip0dagger ˆhellipkTdagger2 fhellipx iexcl 3m2daggerpermil2I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y DaggerI iexcl1

x daggerŠ

Daggerm2I iexcl2z hellipI yI

iexcl1x DaggerI x I iexcl1

y iexcl 2dagger

Dagger12hellipx Dagger6m2x iexcl6m4 iexcl5m2daggerI iexcl1

z hellipI iexcl1x DaggerI iexcl1

y dagger

Dagger12hellipx iexcl m2daggerpermilI iexcl1

z hellipI y I iexcl2x DaggerI x I iexcl2

y dagger

DaggerI zIiexcl1x I iexcl1

y hellipI iexcl1x DaggerI iexcl1

y dagger iexcl 2hellipI iexcl2x DaggerI iexcl2

y daggerŠ

Dagger18 permil3hellipx iexclm2dagger2 iexcl2x Dagger5m2Š

pound permil3I iexcl2x Dagger3I iexcl2

y Dagger2I iexcl1x I iexcl1

y Š Dagger3m4I iexcl2z g

Dagger12hellipx iexcl m2daggerhellip I iexcl2

x hK 2x i0 DaggerI iexcl2

y hK 2yi0dagger

Daggerm2I iexcl2z hK 2

z i0 hellipA 3dagger

R lhellip4daggermsect4 mhellip0dagger ˆ 3hellipkTdagger2

16hellipI iexcl1

x iexcl I iexcl1y dagger2

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 4dagger

R lhellip6daggermmhellip0dagger ˆ iexcl kT

16I3x I3

yI3zhellipR lm

0 DaggerkTR lm1 DaggerhellipkTdagger2R lm

2 dagger hellipA 5dagger

R lhellip6daggermsect4 m ˆ iexcl

3kThellipI x iexcl I ydagger32I3

x I 3yI z

f6I zhellipI2x hK 2

yi0 iexcl I2yhK 2

x i0dagger

Dagger4kT I zhellipI2yhL x K x i0 iexcl I2

x hL yK yi0dagger

Dagger5hellipkTdagger2hellipI x iexcl I ydaggerpermil3xhellipI x DaggerI ydaggerI z

Dagger3hellip2I x I y iexclhellipI x DaggerI ydaggerI zdagger

pound mhellipm sect 4dagger

Dagger2hellipI 2x Dagger22I x I y DaggerI 2

y iexcl12hellipI x DaggerI ydaggerI z DaggerI2zdaggerŠg

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 6dagger

where x ˆ lhellip l Dagger1dagger

1916 Y P Kalmykov and S V Titov

R lm0 ˆ 8f2m2I 2

x I2yhI x I zhellipL yK zdagger2 DaggerI y I zhellipL x K zdagger2

DaggerI x I yhellipL zK zdagger2i0 Daggerhellipx iexclm2daggerI2z permilI2

x hI x I zhellipL yK ydagger2

DaggerI yI zhellipL x K ydagger2 DaggerI x I yhellipL zK ydagger2i0 DaggerI2yhI x I zhellipL yK xdagger2

DaggerI yI zhellipL x K xdagger2 DaggerI x I yhellipL zK xdagger2i0Š

Dagger16I x I yI z fhK zL x K yi0permilhellipx iexclm2daggerI x I zhellipI y DaggerI z iexcl I x dagger

iexcl3m2I y I z Š iexclhK zL yK x i0permilhellipx iexclm2daggerI yI zhellipI x DaggerI z iexcl I ydagger

iexcl3m2I x I z ŠDaggerm2I x I ypermilhK x L yK zi0hellip2I x iexcl2I y iexcl I zdagger

iexclhK yL x K zi0hellip2I y iexcl2I x iexcl I zdaggerŠ

DaggerhK x L zK yi0permilhellipx iexclm2daggerI x I zhellip2I y DaggerI z iexcl I xdagger iexcl3m2I x I yŠ

iexclhK yL zK x i0permilhellipx iexclm2daggerI y I zhellip2I x DaggerI z iexcl I ydagger iexcl3m2I x I y Šg

Dagger2hK 2x i0I yI z fx I zpermil16I x I yhellipI y iexcl I x dagger

iexcl2I yI zhellipI x Dagger9I ydagger Dagger4I xhellipI x iexcl I zdagger2

Dagger9xI yI zhellipI x Dagger3I ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I x I zhellipI z iexcl I xdagger2 DaggerI yI zhellip32I2x

iexcl7I xhellip8I y iexcl I zdagger Dagger45I yI zdagger

Dagger18hellipx iexclm2daggerI y I zhellip2I x I y iexcl I x I z iexcl3I yI zdagger

iexcl9m2I y I2zhellipI x Dagger3I ydaggerŠg

Dagger2hK 2yi0I x I z fx I zpermil16I x I yhellipI x iexcl I ydagger

iexcl2I x I zhellip9I x DaggerI ydagger Dagger4I yhellipI y iexcl I zdagger2

Dagger9xI x I zhellip3I x DaggerI ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I yI zhellipI z iexcl I ydagger2 DaggerI x I zhellip32I2y iexcl7I yhellip8I x iexcl I zdagger

Dagger45I x I zdagger Dagger18hellipx iexclm2daggerI x I zhellip2I x I y iexcl3I x I z iexcl I y I zdagger

iexcl9m2I x I 2zhellip3I x DaggerI ydaggerŠg

Dagger8hK 2zi0I x I y fhellipx iexclm2daggerI zpermilI xhellipI x iexcl I zdagger2

DaggerI yhellip I y iexcl I zdagger2 DaggerI x I yhellip4I z iexcl I x iexcl I ydaggerŠ

Daggerm2I x I ypermil9hellipx iexclm2daggerI zhellipI x DaggerI ydagger Dagger18m2I x I y iexcl8I2z Šg

R lm1 ˆ 16hL zK zi0I2

x I2ym

2 f2I2z iexcl 2hellipIx iexcl I ydagger2 iexcl 3xI zhellipIx Dagger I ydagger

Dagger3m2hellipIx I z Dagger I yI z iexcl 2I x I ydaggerg

Dagger4hLx K x i0I2yI

2z fhellipx iexcl m2daggerpermil4I2

x iexcl 4hellipI y iexcl Izdagger2

iexcl 6I zhellipIx iexcl Iydagger iexcl 3x IzhellipI x Dagger3IydaggerŠ

Dagger3m2permil3I zhellipIx iexcl I ydagger DaggerxhellipI zhellipI x Dagger3I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl Ix I z iexcl 3I yI zdaggerŠg

Dagger4hLyK yi0I2x I2

z fhellipx iexcl m2daggerpermil4I2y iexcl 4hellipI x iexcl I zdagger2

iexcl 6I zhellipIy iexcl I xdagger iexcl 3x Izhellip3Ix Dagger IydaggerŠ

Dagger3m2permil3I zhellipIy iexcl Ixdagger DaggerxhellipI zhellip3I x Dagger I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl 3I x I z iexcl I yI zdaggerŠg

R lm2 ˆ 15permil2Ix Iy iexcl IzhellipI x DaggerI ydaggerŠpermil8I2

x I2y iexcl 8I x I yI zhellipIx Dagger I ydagger

Dagger I2zhellip5I2

x iexcl 2Ix I y Dagger5I2ydaggerŠm6

Dagger15f16Ix I yhellipI x iexcl Iydagger2permilI2x I2

y iexcl I zhellipIx Dagger I ydaggerŠ

Dagger2I2zhellip3I4

x Dagger43I3x Iy iexcl 84I2

x I2y Dagger43I x I3

y Dagger3I4ydagger

Dagger3xI zpermil8I2x I2

yhellipI x Dagger I ydagger iexcl 4I x I yI zhellip3I2x Dagger2I x Iy Dagger3I2

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip5I2

x iexcl 2I x I y Dagger5I2ydaggerŠ

iexcl I3zhellipI x Dagger Iydaggerhellip47I2

x iexcl 78I x Iy Dagger47I2ydagger

Dagger2I4zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggergm4

Dagger f16I x IyhellipI x iexcl I ydagger2hellip3I2x iexcl 2I x I y Dagger3I2

ydagger

iexcl 8I zhellipIx Dagger IydaggerhellipIx iexcl I ydagger2hellip3I2x Dagger34I x I y Dagger3I2

ydagger

Dagger2I2zhellipI x iexcl Iydagger2hellip107I2

x Dagger602I x I y Dagger107I2ydagger

iexcl 32I3zhellipIx Dagger Iydaggerhellip25I2

x iexcl 41Ix I y Dagger25I2ydagger

Dagger2I4zhellip107I2

x Dagger98I x Iy Dagger107I2ydagger iexcl 24I5

zhellipI x DaggerI ydagger

Dagger30xI zpermil16Ix I yhellipI x Dagger IydaggerhellipIx iexcl Iydagger2

iexcl 4I zhellip3I4x Dagger19I3

x Iy iexcl 36I2x I2

y Dagger19I x I3y Dagger3I4

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip69I2

x iexcl 106I x Iy Dagger69I2ydagger

iexcl 4I3zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggerŠ

Dagger90I2zx

2permil2I x Iyhellip3I2x Dagger2Ix I y Dagger3I2

ydagger

Spectral moments of CFs of asymmetric top molecules 1917

iexcl I zhellipI x Dagger I ydaggerhellip5I 2x iexcl 2I x I y Dagger5I2

ydaggerŠgm2

Daggerx I z f4hellipI x iexcl I ydagger2permil6hellipI x Dagger I ydagger3

iexcl I zhellip31I2x Dagger106I x I y Dagger 31I2

ydaggerŠ

Dagger64I2zhellipI x Dagger I ydaggerhellip5I2

x iexcl 7I x I y Dagger5I2ydagger

iexcl 4I3zhellip31I2

x Dagger26I x I y Dagger31I2ydagger Dagger24I4

zhellipI x Dagger I ydagger

Dagger15xI zpermil2hellipI x iexcl I ydagger2hellip3I x Dagger I ydaggerhellip I x Dagger3I ydagger

iexcl 2I zhellipI x Dagger I ydaggerhellip11I2x iexcl 14I x I y Dagger11I2

ydagger

Dagger2I2zhellip3I 2

x Dagger2I x I y Dagger 3I 2ydagger

Daggerx I zhellipI x Dagger I ydaggerhellip5I2x iexcl 2I x I y Dagger5I2

ydaggerŠg

In this derivation we have assumed that

hK ii0 ˆ 0 hellip i ˆ x y zdaggerand

hK iK ji0 ˆ 0 hL iK ji0 ˆ 0 for i 6ˆ j hellip i j ˆ x y zdagger

References[1] BURHSTEIN A I and TEMKIN S I 1994 Spectroscopy

of M olecular Rotation in Gases and Liquids (CambridgeUniversity Press)

[2] ST PIERRE A G and STEELE W A 1981 M olecPhys 43 123

[3] GORDON R G 1963 J chem Phys 39 2788

[4] LEICKMAN JCL and GUISSANIY 1981 M olec Phys42 1105

[5] AGUADO-GOMEZM and LEICKMAN JCL 1986 PhysRev A 34 4195

[6] AGUADO-GOMEZM and LEICKMAN JCL 1987 PhysRev A 35 286

[7] BORISEVICH N A BLOKHINA PZALESSKAYAG ALASTOCHKINA V A and SHUKUROV T 1984 IzvAkad Nauk SSSR Ser Fiz 48 709

[8] KALMYKOV YUP 1985 Opt Spektrosk 58 804 [1985Opt Spectrosc 58 493]

[9] KALMYKOV YU P and TITOV S V 1999 J molecS truct 479 123

[10] GORDON R G 1963 J chem Phys 38 1724[11] GORDON R G 1964 J chem Phys 41 1819[12] GORDON R G 1964 J chem Phys 40 1973[13] BERNE B J and HARP G D 1970 Adv chem Phys

17 63[14] LEICKMAN JCL and GUISSANIY 1984 M olec Phys

53 761[15] VARSHALOVICH D A MOSKALEV A N and

KHERSONSKII V K 1988 Quantum Theory of AngularM omentum (Singapore World Scienti c)

[16] LANDAULD and LIFSHITZEM 1976 M echanics ACourse of theoretical Physics 3rd Edn Vol 1 (OxfordPergamon Press)

[17] BOiumlTTCHERC JF and BORDEWIJK P 1979 Theory ofElectric Polarization Vol 2 (Amsterdam Elsevier)

[18] RODRIGUEZ R and MCHALE J L 1988 J chemPhys 88 2264

[19] BURSHTEINA I and MCCONNELL J R 1989 PhysicaA 157 933

[20] KALMYKOV YU P and MCCONNELL J R 1993Physica A 193 394

1918 Spectral moments of CFs of asymmetric top molecules

D1rmhellip tdagger ˆ D1

rmhelliprsquohellip tdagger sup3hellip tdagger Aacutehellip tdaggerdagger and D2rmhellip tdagger ˆ D2

rmhelliprsquohellip tdagger sup3hellip tdagger Aacutehellip tdaggerdagger are the elements (D functions) of the Wignerrotation matrices Dhellip1daggerhelliprsquohellip tdagger sup3hellip tdagger Aacutehellip tdaggerdagger and Dhellip2daggerhelliprsquohellip tdagger sup3hellip tdagger Aacutehellip tdaggerdagger [15] and the asterisk denotes the complexconjugate (in order to simplify equations (6) and (7)we have used symmetry properties of the D functions[15] and the fact that R l

n mhellip tdagger vanishes if n Daggerm is odd [6])The Euler angles sup3 rsquo and Aacute connecting the molecularframe Oxyz to the laboratory (space- xed) coordinatesystem OXY Z are de ned here as in [16] ie the rela-tions between the unit vectors nx ny nz of the molecularframe axes and the unit vectors nsup3 nrsquo nAacute directed alongthe angular velocities _sup3 _rsquo and _Aacute are given by

nAacute ˆ nz

nsup3 ˆ nx cos Aacute iexcl ny sin Aacute

nrsquo ˆ nx sin Aacute sin sup3 Daggerny cos Aacute sin sup3 Dagger nz cos sup3

The nine elements of the Wigner rotation matrixDhellip1daggerhelliprsquohellip tdagger sup3hellip tdagger Aacutehellip tdaggerdagger can be evaluated from the followingequations [15]

D100hellip tdagger ˆ cos sup3hellip tdagger

D110hellip tdagger ˆ iexcl 1

2p sin sup3hellip tdaggereiexclirsquohelliptdagger

D101hellip tdagger ˆ 1

2p sin sup3hellip tdaggereiexcliAacutehelliptdagger

D11 sect1hellip tdagger ˆ 1

2hellip1 sect cos sup3hellip tdaggerdaggereiexclihelliprsquohelliptdaggersectAacutehelliptdaggerdagger

and

D1curreni jhellip tdagger ˆ hellipiexcl1daggeriexcliiexcljD1

iexcliiexcljhellip tdagger

All the elements D2M Nhellip tdagger of the matrix Dhellip2dagger can be

expressed in terms of D1K Lhellip tdagger (equations relating

D2M Nhellip tdagger and D1

K Lhellip tdagger are available explicitly eg in [6])The values of the CFs C lhellip tdagger at t ˆ 0 are given by

C1hellip0dagger ˆ m2x Dagger m2

y Dagger m2z ˆ m2

C2hellip0dagger ˆ 2hellipa2xy Daggera2

yz Dagger a2xz iexcl ax x ayy iexcl ayyazz iexcl axx azzdagger

According to equations (1) and (2) the spectralmoments M l

2n are determined by the short term behav-iour of the CFs C lhellip tdagger In turn the behaviour of the CFsC lhellip tdagger de ned by equations (6) and (7) is completelydetermined by the CF R l

n mhellip tdagger from equation (10)Thus the evaluation of the spectral moments of asym-metric top molecules requires only the calculation of thetime derivatives of the 2k order at t ˆ 0 of the CFR l

n mhellip tdagger viz

d2k

dt2k R lnmhellip0dagger ˆ hellipiexcl1daggerk

Xl

rˆiexcll

dk

dtk Dlcurrenrnhellip0dagger dk

dtk Dlrmhellip0dagger

+

0

hellip11dagger

For the problem under consideration it is more con-venient to evaluate the equilibrium ensemble averagein equation (11) by using the mixed variables namelythe Euler angles sup3rsquoAacute and the components of theangular velocity vector X ˆ fOx Oy Ozg in the molecu-lar frame instead of the generalized coordinates andimpulses fsup3rsquoAacutepsup3 prsquopAacuteg (here and below we omiteverywhere the argument 0 in the Euler angles andOi) For simplicity we con ne ourselves to the evalua-tion of the second (M l

2dagger fourth (M l4dagger and sixth (M l

6daggermoments only Thus according to equation (11) weneed to evaluate the rst second and third time deriva-tives of the Wigner D functions Dl

n m at t ˆ 0 On takinginto account the Euler relations [16] viz

_sup3 ˆ Ox cos Aacute iexcl Oy sin Aacute hellip12dagger

_rsquo ˆ 1sin sup3

hellipOx sin Aacute DaggerOy cos Aacutedagger hellip13dagger

_Aacute ˆ Oz iexcl ctg sup3hellipOx sin Aacute Dagger Oy cos Aacutedagger hellip14dagger

we can write equations for these derivatives as follows

ddt

Dlnm ˆ _sup3

Dlnm

sup3Dagger _rsquo

Dlnm

rsquoDagger _Aacute

Dlnm

Aacuteˆ

X1

sˆiexcl1

OsLsDlnm hellip15dagger

d2

dt2Dl

nm ˆX1

s1ˆiexcl1

_Os1Ls1

DaggerX1

s1 s2ˆiexcl1

Os2Os1

Ls2L s1

Dl

nm hellip16dagger

d3

dt3Dl

nm ˆX1

s1ˆiexcl1

Os1Ls1

DaggerX1

s1 s2ˆiexcl1

hellip2 _Os1Os2

DaggerOs1_Os2

daggerL s2L s1

DaggerX1

s1 s2 s3ˆiexcl1

Os1Os2

Os3Ls3

L s2Ls1

Dl

nm hellip17dagger

where the new variables Osect1 O0 and the operatorsL sect1 L 0 are de ned as

Osect1 ˆ iexcl 12

p hellipOx uml iOydagger O0 ˆ iOz hellip18dagger

L sect1 ˆ i2

p eumliAacute sect ctg sup3

AacuteDagger i

sup3uml 1

sin sup3

rsquo

micro para

L 0 ˆ iexcli

Aacute hellip19dagger

The operators L sect1 L 0 have the property [15]

Spectral moments of CFs of asymmetric top molecules 1909

L sDln m ˆ iexcl

lhellip l Dagger1dagger

pC lmDaggers

lm1sDln mDaggers

ˆ

iexclmDln m s ˆ 0

sectlhellip l Dagger1dagger iexcl mhellipm sect 1dagger

2

sDl

nmsect1 s ˆ sect1

8gtgtltgtgt

hellip20daggerwhere CL mDaggern

l1 ml2 n are the Clebsch- Gordan coe cients [15]The time derivatives of the angular velocity componentsin equations (19) and (20) can be determined from theEuler equations [16]

I x_Ox ˆ hellipI y iexcl I zdaggerOyOz Dagger K x hellip21dagger

I y_Oy ˆ hellipI z iexcl I xdaggerOx Oz Dagger K y hellip22dagger

I z_Oz ˆ hellipI x iexcl I ydaggerOx Oy DaggerK z hellip23dagger

and their time derivatives

I xOx ˆ hellipI y iexcl I zdaggerhellip _OyOz DaggerOy

_Ozdagger DaggerX1

sˆiexcl1

OsL sK x hellip24dagger

I yOy ˆ hellipI z iexcl I xdaggerhellip _Ox Oz DaggerOx

_Ozdagger DaggerX1

sˆiexcl1

OsL sK y hellip25dagger

I zOz ˆ hellipI x iexcl I ydaggerhellip _Ox Oy DaggerOx

_Oydagger DaggerX1

sˆiexcl1

OsL sK z hellip26dagger

where

I ˆ

I x 0 0

0 I y 0

0 0 I z

0BBB

1CCCA

is the inertia tensor in the principal axis of inertia I x I y I z are the principal components of the molecular inertiatensor I and

K ˆ fK xhellipsup3rsquoAacutedagger K yhellipsup3rsquoAacutedagger K zhellipsup3rsquoAacutedaggerg

is the torque acting on the molecule The components ofK can be expressed [2 12] in terms of the anisotropicpart of the potential energy Vhellipsup3rsquoAacutedagger viz

K x ˆ L x V ˆ iexcl 12

p hellipL 1 DaggerL iexcl1daggerV hellip27dagger

K y ˆ L yV ˆ i2

p hellipL 1 iexcl L iexcl1daggerV hellip28dagger

K z ˆ L zV ˆ iL 0V hellip29dagger

Here L x L y L z are the angular momentum operators inthe molecular coordinate system [14]

On using equations (11) plus (15)- (20) we can obtainequations for the second fourth and sixth derivatives ofR l

nmhellip tdagger at t ˆ 0

R lnmhellip0dagger ˆ iexcllhellip l Dagger1dagger

X1

s 01 s1ˆiexcl1

hOcurrens 01Os1

i0C lnDaggers 01

ln1s 01

pound C lmDaggers1lm1s1

dnDaggers 01 mDaggers1

hellip30dagger

R lhellip4daggernmhellip0dagger ˆ lhellip l Dagger 1dagger

X1

s 01 s1ˆiexcl1

h _Ocurrens 01

_Os1i0C

lnDaggers 01

ln1s 01ClmDaggers1

lm1 s1dnDaggers 0

1 mDaggers1

iexcl l3=2hellip l Dagger1dagger3=2X1

s 01 s2 s1ˆiexcl1

helliph _Ocurrens 01Os2

Os1i0C lnDaggers 0

1ln1 s 0

1

pound C lmDaggers1lm1 s1

C lmDaggers1Daggers2lmDaggers1 1 s2

dnDaggers 01 mDaggers1Daggers2

Dagger h _Os 01Ocurren

s1Ocurren

s2i0

pound C lnDaggers1ln 1s1

C lnDaggers1Daggers2lnDaggers1 1 s2

C lmDaggers 01

lm1s 01dnDaggers1 Daggers2 mDaggers 0

1dagger

Dagger l2hellip l Dagger1dagger2X1

s 02 s 0

1 s2 s1ˆiexcl1

hOcurrens 02Ocurren

s 01Os2

Os1i0

pound C lnDaggers 01

ln 1s 01C lnDaggers 0

1Daggers 02

lnDaggers 01 1 s 0

2C lmDaggers1

lm1s1

pound C lmDaggers1Daggers2lmDaggers1 1 s2

macrnDaggers 01Daggers 0

2 mDaggers1Daggers2 hellip31dagger

R lhellip6daggernmhellip0dagger ˆ iexcllhellipl Dagger1dagger

X1

s 01 s1ˆiexcl1

h Ocurrens 01

Os1i0C

lnDaggers 01

ln1s 01ClmDaggers1

lm1 s1dnDaggers 0

1 mDaggers1

Dagger l3=2hellipl Dagger1dagger3=2X1

s 01 s1 s2ˆiexcl1

pound permilh Ocurrens 01hellip2 _Os1

Os2DaggerOs1

_Os2daggeri0C

lnDaggers 01

ln1 s 01ClmDaggers1

lm1 s1

pound ClmDaggers1Daggers2lmDaggers1 1s2

dnDaggers 01 mDaggers1Daggers2

h Os 01hellip2 _Ocurren

s1Ocurren

s2DaggerOcurren

s1_Ocurren

s2daggeri0

DaggerClnDaggers1ln1 s1

ClnDaggers1Daggers2lnDaggers1 1s2

ClmDaggers 0

1lm1s 0

1dnDaggers1Daggers2 mDaggers 0

1Š

iexcl l2hellip l Dagger1dagger2X1

s 01 s1 s2 s3ˆiexcl1

permilh Ocurrens 01Os1

Os2Os3

i0

pound ClnDaggers 0

1ln1 s 0

1ClmDaggers1

lm1 s1ClmDaggers1Daggers2

lmDaggers1 1 s2ClmDaggers1 Daggers2Daggers3

lmDaggers1 Daggers2 1s3

pound dnDaggers 01 mDaggers1Daggers2Daggers3

Daggerh Os 01Ocurren

s1Ocurren

s2Ocurren

s3i0ClnDaggers1

ln1 s1

pound ClnDaggers1Daggers2lnDaggers1 1s2

ClnDaggers1Daggers2Daggers3lnDaggers1Daggers2 1s3

ClmDaggers 0

1

lm1s 01macrnDaggers1Daggers2Daggers3 mDaggers 0

1Š

iexcl l2hellip l Dagger1dagger2X1

s 01 s 0

2 s1 s2ˆiexcl1

hhellip2 _Ocurrens 01Ocurren

s 02DaggerOcurren

s 01

_Ocurrens 02dagger

1910 Y P Kalmykov and S V Titov

pound hellip2 _Os1Os2

Dagger Os1_Os2

daggeri0C lnDaggers 01

ln1s 01C lnDaggers 0

1Daggers 02

lnDaggers 01 1 s 0

2C lmDaggers1

lm1s1

pound C lmDaggers1 Daggers2lmDaggers1 1s2

dnDaggers 01Daggers 0

2 mDaggers1Daggers2

Dagger l5=2hellip l Dagger1dagger5=2X1

s 01 s 0

2 s1 s2 s3ˆiexcl1

permilhhellip2 _Ocurrens 01Ocurren

s02

DaggerOcurrens 01

_Ocurrens 02daggerOs1

Os2Os3

i0C lnDaggers 01

ln1 s 01

pound C lnDaggers 01Daggers 0

2lnDaggers 0

1 1 s 02ClmDaggers1

lm1s1C lmDaggers1 Daggers2

lmDaggers1 1s2C lmDaggers1 Daggers2 Daggers3

lmDaggers1 Daggers2 1s3

pound dnDaggers 01 Daggers 0

2 mDaggers1 Daggers2 Daggers3

Dagger hhellip2 _Os 01Os 0

2Dagger Os 0

1

_Os 02daggerOcurren

s1Ocurren

s2Ocurren

s3i0C lnDaggers1

ln 1s1C lnDaggers1 Daggers2

lnDaggers1 1 s2

pound C lnDaggers1Daggers2Daggers3lnDaggers1Daggers2 1 s3

ClmDaggers 01

lm1 s 01C lmDaggers 0

1Daggers 02

lmDaggers 01 1s 0

2dnDaggers1 Daggers2 Daggers3 mDaggers 0

1 Daggers 02Š

iexcl l3hellip l Dagger1dagger3X1

s 01 s 0

2 s 03 s1 s2 s3ˆiexcl1

hOcurrens 01Ocurren

s 02Ocurren

s 03Os1

Os2Os3

i0

pound ClnDaggers 0

1

ln1 s 01C

lnDaggers 01 Daggers 0

2

lnDaggers 01 1s 0

2C

lnDaggers 01Daggers 0

2Daggers 03

lnDaggers 01Daggers 0

2 1 s 03C lmDaggers1

lm1s1

pound C lmDaggers1 Daggers2lmDaggers1 1s2

C lmDaggers1 Daggers2 Daggers3lmDaggers1 Daggers2 1s3

dnDaggers 01Daggers 0

2 Daggers 03 mDaggers1Daggers2 Daggers3

hellip32dagger

where macrik is Kronekerrsquo s symbol Here it has been takeninto account that [15]

Xl

rˆiexcll

Dlcurrenr mhellip0daggerDl

rm 0hellip0dagger ˆ macrmm 0 hellip33dagger

Equilibrium averages from combinations of the com-ponents of the angular velocity vector in equations(30)- (32) are easily calculated on noting that

hOiOji0 ˆ kTI i

dij hO2i O2

j i0 ˆ kTI i

hellip dagger2

hellip1 Dagger2dijdagger

hO3i O2

j i0 ˆ 15kTI i

hellip dagger3

dij hellip i j ˆ x y zdagger hellip34dagger

Equations for the time derivatives of the CFs C1hellip tdagger andC2hellip tdagger at t ˆ 0 have a de nite symmetry due to the factthat the values of R lhellip2kdagger

n m hellip0dagger (appearing in those equa-tions) depend on I x I y I z K x K y K z L x L y L z onlyand in the cyclic transformation of indices

fx y zg fy z xg fz x yg

transform into each other as follows

R 1 hellip2kdagger0 0 hellip0dagger R 1 hellip2kdagger

11 hellip0dagger DaggerR 1 hellip2kdagger1 iexcl1 hellip0dagger

R 1 hellip2kdagger11 hellip0dagger iexcl R 1 hellip2kdagger

1 iexcl1 hellip0dagger hellip35dagger

and

R 2 hellip2kdagger22 hellip0dagger iexcl R2 hellip2kdagger

2 iexcl2 hellip0dagger R 2 hellip2kdagger1 1 hellip0dagger iexcl R 2 hellip2kdagger

1iexcl1 hellip0dagger

R 2 hellip2kdagger1 1 hellip0dagger DaggerR 2 hellip2kdagger

1iexcl1 hellip0dagger hellip36dagger

and

R 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger fR 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger

Dagger3R 2 hellip2kdagger00 hellip0dagger iexcl

6

ppermilR 2 hellip2kdagger

20 hellip0dagger

DaggerR 2 hellip2kdagger02 hellip0daggerŠg=4

fR 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger

Dagger3R 2 hellip2kdagger00 hellip0dagger Dagger

6

ppermilR 2 hellip2kdagger

20 hellip0dagger

DaggerR 2 hellip2kdagger02 hellip0daggerŠg=4 hellip37dagger

This circumstance allows us to evaluate in equations(30)- (32) only the time derivatives of R l

m mhellip tdagger and

R lmsect4 mhellip tdagger at t ˆ 0 Having determined R l hellip2kdagger

mm hellip0dagger and

R l hellip2kdaggermsect4 mhelliptdagger we can evaluate R 1 hellip2kdagger

0 0 hellip0dagger R 2 hellip2kdagger2 2 hellip0dagger iexcl

R 2 hellip2kdagger2iexcl2 hellip0dagger and R 2 hellip2kdagger

22 hellip0dagger DaggerR 2 hellip2kdagger2iexcl2 hellip0dagger and then the

time derivatives of all the other functions in equations(35)- (37) by changing the indices in I i K i and L i fromfx y zg to fy z xg and fz x yg respectively

Thus on using explicit equations for the Clebsch-Gordan coe cients [15] and equations (21)- (29) and(34) we can obtain from equations (30)- (32) all thequantities of interest (listed in the appendix)

3 Spectral moments for dipolar absorptionHaving determined R 1hellip2kdagger

00 hellip0dagger R 1 hellip2kdagger1 1 hellip0dagger DaggerR 1 hellip2kdagger

1 iexcl1 hellip0daggerand R 1 hellip2kdagger

1 1 hellip0dagger iexcl R 1 hellip2kdagger1 iexcl1 hellip0dagger for k ˆ 1 2 and 3 from equa-

tions (A 1) (A 3) and (A 5) of the appendix and taking

into account equations (2) and (6) we can obtain thesecond fourth and sixth spectral moments for thedipolar absorpt ion viz

M 12 ˆ kTpermilm2

xhellipI iexcl1y Dagger I iexcl1

z dagger Dagger m2yhellipI iexcl1

x Dagger I iexcl1z dagger

Daggerm2zhellipI iexcl1

x Dagger I iexcl1y daggerŠ hellip38dagger

Spectral moments of CFs of asymmetric top molecules 1911

M 14 ˆ m2

x fhellipkTdagger2permil6I iexcl1y I iexcl1

z iexcl I iexcl1x hellipI iexcl1

y Dagger I iexcl1z dagger

Dagger I iexcl1x hellipI zI

iexcl2y Dagger I yI

iexcl2z dagger

Dagger I x I iexcl1z I iexcl1

y hellipI iexcl1y Dagger I iexcl1

z dagger Dagger I iexcl2y Dagger I iexcl2

z Š

Dagger hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z g

Daggerm2y fhellipkTdagger2permil6I iexcl1

x I iexcl1z iexcl I iexcl1

y hellipI iexcl1x Dagger I iexcl1

z dagger

Dagger I iexcl1y hellipI zI

iexcl2x Dagger I x I iexcl2

z dagger

Dagger I y I iexcl1z I iexcl1

x hellipI iexcl1x Dagger I iexcl1

z dagger Dagger I iexcl2x Dagger I iexcl2

z Š

Dagger hK 2x i0I

iexcl2x Dagger hK 2

z i0I iexcl2z g

Daggerm2z fhellipkTdagger2permil6I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y Dagger I iexcl1

x dagger

Dagger I iexcl1z hellipI yI

iexcl2x Dagger I x I iexcl2

y dagger Dagger I zIiexcl1x I iexcl1

y hellipI iexcl1x Dagger I iexcl1

y dagger

Dagger I iexcl2x Dagger I iexcl2

y Š Dagger hK 2x i0I

iexcl2x Dagger hK 2

yi0I iexcl2y g hellip39dagger

M 16 ˆ kT

I 3x I3

yI3zhellipm2

x M x Dagger m2yM y Daggerm2

zM zdagger hellip40dagger

where

M x ˆ I x I yIzhellipI2xhellipIy DaggerI zdagger DaggerhellipI y DaggerI z iexcl2IxdaggerhellipIy iexcl I zdagger2daggerhK 2

x i0

DaggerI2x I zhellipI3

y iexcl2I2yhellipI x Dagger2Izdagger DaggerIyhellipIx Dagger2I zdagger2 Dagger9I x I2

zdaggerhK 2yi0

DaggerI2x I yhellipI3

z iexcl2I2zhellipIx Dagger2Iydagger DaggerI zhellipI x Dagger2Iydagger2 Dagger9Ix I2

ydaggerhK 2zi0

DaggerI2x I2

zhellipI x IzhhellipL yK ydagger2i0 DaggerIy IzhhellipL x K ydagger2i0

DaggerI x I yhhellipL zK ydagger2i0dagger DaggerI2x I2

yhellipIx IzhhellipL yK zdagger2i0

DaggerI yIzhhellipLx K zdagger2i0 DaggerIx I yhhellipLzK zdagger2i0dagger

Dagger2I2x I yI zpermilhK x L yK zi0IyhellipIx iexclI y DaggerI zdagger

iexclhK xL zK yi0I zhellipI x iexcl I z DaggerI ydagger

DaggerhK yL xK zi0I yhellipI x iexcl Iy Dagger2Izdagger

iexclhK zL xK yi0I zhellipI x iexcl I z Dagger2I ydaggerŠiexcl2hellipkTdaggerI2x

poundpermilhLyK yi0I2zhellipI2

x Dagger3Ix Iy iexcl I2y DaggerI2

z DaggerI x I zdagger

DaggerhLzK zi0I2yhellipI2

x DaggerI x Iy DaggerI2y iexcl I2

z Dagger3I x I zdaggerŠ

DaggerhellipkTdagger2I xpermil3I4xhellipIy DaggerIzdagger DaggerI3

xhellip7I2y Dagger2I yI z Dagger7I2

zdagger

iexclI2xhellipIy DaggerIzdaggerhellip5I2

y iexcl34I yIz Dagger5I2zdagger

DaggerI xhellipIy iexcl Izdagger2hellip7I2y Dagger22I yIz Dagger7I2

zdagger

Dagger3hellipIy iexcl Izdagger2hellipIy DaggerI zdagger3Š hellip41dagger

Equations for M y and M z are obtained by changing allthe indices in equation (41) by cyclic permutation of x y z

4 Moments of Raman spectraUsing the symmetry propert ies of the functions in

equations (36) and (37) we can also obtain from equa-tions (A 1)- (A 6) all the R 2hellip2kdagger

nm hellip0dagger which are needed forthe calculation of the spectral moments M 2

2M24 and M 2

6Thus on taking into account equations (2) (7) and(A 1)- (A 6) we can obtain

M 22 ˆ 2kTpermilhellipa2

xy iexcl axx ayydaggerhellipI iexcl1x Dagger I iexcl1

y Dagger 4I iexcl1z dagger

Daggerhellipa2yz iexcl ayyazzdaggerhellip4I iexcl1

x Dagger I iexcl1y Dagger I iexcl1

z dagger

Daggerhellipa2xz iexcl axx azzdaggerhellipI iexcl1

x Dagger 4I iexcl1y Dagger I iexcl1

z daggerŠ hellip42dagger

M 24 ˆ 2hellipkTdagger2 fhellipa2

xy iexcl axx ayydaggerpermil23I iexcl1z hellipI iexcl1

y Dagger I iexcl1x dagger

Dagger4I iexcl2z hellipI y I iexcl1

x Dagger I x I iexcl1y dagger Dagger I iexcl1

z hellipI yIiexcl2x Dagger I x I iexcl2

y dagger

Dagger I zIiexcl1x I iexcl1

y hellipI iexcl1x Dagger I iexcl1

y dagger DaggerhellipI iexcl1x Dagger I iexcl1

y dagger2

Dagger40I iexcl2z DaggerhellipkTdaggeriexcl2helliphK 2

x i0Iiexcl2x Dagger hK 2

yi0I iexcl2y

Dagger4hK 2z i0I iexcl2

z daggerŠ Daggerhellipa2yz iexcl ayyazzdaggerpermil23I iexcl1

x hellipI iexcl1y Dagger I iexcl1

z dagger

Dagger4I iexcl2x hellipI y I iexcl1

z Dagger I zIiexcl1y dagger Dagger I iexcl1

x hellipI y I iexcl2z Dagger I zI

iexcl2y dagger

Dagger I x I iexcl1y I iexcl1

z hellipI iexcl1y Dagger I iexcl1

z dagger DaggerhellipI iexcl1y Dagger I iexcl1

z dagger2 Dagger40I iexcl2x

DaggerhellipkTdaggeriexcl2hellip4hK 2x i0I iexcl2

x Dagger hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z daggerŠ

Daggerhellipa2xz iexcl axx azzdaggerpermil23I iexcl1

y hellipI iexcl1x Dagger I iexcl1

z dagger

Dagger4I iexcl2y hellipI x I iexcl1

z Dagger I zIiexcl1x dagger Dagger I iexcl1

y hellipI x I iexcl2z Dagger I zI

iexcl2x dagger

Dagger I y I iexcl1x I iexcl1

z hellipI iexcl1x Dagger I iexcl1

z dagger DaggerhellipI iexcl1x Dagger I iexcl1

z dagger2 Dagger40I iexcl2y

DaggerhellipkTdaggeriexcl2helliphK 2x i0I iexcl2

x Dagger 4hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z daggerŠ

iexcl 9axx ayyhellipI iexcl1x iexcl I iexcl1

y dagger2 iexcl 9ayyazzhellipI iexcl1y iexcl I iexcl1

z dagger2

iexcl 9axx azzhellipI iexcl1x iexcl I iexcl1

z dagger2g hellip43dagger

M 26 ˆ 2kT

I3x I3

y I3z

fhellipa2xy iexcl axx ayydaggerM xy Daggerhellipa2

yz iexcl ayyazzdaggerM yz

Daggerhellipa2xz iexcl axx azzdaggerM zx iexcl axx ayyN xy iexcl ayyazzN yz

iexcl axx azzN zxg hellip44dagger

where

1912 Y P Kalmykov and S V Titov

M xy ˆ Ix IzhellipI2x I2

zhhellipLyK ydagger2i0 DaggerI2yI

2zhhellipL yK xdagger2i0

Dagger4I2x I2

yhhellipL yK zdagger2i0dagger DaggerI yIzhellipI2x I2

zhhellipL xK ydagger2i0

DaggerI2yI

2zhhellipL xK xdagger2i0 Dagger4I2

x I2yhhellipL x K zdagger2i0dagger

DaggerI x I yhellipI2x I2

z hhellipL zK ydagger2i0 DaggerI2yI

2zhhellipLzK xdagger2i0

Dagger4I2x I2

yhhellipL zK zdagger2i0dagger DaggerIyI zhellipI3xhellip4I y DaggerI zdagger

iexcl2I2xhellip4I2

y iexcl2I yIz DaggerI2zdagger DaggerI xhellip4I3

y Dagger20I2yIz

Dagger20I yI2z DaggerI3

zdagger Dagger9I2yI

2zdaggerhK 2

xi0 DaggerIx IzhellipI3yhellip4Ix DaggerIzdagger

iexcl2I2yhellip4I2

x iexcl2I x Iz DaggerI2zdagger

DaggerI yhellip4I3x Dagger20I2

x Iz Dagger20Ix I2z DaggerI3

zdagger Dagger9I2x I2

zdaggerhK 2yi0

DaggerI x I yhellipI3zhellipIx DaggerI zdagger iexcl2I2

zhellipI2x Dagger10Ix Iy DaggerI2

ydagger

DaggerI zhellipI x DaggerIydaggerhellipI2x Dagger34Ix Iy DaggerI2

ydagger Dagger144I2x I2

ydaggerhK 2zi0

Dagger2I x I yIzpermil2I x I yhelliphK x LyK zi0hellip2I x iexcl2I y iexcl Izdagger

DaggerhK yLx K zi0hellip2I x iexcl2I y DaggerI zdaggerdagger

DaggerI yIzhelliphK yL zK xi0hellip4Ix DaggerIy iexcl Izdagger

DaggerhK zLyK x i0hellip5I x DaggerIy iexcl I zdaggerdagger

iexclI x I zhelliphK xL zK yi0hellipIx Dagger4Iy iexcl I zdagger

DaggerhK zLx K yi0hellipIx Dagger5Iy iexcl I zdaggerdaggerŠ

iexcl2kTpermilhL yK yi0I2x I2

zhellipI2z Dagger3I yIz iexcl I2

y DaggerI xhellipI x Dagger12I y DaggerIzdaggerdagger

DaggerhLx K xi0I2yI

2zhellipI2

z Dagger3I x I z iexcl I2x DaggerI yhellipIy Dagger12Ix DaggerIzdaggerdagger

Dagger4hLzK zi0I2x I2

yhellipI2x Dagger10Ix Iy DaggerI2

y DaggerIzhellip3Ix Dagger3I y iexcl IzdaggerdaggerŠ

DaggerhellipkTdagger2permil3I5zhellipI x DaggerI ydagger Dagger7I4

zhellipI2x Dagger10I x Iy DaggerI2

ydagger

iexcl5I3zhellipI x DaggerI ydaggerhellipI x iexcl Iydagger2

DaggerI2zhellip7I4

x Dagger68I3x Iy Dagger90I2

x I2y Dagger68Ix I3

y Dagger7I4ydagger

DaggerI zhellipI x DaggerIydagger

poundhellip3I4x Dagger64I3

x Iy Dagger586I2x I2

y Dagger64I x I3y Dagger3I4

ydagger

Dagger4I x I yhellip3I4x Dagger52I3

x Iy Dagger130I2x I2

y Dagger52Ix I3y Dagger3I4

ydaggerŠ hellip45dagger

N xy ˆ 9I2zhellipI x iexcl Iydaggerf3I zhellipI2

xhK 2yi0 iexcl I2

yhK 2x i0dagger

iexcl2kT IzhellipI2xhL yK yi0 iexcl I2

yhLx K xi0dagger

DaggerhellipkTdagger2permil5hellipIx iexcl IydaggerhellipI2x Dagger10Ix Iy DaggerI2

y

Dagger3I zhellipIx DaggerI ydagger DaggerI2zdaggerŠg hellip46dagger

Equations for M yz M zx and N yz N zx in equation (44)can be obtained readily by changing all the indices inequations (45) and (46) from fx y zg to fy z xg andfz x yg respectively

5 Linear spherical and symmetric top moleculesThe results we have obtained contain the particular

cases of classical ensembles of symmetric tops sphericaltops and linear molecules In these cases the theory isconsiderably simpli ed as the calculation of all the spec-tral moments M l

2k needs only that of R l hellip2kdaggermm hellip0dagger [2] Thus

for symmetric top molecules (I x ˆ I y ˆ I 6ˆ I zdagger equa-tions (A 1) (A 3) and (A 5) from the appendix yield

R lmmhellip0daggerˆiexclkT

Ipermilx Daggersup2m2Š hellip47dagger

R l hellip4daggermm hellip0daggerˆ kT

Ihellip dagger2

xhellip3x iexcl1daggerDaggersup2 6xm2 Daggerm2 iexclx1Daggersup2hellip daggerDagger3sup22m4

Dagger 12I2permilhellipx iexclm2daggerhK 2

i0 Dagger2hellip1Daggersup2dagger2m2hK 2zi0Š hellip48dagger

R l hellip6daggermm hellip0daggerˆiexcl kT

Ihellip dagger3 11Daggersup2

copy5xhellip3x2 iexcl3x Dagger1dagger

Dagger3sup2hellipxhellip5x2 iexcl10x Dagger8daggerDagger3m2hellip5x2 iexcl4daggerdaggerDagger15sup22m2hellipxhellip3x iexcl2daggerDaggerm2hellip3x Dagger1daggerdagger

Dagger15sup23m4hellip3x Daggerm2daggerDagger15sup24m6 iexcl3sup2hellipx iexclm2dagger1Daggersup2

Dagger 1

hellipkTdagger2permil12hK 2

i0hellipxhellip9x iexcl5daggeriexclm2hellip9x iexcl7dagger

Daggersup2permilxhellip9x iexcl5daggeriexclm2hellip9m2 iexcl1daggerŠDaggersup22permilx Daggerm2hellip9x iexcl9m2 iexcl7daggerŠdaggerDaggerhK 2

zi0hellip2x Daggerm2hellip9x iexcl11daggerDaggersup2permil2x Daggerm2hellip27x Dagger9m2 iexcl20daggerŠDagger9sup22m2hellip3x iexcl1Dagger3m2daggerDagger9sup23m2hellipx Dagger3m2daggerDagger9sup24m4daggerDagger1

2helliphhellipLx K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipLyK xdagger2

DaggerhellipL yK ydagger2i0 Daggerhellip1Daggersup2daggerhhellipLzK xdagger2

DaggerhellipL zK ydagger2i0daggerhellip1Daggersup2daggerhellipx iexclm2daggerDaggerhelliphhellipL xK zdagger2

DaggerhellipL yK zdagger2i0 Daggerhellip1Daggersup2daggerhhellipL zK zdagger2i0daggerhellip1Daggersup2dagger3m2

DaggerhK yLx K z iexclK x LyK zi0m2hellip1Daggersup2dagger2

DaggerhK yLzK x iexclK x LzK yi0permilm2hellip5Dagger4sup2daggeriexclxhellip2Daggersup2daggerŠpoundhellip1Daggersup2daggerŠ

Dagger 1kT

permilhLxK x DaggerL yK yi0hellipx iexclm2daggerhellip1iexcl3hellip1Daggersup2dagger

poundhellipx Daggersup2m2daggerDagger2sup2daggerDagger2hL zK zi0hellip1Daggersup2dagger2m2

poundhellip1iexcl3hellip1Daggersup2daggerhellipx Daggersup2m2daggerdaggerŠordf

hellip49dagger

Spectral moments of CFs of asymmetric top molecules 1913

where

x ˆ lhellip l Dagger1dagger sup2 ˆ I=I z iexcl 1 K 2 ˆ K 2

x DaggerK 2y

Here we have taken into account that [14]

L x L y iexcl L yL x ˆ L z

Equations (47) and (48) for the second and the fourthmoments coincide with the results of [2] which wereobtained from the consideration of an ensemble of hin-dered symmetric tops Equation (49) for freely hellipV ˆ 0daggerrotating symmetric tops was obtained in [2] However itwas presented there with several misprints

For spherical tops hellipI x ˆ I y ˆ I z ˆ Idagger we simplyput sup2 ˆ 0 in equations (47)- (49) Thus for m ˆ 0 wehave

R l00hellip0dagger ˆ iexcl

kTI

x hellip50dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 1dagger Dagger 12I2

hK 2i0 hellip51dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 15x Dagger 5 iexcl 3x iexcl 1kT

pound hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 5dagger Dagger4hK 2

z i0

Dagger4hK x L zK y iexcl K yL zK x i0

Dagger hhellipL x K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipL yK xdagger2

DaggerhellipL yK ydagger2 DaggerhellipL zK xdagger2 DaggerhellipL zK ydagger2i0Šordf

hellip52dagger

Equations (50)- (52) are in accordance with the results of[2]

For linear molecules hellipI x ˆ I y ˆ I I z ˆ 0dagger one has

R l00hellip0dagger ˆ iexcl kT

Ix hellip53dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 2dagger Dagger 12I2

hK 2i0 hellip54dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 30x Dagger 24

iexcl 3x iexcl 2kT

hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 1dagger Dagger hhellipL x K xdagger2

DaggerhellipL x K ydagger2 DaggerhellipL yK x dagger2 DaggerhellipL yK ydagger2i0Šordf

hellip55dagger

Equations (53)- (55) are also in agreement with theresults of [2]

6 Spectral moments and coe cients in the Taylorseries expansion of the memory functions

As has been shown on many occasions (eg [4- 6 8])the calculation of the CFs C lhellip tdagger for asymmetric topmolecules can be simpli ed considerably by makinguse of the memory function approach [13] Thisapproach allows one to express the absorpt ion andRaman spectra in terms of the memory functionsK l

nhellip tdagger of arbitrary order n [2 13] The coe cients inthe Taylor series expansions of the memory functionsK l

nhellip tdagger are also related to the spectral moments M ln as

the CFs C lhellip tdagger are connected to the correspondingmemory functions K l

nhellip tdagger by equations (see eg [13])

ddt

C lhellip tdagger ˆ iexclhellip t

0K l

1hellip t iexcl t 0daggerChellip t 0daggerdt 0 hellip56dagger

ddt

K lnhellip tdagger ˆ iexcl

hellip t

0K l

nDagger1hellip t iexcl t 0daggerK lnhellip t 0daggerdt 0

hellip57dagger

In the classical limit the CF C lhellip tdagger and the memory func-tions K l

nhellip tdagger which are both even functions of time t canbe expanded in Taylor series as [13]

C lhellip tdagger ˆX1

kˆ0

Chellip2kdaggerl hellip0dagger t2k

hellip2kdaggerˆ

X1

kˆ0

M l2k

hellip itdagger2k

hellip2kdagger hellip58dagger

K lnhellip tdagger ˆ

X1

kˆ0

K lhellip2kdaggern hellip0dagger t2k

hellip2kdagger hellip59dagger

Here we have taken into account equation (2)Substituting equations (58) and (59) in (56) and (57)one can obtain after some algebraic transformations

C lhellip0daggerK lhellip2ndagger1 hellip0dagger ˆ iexclChellip2nDagger2dagger

l hellip0dagger

iexclXn

sˆ1

Chellip2sdaggerl hellip0daggerK lhellip2hellipniexclsdaggerdagger

1 hellip0dagger hellip60dagger

K lihellip0daggerK lhellip2ndagger

iDagger1 hellip0dagger ˆ iexclK lhellip2nDagger2daggeri hellip0dagger

iexclXn

sˆ1

K lhellip2sdaggeri hellip0daggerK lhellip2hellipniexclsdaggerdagger

iDagger1 hellip0dagger

hellip61dagger

It is convenient to present equations (60) and (61) interms of determinants

1914 Y P Kalmykov and S V Titov

K lhellip2ndagger1 hellip0dagger ˆdet

M l2=Clhellip0dagger 1 0 centcent cent 0

M l4=Clhellip0dagger M l

2=Clhellip0dagger 1

0

M l6=Clhellip0dagger M l

4=Clhellip0dagger M l2=Clhellip0dagger

1

M l2hellipnDagger1dagger=Clhellip0dagger M l

2n=Clhellip0dagger cent centcent M l4=Clhellip0dagger M l

2=Clhellip0dagger

7777777777777777777

7777777777777777777

hellip62dagger

K lihellip0daggerK l hellip2ndagger

iDagger1 hellip0dagger ˆ i2n det

K lhellip2daggeri hellip0dagger 1 0 cent cent cent 0

K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger 1

1

K lhellip2hellipnDagger1daggerdaggeri hellip0dagger K lhellip2ndagger

i hellip0dagger cent cent cent K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger

777777777777777

777777777777777

hellip63dagger

where n ˆ 0 1 2 and

M l2n ˆClhellip0daggerdet

K l1hellip0dagger 1 0 centcentcent 0

K lhellip2dagger1 hellip0dagger K l

1hellip0dagger 1

1

K lhellip2niexcl2dagger1 hellip0dagger K lhellip2hellipniexcl3daggerdagger

1 hellip0dagger centcent cent K lhellip2dagger1 hellip0dagger K l

1hellip0dagger

777777777777777

777777777777777

hellip64dagger

Thus if the moments M l2n are known it is possible to

calculate all the memory functions and their derivativesat t ˆ 0 Conversely if the derivatives of the memoryfunctions K lhellip2ndagger

i hellip0dagger are known it is possible to calculateall the spectral moments M l

2n

7 DiscussionAs we have already mentioned in the analysis of

absorption and scattering spectra of molecular uidsin terms of the moments of the CFs the key problemis the evaluation of the short term behaviour of thecorrelation functions R l

nmhellip tdagger This problem is relatedin turn to the calculation of the coe cients in aTaylor series expansion of R l

n mhellip tdagger The results obtainedin the present work allow one to calculate these coe -cients and hence the spectral moments in a systematicmanner Equations for M l

n obtained in the present paperpermit one also to calculate from equation (64) the co-e cients in Taylor series expansions of the appropriatememory functions Here for simplicity we have con- ned ourselves to the evaluation of the second (M l

2)fourth (M l

4) and sixth (M l6) moments only However

the method used is quite general and it may be extendedreadily for evaluating higher spectral moments as thehigher time derivatives of R l

n mhellip tdagger at t ˆ 0 can be calcu-lated by a similar way For example according to equa-tion (11) on evaluating the fourth derivative of theWigner D function Dl

n mhellipsup3rsquoAacutedagger at t ˆ 0 one can calcu-late the eighth derivative of R lhellip8dagger

n mhellip0dagger etcThe second moment M 1

2 for the dipole absorption ofasymmetric top molecules has been obtained in many

works as within the framework of classical andquantum theory (eg [10 17]) The derivation of thefourth moment M 1

4 for molecules of C2v symmetry wasgiven in [17] The second and fourth moments for anarbitrary asymmetric top molecule have been calculatedby Gordon [11] on using quantum-mechanical methodswith the subsequent transition to the classical limithellip -h 0dagger Our results (equations (38) and (39)) are incomplete agreement with those of Gordon [11] Inmatrix notations of [11] equations (38) and (39) canbe written as

M 12 ˆ kTpermilm2Tr fBg iexcl m cent B cent mTŠ hellip65dagger

M 14 ˆ hellipkTdagger2 fm2permil2hellipTr fBgdagger2 DaggerTr fB2ghellipTr fBgTr fBiexcl1g iexcl 3dagger

iexcl Tr fB3gTr fBiexcl1gŠ

Daggerm cent B2 cent mThellip7 iexcl Tr fBgTr fBiexcl1gdagger

iexcl 6 m cent B cent mTTr fBg Daggerm cent B3 cent mTTr fBiexcl1gg

Daggerm2hK cent B2 cent KTi0 iexcl hhellipm cent B cent KTdagger2i0 hellip66dagger

where B ˆ Iiexcl1 and the upper index T denotes the trans-position

Equations (42) and (43) for the second and fourthRaman spectral moments M 2

2 and M 24 coincide with

the results of Gordon [12] and can be rewritten in thematrix form as follows

M 22 ˆ 2kTpermil2Tr fBgTr fa2g iexcl 3Tr fa cent B cent agŠ hellip67dagger

M 24 ˆ hellipkTdagger2permil27hellipTr fBgdagger2Tr fa2g iexcl 72Tr fa cent B cent agTr fBg

Dagger18Tr fa cent B cent a cent Bg

Dagger6Tr fa cent B2 cent aghellip1 iexcl Tr fBgTr fBiexcl1gdagger

DaggerTr fa2gTr fB2ghellip5 Dagger4Tr fBgTr fBiexcl1gdagger

iexcl 4Tr fa2gTr fB3gTr fBiexcl1g

Dagger6Tr fa cent B3 cent agTr fBiexcl1gŠ

Dagger4hK cent B2 cent KTTr fa2gi0 iexcl 6hK cent B cent a2 cent B cent KT i0

hellip68dagger

It should be noted that there was a misprint in theequation for the fourth moment M 2

4 in [12] equation(58) To our knowledge the sixth moments forthe dipole absorpt ion (equation (40) and Ramanscattering (equation (44)) spectra of an arbitrary asym-metric top molecule have been calculated for the rsttime

Spectral moments of CFs of asymmetric top molecules 1915

The spectral moments allow one to carry out thequantitative analysis of experimental data on IR absorp-tion and Raman spectra of uids with asymmetrictop molecules As is well known the spectral momentM l

2 does not depend on intermolecular interactions Thespectral moment M l

4 depends on the mean-squaretorque acting on the molecule The spectral momentM l

6 includes also averages of angular derivatives ofthe torque Thus the spectral moments allow one toobtain information both about molecular motions andintermolecular interactions Another important prop-erty of spectral moments is the possibility of evaluatingasymptotic properties of spectra For example ifspectral moments M l

2 M l4 and M l

6 exist the integral inthe right hand side of equation (1) exists for n ˆ 2 4and 6 As a result the spectral function I lhellipdagger mustdecrease faster than iexcl6 Moreover the spectralmoments play a rather important role in the theoreticalcalculation of the absorption and scattering spectraand in the modelling of molecular rotation in uidsAs the second spectral moment M l

2 does not dependon the torques acting on molecules (it is exclusivelydetermined by molecular parameters) this momentcalculated in the context of any particular modelshould not depend on parameters describing inter-actions of molecules In other words the secondmoment can be used as a criterion of the correctnessof the model assumptions As the higher momentsdepend on angular derivatives of the intermolecularpotential energy they can be used for the evaluationof model parameters in systems of interacting moleculesFor example the mean-square torque acting on themolecule can be determined from the fourth spectralmoment M l

4 [18] With the help of the spectral momentsM l

2 and M l4 one can evaluate the mean time of molecular

collisions [19 20] Examples of such evaluations forasymmetric top molecules like H 2O and D 2O weregiven in [9]

We thank Professor J L Dejardin for usefulcomments and suggestions The support of this workby the International Association for the Promotionof Co-operation with Scientists from the NewIndependent States of the Former Soviet Union(Project INTAS 96-1411) is gratefully acknowledged

Appendix

Equat ions for R l hellip2kdaggermm hellip0dagger and R l hellip2kdagger

mmsect4hellip0daggerEquations (21)- (32) and (34) yield

R l hellip2daggermm hellip0dagger ˆ iexcl

kT2

permilhellipx iexcl m2daggerhellipI iexcl1x Dagger I iexcl1

y dagger Dagger 2m2I iexcl1z Š hellipA 1dagger

R l hellip2daggermmsect4hellip0dagger ˆ 0 hellipA 2dagger

R l hellip4daggermmhellip0dagger ˆhellipkTdagger2 fhellipx iexcl 3m2daggerpermil2I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y DaggerI iexcl1

x daggerŠ

Daggerm2I iexcl2z hellipI yI

iexcl1x DaggerI x I iexcl1

y iexcl 2dagger

Dagger12hellipx Dagger6m2x iexcl6m4 iexcl5m2daggerI iexcl1

z hellipI iexcl1x DaggerI iexcl1

y dagger

Dagger12hellipx iexcl m2daggerpermilI iexcl1

z hellipI y I iexcl2x DaggerI x I iexcl2

y dagger

DaggerI zIiexcl1x I iexcl1

y hellipI iexcl1x DaggerI iexcl1

y dagger iexcl 2hellipI iexcl2x DaggerI iexcl2

y daggerŠ

Dagger18 permil3hellipx iexclm2dagger2 iexcl2x Dagger5m2Š

pound permil3I iexcl2x Dagger3I iexcl2

y Dagger2I iexcl1x I iexcl1

y Š Dagger3m4I iexcl2z g

Dagger12hellipx iexcl m2daggerhellip I iexcl2

x hK 2x i0 DaggerI iexcl2

y hK 2yi0dagger

Daggerm2I iexcl2z hK 2

z i0 hellipA 3dagger

R lhellip4daggermsect4 mhellip0dagger ˆ 3hellipkTdagger2

16hellipI iexcl1

x iexcl I iexcl1y dagger2

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 4dagger

R lhellip6daggermmhellip0dagger ˆ iexcl kT

16I3x I3

yI3zhellipR lm

0 DaggerkTR lm1 DaggerhellipkTdagger2R lm

2 dagger hellipA 5dagger

R lhellip6daggermsect4 m ˆ iexcl

3kThellipI x iexcl I ydagger32I3

x I 3yI z

f6I zhellipI2x hK 2

yi0 iexcl I2yhK 2

x i0dagger

Dagger4kT I zhellipI2yhL x K x i0 iexcl I2

x hL yK yi0dagger

Dagger5hellipkTdagger2hellipI x iexcl I ydaggerpermil3xhellipI x DaggerI ydaggerI z

Dagger3hellip2I x I y iexclhellipI x DaggerI ydaggerI zdagger

pound mhellipm sect 4dagger

Dagger2hellipI 2x Dagger22I x I y DaggerI 2

y iexcl12hellipI x DaggerI ydaggerI z DaggerI2zdaggerŠg

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 6dagger

where x ˆ lhellip l Dagger1dagger

1916 Y P Kalmykov and S V Titov

R lm0 ˆ 8f2m2I 2

x I2yhI x I zhellipL yK zdagger2 DaggerI y I zhellipL x K zdagger2

DaggerI x I yhellipL zK zdagger2i0 Daggerhellipx iexclm2daggerI2z permilI2

x hI x I zhellipL yK ydagger2

DaggerI yI zhellipL x K ydagger2 DaggerI x I yhellipL zK ydagger2i0 DaggerI2yhI x I zhellipL yK xdagger2

DaggerI yI zhellipL x K xdagger2 DaggerI x I yhellipL zK xdagger2i0Š

Dagger16I x I yI z fhK zL x K yi0permilhellipx iexclm2daggerI x I zhellipI y DaggerI z iexcl I x dagger

iexcl3m2I y I z Š iexclhK zL yK x i0permilhellipx iexclm2daggerI yI zhellipI x DaggerI z iexcl I ydagger

iexcl3m2I x I z ŠDaggerm2I x I ypermilhK x L yK zi0hellip2I x iexcl2I y iexcl I zdagger

iexclhK yL x K zi0hellip2I y iexcl2I x iexcl I zdaggerŠ

DaggerhK x L zK yi0permilhellipx iexclm2daggerI x I zhellip2I y DaggerI z iexcl I xdagger iexcl3m2I x I yŠ

iexclhK yL zK x i0permilhellipx iexclm2daggerI y I zhellip2I x DaggerI z iexcl I ydagger iexcl3m2I x I y Šg

Dagger2hK 2x i0I yI z fx I zpermil16I x I yhellipI y iexcl I x dagger

iexcl2I yI zhellipI x Dagger9I ydagger Dagger4I xhellipI x iexcl I zdagger2

Dagger9xI yI zhellipI x Dagger3I ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I x I zhellipI z iexcl I xdagger2 DaggerI yI zhellip32I2x

iexcl7I xhellip8I y iexcl I zdagger Dagger45I yI zdagger

Dagger18hellipx iexclm2daggerI y I zhellip2I x I y iexcl I x I z iexcl3I yI zdagger

iexcl9m2I y I2zhellipI x Dagger3I ydaggerŠg

Dagger2hK 2yi0I x I z fx I zpermil16I x I yhellipI x iexcl I ydagger

iexcl2I x I zhellip9I x DaggerI ydagger Dagger4I yhellipI y iexcl I zdagger2

Dagger9xI x I zhellip3I x DaggerI ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I yI zhellipI z iexcl I ydagger2 DaggerI x I zhellip32I2y iexcl7I yhellip8I x iexcl I zdagger

Dagger45I x I zdagger Dagger18hellipx iexclm2daggerI x I zhellip2I x I y iexcl3I x I z iexcl I y I zdagger

iexcl9m2I x I 2zhellip3I x DaggerI ydaggerŠg

Dagger8hK 2zi0I x I y fhellipx iexclm2daggerI zpermilI xhellipI x iexcl I zdagger2

DaggerI yhellip I y iexcl I zdagger2 DaggerI x I yhellip4I z iexcl I x iexcl I ydaggerŠ

Daggerm2I x I ypermil9hellipx iexclm2daggerI zhellipI x DaggerI ydagger Dagger18m2I x I y iexcl8I2z Šg

R lm1 ˆ 16hL zK zi0I2

x I2ym

2 f2I2z iexcl 2hellipIx iexcl I ydagger2 iexcl 3xI zhellipIx Dagger I ydagger

Dagger3m2hellipIx I z Dagger I yI z iexcl 2I x I ydaggerg

Dagger4hLx K x i0I2yI

2z fhellipx iexcl m2daggerpermil4I2

x iexcl 4hellipI y iexcl Izdagger2

iexcl 6I zhellipIx iexcl Iydagger iexcl 3x IzhellipI x Dagger3IydaggerŠ

Dagger3m2permil3I zhellipIx iexcl I ydagger DaggerxhellipI zhellipI x Dagger3I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl Ix I z iexcl 3I yI zdaggerŠg

Dagger4hLyK yi0I2x I2

z fhellipx iexcl m2daggerpermil4I2y iexcl 4hellipI x iexcl I zdagger2

iexcl 6I zhellipIy iexcl I xdagger iexcl 3x Izhellip3Ix Dagger IydaggerŠ

Dagger3m2permil3I zhellipIy iexcl Ixdagger DaggerxhellipI zhellip3I x Dagger I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl 3I x I z iexcl I yI zdaggerŠg

R lm2 ˆ 15permil2Ix Iy iexcl IzhellipI x DaggerI ydaggerŠpermil8I2

x I2y iexcl 8I x I yI zhellipIx Dagger I ydagger

Dagger I2zhellip5I2

x iexcl 2Ix I y Dagger5I2ydaggerŠm6

Dagger15f16Ix I yhellipI x iexcl Iydagger2permilI2x I2

y iexcl I zhellipIx Dagger I ydaggerŠ

Dagger2I2zhellip3I4

x Dagger43I3x Iy iexcl 84I2

x I2y Dagger43I x I3

y Dagger3I4ydagger

Dagger3xI zpermil8I2x I2

yhellipI x Dagger I ydagger iexcl 4I x I yI zhellip3I2x Dagger2I x Iy Dagger3I2

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip5I2

x iexcl 2I x I y Dagger5I2ydaggerŠ

iexcl I3zhellipI x Dagger Iydaggerhellip47I2

x iexcl 78I x Iy Dagger47I2ydagger

Dagger2I4zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggergm4

Dagger f16I x IyhellipI x iexcl I ydagger2hellip3I2x iexcl 2I x I y Dagger3I2

ydagger

iexcl 8I zhellipIx Dagger IydaggerhellipIx iexcl I ydagger2hellip3I2x Dagger34I x I y Dagger3I2

ydagger

Dagger2I2zhellipI x iexcl Iydagger2hellip107I2

x Dagger602I x I y Dagger107I2ydagger

iexcl 32I3zhellipIx Dagger Iydaggerhellip25I2

x iexcl 41Ix I y Dagger25I2ydagger

Dagger2I4zhellip107I2

x Dagger98I x Iy Dagger107I2ydagger iexcl 24I5

zhellipI x DaggerI ydagger

Dagger30xI zpermil16Ix I yhellipI x Dagger IydaggerhellipIx iexcl Iydagger2

iexcl 4I zhellip3I4x Dagger19I3

x Iy iexcl 36I2x I2

y Dagger19I x I3y Dagger3I4

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip69I2

x iexcl 106I x Iy Dagger69I2ydagger

iexcl 4I3zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggerŠ

Dagger90I2zx

2permil2I x Iyhellip3I2x Dagger2Ix I y Dagger3I2

ydagger

Spectral moments of CFs of asymmetric top molecules 1917

iexcl I zhellipI x Dagger I ydaggerhellip5I 2x iexcl 2I x I y Dagger5I2

ydaggerŠgm2

Daggerx I z f4hellipI x iexcl I ydagger2permil6hellipI x Dagger I ydagger3

iexcl I zhellip31I2x Dagger106I x I y Dagger 31I2

ydaggerŠ

Dagger64I2zhellipI x Dagger I ydaggerhellip5I2

x iexcl 7I x I y Dagger5I2ydagger

iexcl 4I3zhellip31I2

x Dagger26I x I y Dagger31I2ydagger Dagger24I4

zhellipI x Dagger I ydagger

Dagger15xI zpermil2hellipI x iexcl I ydagger2hellip3I x Dagger I ydaggerhellip I x Dagger3I ydagger

iexcl 2I zhellipI x Dagger I ydaggerhellip11I2x iexcl 14I x I y Dagger11I2

ydagger

Dagger2I2zhellip3I 2

x Dagger2I x I y Dagger 3I 2ydagger

Daggerx I zhellipI x Dagger I ydaggerhellip5I2x iexcl 2I x I y Dagger5I2

ydaggerŠg

In this derivation we have assumed that

hK ii0 ˆ 0 hellip i ˆ x y zdaggerand

hK iK ji0 ˆ 0 hL iK ji0 ˆ 0 for i 6ˆ j hellip i j ˆ x y zdagger

References[1] BURHSTEIN A I and TEMKIN S I 1994 Spectroscopy

of M olecular Rotation in Gases and Liquids (CambridgeUniversity Press)

[2] ST PIERRE A G and STEELE W A 1981 M olecPhys 43 123

[3] GORDON R G 1963 J chem Phys 39 2788

[4] LEICKMAN JCL and GUISSANIY 1981 M olec Phys42 1105

[5] AGUADO-GOMEZM and LEICKMAN JCL 1986 PhysRev A 34 4195

[6] AGUADO-GOMEZM and LEICKMAN JCL 1987 PhysRev A 35 286

[7] BORISEVICH N A BLOKHINA PZALESSKAYAG ALASTOCHKINA V A and SHUKUROV T 1984 IzvAkad Nauk SSSR Ser Fiz 48 709

[8] KALMYKOV YUP 1985 Opt Spektrosk 58 804 [1985Opt Spectrosc 58 493]

[9] KALMYKOV YU P and TITOV S V 1999 J molecS truct 479 123

[10] GORDON R G 1963 J chem Phys 38 1724[11] GORDON R G 1964 J chem Phys 41 1819[12] GORDON R G 1964 J chem Phys 40 1973[13] BERNE B J and HARP G D 1970 Adv chem Phys

17 63[14] LEICKMAN JCL and GUISSANIY 1984 M olec Phys

53 761[15] VARSHALOVICH D A MOSKALEV A N and

KHERSONSKII V K 1988 Quantum Theory of AngularM omentum (Singapore World Scienti c)

[16] LANDAULD and LIFSHITZEM 1976 M echanics ACourse of theoretical Physics 3rd Edn Vol 1 (OxfordPergamon Press)

[17] BOiumlTTCHERC JF and BORDEWIJK P 1979 Theory ofElectric Polarization Vol 2 (Amsterdam Elsevier)

[18] RODRIGUEZ R and MCHALE J L 1988 J chemPhys 88 2264

[19] BURSHTEINA I and MCCONNELL J R 1989 PhysicaA 157 933

[20] KALMYKOV YU P and MCCONNELL J R 1993Physica A 193 394

1918 Spectral moments of CFs of asymmetric top molecules

L sDln m ˆ iexcl

lhellip l Dagger1dagger

pC lmDaggers

lm1sDln mDaggers

ˆ

iexclmDln m s ˆ 0

sectlhellip l Dagger1dagger iexcl mhellipm sect 1dagger

2

sDl

nmsect1 s ˆ sect1

8gtgtltgtgt

hellip20daggerwhere CL mDaggern

l1 ml2 n are the Clebsch- Gordan coe cients [15]The time derivatives of the angular velocity componentsin equations (19) and (20) can be determined from theEuler equations [16]

I x_Ox ˆ hellipI y iexcl I zdaggerOyOz Dagger K x hellip21dagger

I y_Oy ˆ hellipI z iexcl I xdaggerOx Oz Dagger K y hellip22dagger

I z_Oz ˆ hellipI x iexcl I ydaggerOx Oy DaggerK z hellip23dagger

and their time derivatives

I xOx ˆ hellipI y iexcl I zdaggerhellip _OyOz DaggerOy

_Ozdagger DaggerX1

sˆiexcl1

OsL sK x hellip24dagger

I yOy ˆ hellipI z iexcl I xdaggerhellip _Ox Oz DaggerOx

_Ozdagger DaggerX1

sˆiexcl1

OsL sK y hellip25dagger

I zOz ˆ hellipI x iexcl I ydaggerhellip _Ox Oy DaggerOx

_Oydagger DaggerX1

sˆiexcl1

OsL sK z hellip26dagger

where

I ˆ

I x 0 0

0 I y 0

0 0 I z

0BBB

1CCCA

is the inertia tensor in the principal axis of inertia I x I y I z are the principal components of the molecular inertiatensor I and

K ˆ fK xhellipsup3rsquoAacutedagger K yhellipsup3rsquoAacutedagger K zhellipsup3rsquoAacutedaggerg

is the torque acting on the molecule The components ofK can be expressed [2 12] in terms of the anisotropicpart of the potential energy Vhellipsup3rsquoAacutedagger viz

K x ˆ L x V ˆ iexcl 12

p hellipL 1 DaggerL iexcl1daggerV hellip27dagger

K y ˆ L yV ˆ i2

p hellipL 1 iexcl L iexcl1daggerV hellip28dagger

K z ˆ L zV ˆ iL 0V hellip29dagger

Here L x L y L z are the angular momentum operators inthe molecular coordinate system [14]

On using equations (11) plus (15)- (20) we can obtainequations for the second fourth and sixth derivatives ofR l

nmhellip tdagger at t ˆ 0

R lnmhellip0dagger ˆ iexcllhellip l Dagger1dagger

X1

s 01 s1ˆiexcl1

hOcurrens 01Os1

i0C lnDaggers 01

ln1s 01

pound C lmDaggers1lm1s1

dnDaggers 01 mDaggers1

hellip30dagger

R lhellip4daggernmhellip0dagger ˆ lhellip l Dagger 1dagger

X1

s 01 s1ˆiexcl1

h _Ocurrens 01

_Os1i0C

lnDaggers 01

ln1s 01ClmDaggers1

lm1 s1dnDaggers 0

1 mDaggers1

iexcl l3=2hellip l Dagger1dagger3=2X1

s 01 s2 s1ˆiexcl1

helliph _Ocurrens 01Os2

Os1i0C lnDaggers 0

1ln1 s 0

1

pound C lmDaggers1lm1 s1

C lmDaggers1Daggers2lmDaggers1 1 s2

dnDaggers 01 mDaggers1Daggers2

Dagger h _Os 01Ocurren

s1Ocurren

s2i0

pound C lnDaggers1ln 1s1

C lnDaggers1Daggers2lnDaggers1 1 s2

C lmDaggers 01

lm1s 01dnDaggers1 Daggers2 mDaggers 0

1dagger

Dagger l2hellip l Dagger1dagger2X1

s 02 s 0

1 s2 s1ˆiexcl1

hOcurrens 02Ocurren

s 01Os2

Os1i0

pound C lnDaggers 01

ln 1s 01C lnDaggers 0

1Daggers 02

lnDaggers 01 1 s 0

2C lmDaggers1

lm1s1

pound C lmDaggers1Daggers2lmDaggers1 1 s2

macrnDaggers 01Daggers 0

2 mDaggers1Daggers2 hellip31dagger

R lhellip6daggernmhellip0dagger ˆ iexcllhellipl Dagger1dagger

X1

s 01 s1ˆiexcl1

h Ocurrens 01

Os1i0C

lnDaggers 01

ln1s 01ClmDaggers1

lm1 s1dnDaggers 0

1 mDaggers1

Dagger l3=2hellipl Dagger1dagger3=2X1

s 01 s1 s2ˆiexcl1

pound permilh Ocurrens 01hellip2 _Os1

Os2DaggerOs1

_Os2daggeri0C

lnDaggers 01

ln1 s 01ClmDaggers1

lm1 s1

pound ClmDaggers1Daggers2lmDaggers1 1s2

dnDaggers 01 mDaggers1Daggers2

h Os 01hellip2 _Ocurren

s1Ocurren

s2DaggerOcurren

s1_Ocurren

s2daggeri0

DaggerClnDaggers1ln1 s1

ClnDaggers1Daggers2lnDaggers1 1s2

ClmDaggers 0

1lm1s 0

1dnDaggers1Daggers2 mDaggers 0

1Š

iexcl l2hellip l Dagger1dagger2X1

s 01 s1 s2 s3ˆiexcl1

permilh Ocurrens 01Os1

Os2Os3

i0

pound ClnDaggers 0

1ln1 s 0

1ClmDaggers1

lm1 s1ClmDaggers1Daggers2

lmDaggers1 1 s2ClmDaggers1 Daggers2Daggers3

lmDaggers1 Daggers2 1s3

pound dnDaggers 01 mDaggers1Daggers2Daggers3

Daggerh Os 01Ocurren

s1Ocurren

s2Ocurren

s3i0ClnDaggers1

ln1 s1

pound ClnDaggers1Daggers2lnDaggers1 1s2

ClnDaggers1Daggers2Daggers3lnDaggers1Daggers2 1s3

ClmDaggers 0

1

lm1s 01macrnDaggers1Daggers2Daggers3 mDaggers 0

1Š

iexcl l2hellip l Dagger1dagger2X1

s 01 s 0

2 s1 s2ˆiexcl1

hhellip2 _Ocurrens 01Ocurren

s 02DaggerOcurren

s 01

_Ocurrens 02dagger

1910 Y P Kalmykov and S V Titov

pound hellip2 _Os1Os2

Dagger Os1_Os2

daggeri0C lnDaggers 01

ln1s 01C lnDaggers 0

1Daggers 02

lnDaggers 01 1 s 0

2C lmDaggers1

lm1s1

pound C lmDaggers1 Daggers2lmDaggers1 1s2

dnDaggers 01Daggers 0

2 mDaggers1Daggers2

Dagger l5=2hellip l Dagger1dagger5=2X1

s 01 s 0

2 s1 s2 s3ˆiexcl1

permilhhellip2 _Ocurrens 01Ocurren

s02

DaggerOcurrens 01

_Ocurrens 02daggerOs1

Os2Os3

i0C lnDaggers 01

ln1 s 01

pound C lnDaggers 01Daggers 0

2lnDaggers 0

1 1 s 02ClmDaggers1

lm1s1C lmDaggers1 Daggers2

lmDaggers1 1s2C lmDaggers1 Daggers2 Daggers3

lmDaggers1 Daggers2 1s3

pound dnDaggers 01 Daggers 0

2 mDaggers1 Daggers2 Daggers3

Dagger hhellip2 _Os 01Os 0

2Dagger Os 0

1

_Os 02daggerOcurren

s1Ocurren

s2Ocurren

s3i0C lnDaggers1

ln 1s1C lnDaggers1 Daggers2

lnDaggers1 1 s2

pound C lnDaggers1Daggers2Daggers3lnDaggers1Daggers2 1 s3

ClmDaggers 01

lm1 s 01C lmDaggers 0

1Daggers 02

lmDaggers 01 1s 0

2dnDaggers1 Daggers2 Daggers3 mDaggers 0

1 Daggers 02Š

iexcl l3hellip l Dagger1dagger3X1

s 01 s 0

2 s 03 s1 s2 s3ˆiexcl1

hOcurrens 01Ocurren

s 02Ocurren

s 03Os1

Os2Os3

i0

pound ClnDaggers 0

1

ln1 s 01C

lnDaggers 01 Daggers 0

2

lnDaggers 01 1s 0

2C

lnDaggers 01Daggers 0

2Daggers 03

lnDaggers 01Daggers 0

2 1 s 03C lmDaggers1

lm1s1

pound C lmDaggers1 Daggers2lmDaggers1 1s2

C lmDaggers1 Daggers2 Daggers3lmDaggers1 Daggers2 1s3

dnDaggers 01Daggers 0

2 Daggers 03 mDaggers1Daggers2 Daggers3

hellip32dagger

where macrik is Kronekerrsquo s symbol Here it has been takeninto account that [15]

Xl

rˆiexcll

Dlcurrenr mhellip0daggerDl

rm 0hellip0dagger ˆ macrmm 0 hellip33dagger

Equilibrium averages from combinations of the com-ponents of the angular velocity vector in equations(30)- (32) are easily calculated on noting that

hOiOji0 ˆ kTI i

dij hO2i O2

j i0 ˆ kTI i

hellip dagger2

hellip1 Dagger2dijdagger

hO3i O2

j i0 ˆ 15kTI i

hellip dagger3

dij hellip i j ˆ x y zdagger hellip34dagger

Equations for the time derivatives of the CFs C1hellip tdagger andC2hellip tdagger at t ˆ 0 have a de nite symmetry due to the factthat the values of R lhellip2kdagger

n m hellip0dagger (appearing in those equa-tions) depend on I x I y I z K x K y K z L x L y L z onlyand in the cyclic transformation of indices

fx y zg fy z xg fz x yg

transform into each other as follows

R 1 hellip2kdagger0 0 hellip0dagger R 1 hellip2kdagger

11 hellip0dagger DaggerR 1 hellip2kdagger1 iexcl1 hellip0dagger

R 1 hellip2kdagger11 hellip0dagger iexcl R 1 hellip2kdagger

1 iexcl1 hellip0dagger hellip35dagger

and

R 2 hellip2kdagger22 hellip0dagger iexcl R2 hellip2kdagger

2 iexcl2 hellip0dagger R 2 hellip2kdagger1 1 hellip0dagger iexcl R 2 hellip2kdagger

1iexcl1 hellip0dagger

R 2 hellip2kdagger1 1 hellip0dagger DaggerR 2 hellip2kdagger

1iexcl1 hellip0dagger hellip36dagger

and

R 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger fR 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger

Dagger3R 2 hellip2kdagger00 hellip0dagger iexcl

6

ppermilR 2 hellip2kdagger

20 hellip0dagger

DaggerR 2 hellip2kdagger02 hellip0daggerŠg=4

fR 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger

Dagger3R 2 hellip2kdagger00 hellip0dagger Dagger

6

ppermilR 2 hellip2kdagger

20 hellip0dagger

DaggerR 2 hellip2kdagger02 hellip0daggerŠg=4 hellip37dagger

This circumstance allows us to evaluate in equations(30)- (32) only the time derivatives of R l

m mhellip tdagger and

R lmsect4 mhellip tdagger at t ˆ 0 Having determined R l hellip2kdagger

mm hellip0dagger and

R l hellip2kdaggermsect4 mhelliptdagger we can evaluate R 1 hellip2kdagger

0 0 hellip0dagger R 2 hellip2kdagger2 2 hellip0dagger iexcl

R 2 hellip2kdagger2iexcl2 hellip0dagger and R 2 hellip2kdagger

22 hellip0dagger DaggerR 2 hellip2kdagger2iexcl2 hellip0dagger and then the

time derivatives of all the other functions in equations(35)- (37) by changing the indices in I i K i and L i fromfx y zg to fy z xg and fz x yg respectively

Thus on using explicit equations for the Clebsch-Gordan coe cients [15] and equations (21)- (29) and(34) we can obtain from equations (30)- (32) all thequantities of interest (listed in the appendix)

3 Spectral moments for dipolar absorptionHaving determined R 1hellip2kdagger

00 hellip0dagger R 1 hellip2kdagger1 1 hellip0dagger DaggerR 1 hellip2kdagger

1 iexcl1 hellip0daggerand R 1 hellip2kdagger

1 1 hellip0dagger iexcl R 1 hellip2kdagger1 iexcl1 hellip0dagger for k ˆ 1 2 and 3 from equa-

tions (A 1) (A 3) and (A 5) of the appendix and taking

into account equations (2) and (6) we can obtain thesecond fourth and sixth spectral moments for thedipolar absorpt ion viz

M 12 ˆ kTpermilm2

xhellipI iexcl1y Dagger I iexcl1

z dagger Dagger m2yhellipI iexcl1

x Dagger I iexcl1z dagger

Daggerm2zhellipI iexcl1

x Dagger I iexcl1y daggerŠ hellip38dagger

Spectral moments of CFs of asymmetric top molecules 1911

M 14 ˆ m2

x fhellipkTdagger2permil6I iexcl1y I iexcl1

z iexcl I iexcl1x hellipI iexcl1

y Dagger I iexcl1z dagger

Dagger I iexcl1x hellipI zI

iexcl2y Dagger I yI

iexcl2z dagger

Dagger I x I iexcl1z I iexcl1

y hellipI iexcl1y Dagger I iexcl1

z dagger Dagger I iexcl2y Dagger I iexcl2

z Š

Dagger hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z g

Daggerm2y fhellipkTdagger2permil6I iexcl1

x I iexcl1z iexcl I iexcl1

y hellipI iexcl1x Dagger I iexcl1

z dagger

Dagger I iexcl1y hellipI zI

iexcl2x Dagger I x I iexcl2

z dagger

Dagger I y I iexcl1z I iexcl1

x hellipI iexcl1x Dagger I iexcl1

z dagger Dagger I iexcl2x Dagger I iexcl2

z Š

Dagger hK 2x i0I

iexcl2x Dagger hK 2

z i0I iexcl2z g

Daggerm2z fhellipkTdagger2permil6I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y Dagger I iexcl1

x dagger

Dagger I iexcl1z hellipI yI

iexcl2x Dagger I x I iexcl2

y dagger Dagger I zIiexcl1x I iexcl1

y hellipI iexcl1x Dagger I iexcl1

y dagger

Dagger I iexcl2x Dagger I iexcl2

y Š Dagger hK 2x i0I

iexcl2x Dagger hK 2

yi0I iexcl2y g hellip39dagger

M 16 ˆ kT

I 3x I3

yI3zhellipm2

x M x Dagger m2yM y Daggerm2

zM zdagger hellip40dagger

where

M x ˆ I x I yIzhellipI2xhellipIy DaggerI zdagger DaggerhellipI y DaggerI z iexcl2IxdaggerhellipIy iexcl I zdagger2daggerhK 2

x i0

DaggerI2x I zhellipI3

y iexcl2I2yhellipI x Dagger2Izdagger DaggerIyhellipIx Dagger2I zdagger2 Dagger9I x I2

zdaggerhK 2yi0

DaggerI2x I yhellipI3

z iexcl2I2zhellipIx Dagger2Iydagger DaggerI zhellipI x Dagger2Iydagger2 Dagger9Ix I2

ydaggerhK 2zi0

DaggerI2x I2

zhellipI x IzhhellipL yK ydagger2i0 DaggerIy IzhhellipL x K ydagger2i0

DaggerI x I yhhellipL zK ydagger2i0dagger DaggerI2x I2

yhellipIx IzhhellipL yK zdagger2i0

DaggerI yIzhhellipLx K zdagger2i0 DaggerIx I yhhellipLzK zdagger2i0dagger

Dagger2I2x I yI zpermilhK x L yK zi0IyhellipIx iexclI y DaggerI zdagger

iexclhK xL zK yi0I zhellipI x iexcl I z DaggerI ydagger

DaggerhK yL xK zi0I yhellipI x iexcl Iy Dagger2Izdagger

iexclhK zL xK yi0I zhellipI x iexcl I z Dagger2I ydaggerŠiexcl2hellipkTdaggerI2x

poundpermilhLyK yi0I2zhellipI2

x Dagger3Ix Iy iexcl I2y DaggerI2

z DaggerI x I zdagger

DaggerhLzK zi0I2yhellipI2

x DaggerI x Iy DaggerI2y iexcl I2

z Dagger3I x I zdaggerŠ

DaggerhellipkTdagger2I xpermil3I4xhellipIy DaggerIzdagger DaggerI3

xhellip7I2y Dagger2I yI z Dagger7I2

zdagger

iexclI2xhellipIy DaggerIzdaggerhellip5I2

y iexcl34I yIz Dagger5I2zdagger

DaggerI xhellipIy iexcl Izdagger2hellip7I2y Dagger22I yIz Dagger7I2

zdagger

Dagger3hellipIy iexcl Izdagger2hellipIy DaggerI zdagger3Š hellip41dagger

Equations for M y and M z are obtained by changing allthe indices in equation (41) by cyclic permutation of x y z

4 Moments of Raman spectraUsing the symmetry propert ies of the functions in

equations (36) and (37) we can also obtain from equa-tions (A 1)- (A 6) all the R 2hellip2kdagger

nm hellip0dagger which are needed forthe calculation of the spectral moments M 2

2M24 and M 2

6Thus on taking into account equations (2) (7) and(A 1)- (A 6) we can obtain

M 22 ˆ 2kTpermilhellipa2

xy iexcl axx ayydaggerhellipI iexcl1x Dagger I iexcl1

y Dagger 4I iexcl1z dagger

Daggerhellipa2yz iexcl ayyazzdaggerhellip4I iexcl1

x Dagger I iexcl1y Dagger I iexcl1

z dagger

Daggerhellipa2xz iexcl axx azzdaggerhellipI iexcl1

x Dagger 4I iexcl1y Dagger I iexcl1

z daggerŠ hellip42dagger

M 24 ˆ 2hellipkTdagger2 fhellipa2

xy iexcl axx ayydaggerpermil23I iexcl1z hellipI iexcl1

y Dagger I iexcl1x dagger

Dagger4I iexcl2z hellipI y I iexcl1

x Dagger I x I iexcl1y dagger Dagger I iexcl1

z hellipI yIiexcl2x Dagger I x I iexcl2

y dagger

Dagger I zIiexcl1x I iexcl1

y hellipI iexcl1x Dagger I iexcl1

y dagger DaggerhellipI iexcl1x Dagger I iexcl1

y dagger2

Dagger40I iexcl2z DaggerhellipkTdaggeriexcl2helliphK 2

x i0Iiexcl2x Dagger hK 2

yi0I iexcl2y

Dagger4hK 2z i0I iexcl2

z daggerŠ Daggerhellipa2yz iexcl ayyazzdaggerpermil23I iexcl1

x hellipI iexcl1y Dagger I iexcl1

z dagger

Dagger4I iexcl2x hellipI y I iexcl1

z Dagger I zIiexcl1y dagger Dagger I iexcl1

x hellipI y I iexcl2z Dagger I zI

iexcl2y dagger

Dagger I x I iexcl1y I iexcl1

z hellipI iexcl1y Dagger I iexcl1

z dagger DaggerhellipI iexcl1y Dagger I iexcl1

z dagger2 Dagger40I iexcl2x

DaggerhellipkTdaggeriexcl2hellip4hK 2x i0I iexcl2

x Dagger hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z daggerŠ

Daggerhellipa2xz iexcl axx azzdaggerpermil23I iexcl1

y hellipI iexcl1x Dagger I iexcl1

z dagger

Dagger4I iexcl2y hellipI x I iexcl1

z Dagger I zIiexcl1x dagger Dagger I iexcl1

y hellipI x I iexcl2z Dagger I zI

iexcl2x dagger

Dagger I y I iexcl1x I iexcl1

z hellipI iexcl1x Dagger I iexcl1

z dagger DaggerhellipI iexcl1x Dagger I iexcl1

z dagger2 Dagger40I iexcl2y

DaggerhellipkTdaggeriexcl2helliphK 2x i0I iexcl2

x Dagger 4hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z daggerŠ

iexcl 9axx ayyhellipI iexcl1x iexcl I iexcl1

y dagger2 iexcl 9ayyazzhellipI iexcl1y iexcl I iexcl1

z dagger2

iexcl 9axx azzhellipI iexcl1x iexcl I iexcl1

z dagger2g hellip43dagger

M 26 ˆ 2kT

I3x I3

y I3z

fhellipa2xy iexcl axx ayydaggerM xy Daggerhellipa2

yz iexcl ayyazzdaggerM yz

Daggerhellipa2xz iexcl axx azzdaggerM zx iexcl axx ayyN xy iexcl ayyazzN yz

iexcl axx azzN zxg hellip44dagger

where

1912 Y P Kalmykov and S V Titov

M xy ˆ Ix IzhellipI2x I2

zhhellipLyK ydagger2i0 DaggerI2yI

2zhhellipL yK xdagger2i0

Dagger4I2x I2

yhhellipL yK zdagger2i0dagger DaggerI yIzhellipI2x I2

zhhellipL xK ydagger2i0

DaggerI2yI

2zhhellipL xK xdagger2i0 Dagger4I2

x I2yhhellipL x K zdagger2i0dagger

DaggerI x I yhellipI2x I2

z hhellipL zK ydagger2i0 DaggerI2yI

2zhhellipLzK xdagger2i0

Dagger4I2x I2

yhhellipL zK zdagger2i0dagger DaggerIyI zhellipI3xhellip4I y DaggerI zdagger

iexcl2I2xhellip4I2

y iexcl2I yIz DaggerI2zdagger DaggerI xhellip4I3

y Dagger20I2yIz

Dagger20I yI2z DaggerI3

zdagger Dagger9I2yI

2zdaggerhK 2

xi0 DaggerIx IzhellipI3yhellip4Ix DaggerIzdagger

iexcl2I2yhellip4I2

x iexcl2I x Iz DaggerI2zdagger

DaggerI yhellip4I3x Dagger20I2

x Iz Dagger20Ix I2z DaggerI3

zdagger Dagger9I2x I2

zdaggerhK 2yi0

DaggerI x I yhellipI3zhellipIx DaggerI zdagger iexcl2I2

zhellipI2x Dagger10Ix Iy DaggerI2

ydagger

DaggerI zhellipI x DaggerIydaggerhellipI2x Dagger34Ix Iy DaggerI2

ydagger Dagger144I2x I2

ydaggerhK 2zi0

Dagger2I x I yIzpermil2I x I yhelliphK x LyK zi0hellip2I x iexcl2I y iexcl Izdagger

DaggerhK yLx K zi0hellip2I x iexcl2I y DaggerI zdaggerdagger

DaggerI yIzhelliphK yL zK xi0hellip4Ix DaggerIy iexcl Izdagger

DaggerhK zLyK x i0hellip5I x DaggerIy iexcl I zdaggerdagger

iexclI x I zhelliphK xL zK yi0hellipIx Dagger4Iy iexcl I zdagger

DaggerhK zLx K yi0hellipIx Dagger5Iy iexcl I zdaggerdaggerŠ

iexcl2kTpermilhL yK yi0I2x I2

zhellipI2z Dagger3I yIz iexcl I2

y DaggerI xhellipI x Dagger12I y DaggerIzdaggerdagger

DaggerhLx K xi0I2yI

2zhellipI2

z Dagger3I x I z iexcl I2x DaggerI yhellipIy Dagger12Ix DaggerIzdaggerdagger

Dagger4hLzK zi0I2x I2

yhellipI2x Dagger10Ix Iy DaggerI2

y DaggerIzhellip3Ix Dagger3I y iexcl IzdaggerdaggerŠ

DaggerhellipkTdagger2permil3I5zhellipI x DaggerI ydagger Dagger7I4

zhellipI2x Dagger10I x Iy DaggerI2

ydagger

iexcl5I3zhellipI x DaggerI ydaggerhellipI x iexcl Iydagger2

DaggerI2zhellip7I4

x Dagger68I3x Iy Dagger90I2

x I2y Dagger68Ix I3

y Dagger7I4ydagger

DaggerI zhellipI x DaggerIydagger

poundhellip3I4x Dagger64I3

x Iy Dagger586I2x I2

y Dagger64I x I3y Dagger3I4

ydagger

Dagger4I x I yhellip3I4x Dagger52I3

x Iy Dagger130I2x I2

y Dagger52Ix I3y Dagger3I4

ydaggerŠ hellip45dagger

N xy ˆ 9I2zhellipI x iexcl Iydaggerf3I zhellipI2

xhK 2yi0 iexcl I2

yhK 2x i0dagger

iexcl2kT IzhellipI2xhL yK yi0 iexcl I2

yhLx K xi0dagger

DaggerhellipkTdagger2permil5hellipIx iexcl IydaggerhellipI2x Dagger10Ix Iy DaggerI2

y

Dagger3I zhellipIx DaggerI ydagger DaggerI2zdaggerŠg hellip46dagger

Equations for M yz M zx and N yz N zx in equation (44)can be obtained readily by changing all the indices inequations (45) and (46) from fx y zg to fy z xg andfz x yg respectively

5 Linear spherical and symmetric top moleculesThe results we have obtained contain the particular

cases of classical ensembles of symmetric tops sphericaltops and linear molecules In these cases the theory isconsiderably simpli ed as the calculation of all the spec-tral moments M l

2k needs only that of R l hellip2kdaggermm hellip0dagger [2] Thus

for symmetric top molecules (I x ˆ I y ˆ I 6ˆ I zdagger equa-tions (A 1) (A 3) and (A 5) from the appendix yield

R lmmhellip0daggerˆiexclkT

Ipermilx Daggersup2m2Š hellip47dagger

R l hellip4daggermm hellip0daggerˆ kT

Ihellip dagger2

xhellip3x iexcl1daggerDaggersup2 6xm2 Daggerm2 iexclx1Daggersup2hellip daggerDagger3sup22m4

Dagger 12I2permilhellipx iexclm2daggerhK 2

i0 Dagger2hellip1Daggersup2dagger2m2hK 2zi0Š hellip48dagger

R l hellip6daggermm hellip0daggerˆiexcl kT

Ihellip dagger3 11Daggersup2

copy5xhellip3x2 iexcl3x Dagger1dagger

Dagger3sup2hellipxhellip5x2 iexcl10x Dagger8daggerDagger3m2hellip5x2 iexcl4daggerdaggerDagger15sup22m2hellipxhellip3x iexcl2daggerDaggerm2hellip3x Dagger1daggerdagger

Dagger15sup23m4hellip3x Daggerm2daggerDagger15sup24m6 iexcl3sup2hellipx iexclm2dagger1Daggersup2

Dagger 1

hellipkTdagger2permil12hK 2

i0hellipxhellip9x iexcl5daggeriexclm2hellip9x iexcl7dagger

Daggersup2permilxhellip9x iexcl5daggeriexclm2hellip9m2 iexcl1daggerŠDaggersup22permilx Daggerm2hellip9x iexcl9m2 iexcl7daggerŠdaggerDaggerhK 2

zi0hellip2x Daggerm2hellip9x iexcl11daggerDaggersup2permil2x Daggerm2hellip27x Dagger9m2 iexcl20daggerŠDagger9sup22m2hellip3x iexcl1Dagger3m2daggerDagger9sup23m2hellipx Dagger3m2daggerDagger9sup24m4daggerDagger1

2helliphhellipLx K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipLyK xdagger2

DaggerhellipL yK ydagger2i0 Daggerhellip1Daggersup2daggerhhellipLzK xdagger2

DaggerhellipL zK ydagger2i0daggerhellip1Daggersup2daggerhellipx iexclm2daggerDaggerhelliphhellipL xK zdagger2

DaggerhellipL yK zdagger2i0 Daggerhellip1Daggersup2daggerhhellipL zK zdagger2i0daggerhellip1Daggersup2dagger3m2

DaggerhK yLx K z iexclK x LyK zi0m2hellip1Daggersup2dagger2

DaggerhK yLzK x iexclK x LzK yi0permilm2hellip5Dagger4sup2daggeriexclxhellip2Daggersup2daggerŠpoundhellip1Daggersup2daggerŠ

Dagger 1kT

permilhLxK x DaggerL yK yi0hellipx iexclm2daggerhellip1iexcl3hellip1Daggersup2dagger

poundhellipx Daggersup2m2daggerDagger2sup2daggerDagger2hL zK zi0hellip1Daggersup2dagger2m2

poundhellip1iexcl3hellip1Daggersup2daggerhellipx Daggersup2m2daggerdaggerŠordf

hellip49dagger

Spectral moments of CFs of asymmetric top molecules 1913

where

x ˆ lhellip l Dagger1dagger sup2 ˆ I=I z iexcl 1 K 2 ˆ K 2

x DaggerK 2y

Here we have taken into account that [14]

L x L y iexcl L yL x ˆ L z

Equations (47) and (48) for the second and the fourthmoments coincide with the results of [2] which wereobtained from the consideration of an ensemble of hin-dered symmetric tops Equation (49) for freely hellipV ˆ 0daggerrotating symmetric tops was obtained in [2] However itwas presented there with several misprints

For spherical tops hellipI x ˆ I y ˆ I z ˆ Idagger we simplyput sup2 ˆ 0 in equations (47)- (49) Thus for m ˆ 0 wehave

R l00hellip0dagger ˆ iexcl

kTI

x hellip50dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 1dagger Dagger 12I2

hK 2i0 hellip51dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 15x Dagger 5 iexcl 3x iexcl 1kT

pound hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 5dagger Dagger4hK 2

z i0

Dagger4hK x L zK y iexcl K yL zK x i0

Dagger hhellipL x K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipL yK xdagger2

DaggerhellipL yK ydagger2 DaggerhellipL zK xdagger2 DaggerhellipL zK ydagger2i0Šordf

hellip52dagger

Equations (50)- (52) are in accordance with the results of[2]

For linear molecules hellipI x ˆ I y ˆ I I z ˆ 0dagger one has

R l00hellip0dagger ˆ iexcl kT

Ix hellip53dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 2dagger Dagger 12I2

hK 2i0 hellip54dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 30x Dagger 24

iexcl 3x iexcl 2kT

hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 1dagger Dagger hhellipL x K xdagger2

DaggerhellipL x K ydagger2 DaggerhellipL yK x dagger2 DaggerhellipL yK ydagger2i0Šordf

hellip55dagger

Equations (53)- (55) are also in agreement with theresults of [2]

6 Spectral moments and coe cients in the Taylorseries expansion of the memory functions

As has been shown on many occasions (eg [4- 6 8])the calculation of the CFs C lhellip tdagger for asymmetric topmolecules can be simpli ed considerably by makinguse of the memory function approach [13] Thisapproach allows one to express the absorpt ion andRaman spectra in terms of the memory functionsK l

nhellip tdagger of arbitrary order n [2 13] The coe cients inthe Taylor series expansions of the memory functionsK l

nhellip tdagger are also related to the spectral moments M ln as

the CFs C lhellip tdagger are connected to the correspondingmemory functions K l

nhellip tdagger by equations (see eg [13])

ddt

C lhellip tdagger ˆ iexclhellip t

0K l

1hellip t iexcl t 0daggerChellip t 0daggerdt 0 hellip56dagger

ddt

K lnhellip tdagger ˆ iexcl

hellip t

0K l

nDagger1hellip t iexcl t 0daggerK lnhellip t 0daggerdt 0

hellip57dagger

In the classical limit the CF C lhellip tdagger and the memory func-tions K l

nhellip tdagger which are both even functions of time t canbe expanded in Taylor series as [13]

C lhellip tdagger ˆX1

kˆ0

Chellip2kdaggerl hellip0dagger t2k

hellip2kdaggerˆ

X1

kˆ0

M l2k

hellip itdagger2k

hellip2kdagger hellip58dagger

K lnhellip tdagger ˆ

X1

kˆ0

K lhellip2kdaggern hellip0dagger t2k

hellip2kdagger hellip59dagger

Here we have taken into account equation (2)Substituting equations (58) and (59) in (56) and (57)one can obtain after some algebraic transformations

C lhellip0daggerK lhellip2ndagger1 hellip0dagger ˆ iexclChellip2nDagger2dagger

l hellip0dagger

iexclXn

sˆ1

Chellip2sdaggerl hellip0daggerK lhellip2hellipniexclsdaggerdagger

1 hellip0dagger hellip60dagger

K lihellip0daggerK lhellip2ndagger

iDagger1 hellip0dagger ˆ iexclK lhellip2nDagger2daggeri hellip0dagger

iexclXn

sˆ1

K lhellip2sdaggeri hellip0daggerK lhellip2hellipniexclsdaggerdagger

iDagger1 hellip0dagger

hellip61dagger

It is convenient to present equations (60) and (61) interms of determinants

1914 Y P Kalmykov and S V Titov

K lhellip2ndagger1 hellip0dagger ˆdet

M l2=Clhellip0dagger 1 0 centcent cent 0

M l4=Clhellip0dagger M l

2=Clhellip0dagger 1

0

M l6=Clhellip0dagger M l

4=Clhellip0dagger M l2=Clhellip0dagger

1

M l2hellipnDagger1dagger=Clhellip0dagger M l

2n=Clhellip0dagger cent centcent M l4=Clhellip0dagger M l

2=Clhellip0dagger

7777777777777777777

7777777777777777777

hellip62dagger

K lihellip0daggerK l hellip2ndagger

iDagger1 hellip0dagger ˆ i2n det

K lhellip2daggeri hellip0dagger 1 0 cent cent cent 0

K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger 1

1

K lhellip2hellipnDagger1daggerdaggeri hellip0dagger K lhellip2ndagger

i hellip0dagger cent cent cent K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger

777777777777777

777777777777777

hellip63dagger

where n ˆ 0 1 2 and

M l2n ˆClhellip0daggerdet

K l1hellip0dagger 1 0 centcentcent 0

K lhellip2dagger1 hellip0dagger K l

1hellip0dagger 1

1

K lhellip2niexcl2dagger1 hellip0dagger K lhellip2hellipniexcl3daggerdagger

1 hellip0dagger centcent cent K lhellip2dagger1 hellip0dagger K l

1hellip0dagger

777777777777777

777777777777777

hellip64dagger

Thus if the moments M l2n are known it is possible to

calculate all the memory functions and their derivativesat t ˆ 0 Conversely if the derivatives of the memoryfunctions K lhellip2ndagger

i hellip0dagger are known it is possible to calculateall the spectral moments M l

2n

7 DiscussionAs we have already mentioned in the analysis of

absorption and scattering spectra of molecular uidsin terms of the moments of the CFs the key problemis the evaluation of the short term behaviour of thecorrelation functions R l

nmhellip tdagger This problem is relatedin turn to the calculation of the coe cients in aTaylor series expansion of R l

n mhellip tdagger The results obtainedin the present work allow one to calculate these coe -cients and hence the spectral moments in a systematicmanner Equations for M l

n obtained in the present paperpermit one also to calculate from equation (64) the co-e cients in Taylor series expansions of the appropriatememory functions Here for simplicity we have con- ned ourselves to the evaluation of the second (M l

2)fourth (M l

4) and sixth (M l6) moments only However

the method used is quite general and it may be extendedreadily for evaluating higher spectral moments as thehigher time derivatives of R l

n mhellip tdagger at t ˆ 0 can be calcu-lated by a similar way For example according to equa-tion (11) on evaluating the fourth derivative of theWigner D function Dl

n mhellipsup3rsquoAacutedagger at t ˆ 0 one can calcu-late the eighth derivative of R lhellip8dagger

n mhellip0dagger etcThe second moment M 1

2 for the dipole absorption ofasymmetric top molecules has been obtained in many

works as within the framework of classical andquantum theory (eg [10 17]) The derivation of thefourth moment M 1

4 for molecules of C2v symmetry wasgiven in [17] The second and fourth moments for anarbitrary asymmetric top molecule have been calculatedby Gordon [11] on using quantum-mechanical methodswith the subsequent transition to the classical limithellip -h 0dagger Our results (equations (38) and (39)) are incomplete agreement with those of Gordon [11] Inmatrix notations of [11] equations (38) and (39) canbe written as

M 12 ˆ kTpermilm2Tr fBg iexcl m cent B cent mTŠ hellip65dagger

M 14 ˆ hellipkTdagger2 fm2permil2hellipTr fBgdagger2 DaggerTr fB2ghellipTr fBgTr fBiexcl1g iexcl 3dagger

iexcl Tr fB3gTr fBiexcl1gŠ

Daggerm cent B2 cent mThellip7 iexcl Tr fBgTr fBiexcl1gdagger

iexcl 6 m cent B cent mTTr fBg Daggerm cent B3 cent mTTr fBiexcl1gg

Daggerm2hK cent B2 cent KTi0 iexcl hhellipm cent B cent KTdagger2i0 hellip66dagger

where B ˆ Iiexcl1 and the upper index T denotes the trans-position

Equations (42) and (43) for the second and fourthRaman spectral moments M 2

2 and M 24 coincide with

the results of Gordon [12] and can be rewritten in thematrix form as follows

M 22 ˆ 2kTpermil2Tr fBgTr fa2g iexcl 3Tr fa cent B cent agŠ hellip67dagger

M 24 ˆ hellipkTdagger2permil27hellipTr fBgdagger2Tr fa2g iexcl 72Tr fa cent B cent agTr fBg

Dagger18Tr fa cent B cent a cent Bg

Dagger6Tr fa cent B2 cent aghellip1 iexcl Tr fBgTr fBiexcl1gdagger

DaggerTr fa2gTr fB2ghellip5 Dagger4Tr fBgTr fBiexcl1gdagger

iexcl 4Tr fa2gTr fB3gTr fBiexcl1g

Dagger6Tr fa cent B3 cent agTr fBiexcl1gŠ

Dagger4hK cent B2 cent KTTr fa2gi0 iexcl 6hK cent B cent a2 cent B cent KT i0

hellip68dagger

It should be noted that there was a misprint in theequation for the fourth moment M 2

4 in [12] equation(58) To our knowledge the sixth moments forthe dipole absorpt ion (equation (40) and Ramanscattering (equation (44)) spectra of an arbitrary asym-metric top molecule have been calculated for the rsttime

Spectral moments of CFs of asymmetric top molecules 1915

The spectral moments allow one to carry out thequantitative analysis of experimental data on IR absorp-tion and Raman spectra of uids with asymmetrictop molecules As is well known the spectral momentM l

2 does not depend on intermolecular interactions Thespectral moment M l

4 depends on the mean-squaretorque acting on the molecule The spectral momentM l

6 includes also averages of angular derivatives ofthe torque Thus the spectral moments allow one toobtain information both about molecular motions andintermolecular interactions Another important prop-erty of spectral moments is the possibility of evaluatingasymptotic properties of spectra For example ifspectral moments M l

2 M l4 and M l

6 exist the integral inthe right hand side of equation (1) exists for n ˆ 2 4and 6 As a result the spectral function I lhellipdagger mustdecrease faster than iexcl6 Moreover the spectralmoments play a rather important role in the theoreticalcalculation of the absorption and scattering spectraand in the modelling of molecular rotation in uidsAs the second spectral moment M l

2 does not dependon the torques acting on molecules (it is exclusivelydetermined by molecular parameters) this momentcalculated in the context of any particular modelshould not depend on parameters describing inter-actions of molecules In other words the secondmoment can be used as a criterion of the correctnessof the model assumptions As the higher momentsdepend on angular derivatives of the intermolecularpotential energy they can be used for the evaluationof model parameters in systems of interacting moleculesFor example the mean-square torque acting on themolecule can be determined from the fourth spectralmoment M l

4 [18] With the help of the spectral momentsM l

2 and M l4 one can evaluate the mean time of molecular

collisions [19 20] Examples of such evaluations forasymmetric top molecules like H 2O and D 2O weregiven in [9]

We thank Professor J L Dejardin for usefulcomments and suggestions The support of this workby the International Association for the Promotionof Co-operation with Scientists from the NewIndependent States of the Former Soviet Union(Project INTAS 96-1411) is gratefully acknowledged

Appendix

Equat ions for R l hellip2kdaggermm hellip0dagger and R l hellip2kdagger

mmsect4hellip0daggerEquations (21)- (32) and (34) yield

R l hellip2daggermm hellip0dagger ˆ iexcl

kT2

permilhellipx iexcl m2daggerhellipI iexcl1x Dagger I iexcl1

y dagger Dagger 2m2I iexcl1z Š hellipA 1dagger

R l hellip2daggermmsect4hellip0dagger ˆ 0 hellipA 2dagger

R l hellip4daggermmhellip0dagger ˆhellipkTdagger2 fhellipx iexcl 3m2daggerpermil2I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y DaggerI iexcl1

x daggerŠ

Daggerm2I iexcl2z hellipI yI

iexcl1x DaggerI x I iexcl1

y iexcl 2dagger

Dagger12hellipx Dagger6m2x iexcl6m4 iexcl5m2daggerI iexcl1

z hellipI iexcl1x DaggerI iexcl1

y dagger

Dagger12hellipx iexcl m2daggerpermilI iexcl1

z hellipI y I iexcl2x DaggerI x I iexcl2

y dagger

DaggerI zIiexcl1x I iexcl1

y hellipI iexcl1x DaggerI iexcl1

y dagger iexcl 2hellipI iexcl2x DaggerI iexcl2

y daggerŠ

Dagger18 permil3hellipx iexclm2dagger2 iexcl2x Dagger5m2Š

pound permil3I iexcl2x Dagger3I iexcl2

y Dagger2I iexcl1x I iexcl1

y Š Dagger3m4I iexcl2z g

Dagger12hellipx iexcl m2daggerhellip I iexcl2

x hK 2x i0 DaggerI iexcl2

y hK 2yi0dagger

Daggerm2I iexcl2z hK 2

z i0 hellipA 3dagger

R lhellip4daggermsect4 mhellip0dagger ˆ 3hellipkTdagger2

16hellipI iexcl1

x iexcl I iexcl1y dagger2

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 4dagger

R lhellip6daggermmhellip0dagger ˆ iexcl kT

16I3x I3

yI3zhellipR lm

0 DaggerkTR lm1 DaggerhellipkTdagger2R lm

2 dagger hellipA 5dagger

R lhellip6daggermsect4 m ˆ iexcl

3kThellipI x iexcl I ydagger32I3

x I 3yI z

f6I zhellipI2x hK 2

yi0 iexcl I2yhK 2

x i0dagger

Dagger4kT I zhellipI2yhL x K x i0 iexcl I2

x hL yK yi0dagger

Dagger5hellipkTdagger2hellipI x iexcl I ydaggerpermil3xhellipI x DaggerI ydaggerI z

Dagger3hellip2I x I y iexclhellipI x DaggerI ydaggerI zdagger

pound mhellipm sect 4dagger

Dagger2hellipI 2x Dagger22I x I y DaggerI 2

y iexcl12hellipI x DaggerI ydaggerI z DaggerI2zdaggerŠg

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 6dagger

where x ˆ lhellip l Dagger1dagger

1916 Y P Kalmykov and S V Titov

R lm0 ˆ 8f2m2I 2

x I2yhI x I zhellipL yK zdagger2 DaggerI y I zhellipL x K zdagger2

DaggerI x I yhellipL zK zdagger2i0 Daggerhellipx iexclm2daggerI2z permilI2

x hI x I zhellipL yK ydagger2

DaggerI yI zhellipL x K ydagger2 DaggerI x I yhellipL zK ydagger2i0 DaggerI2yhI x I zhellipL yK xdagger2

DaggerI yI zhellipL x K xdagger2 DaggerI x I yhellipL zK xdagger2i0Š

Dagger16I x I yI z fhK zL x K yi0permilhellipx iexclm2daggerI x I zhellipI y DaggerI z iexcl I x dagger

iexcl3m2I y I z Š iexclhK zL yK x i0permilhellipx iexclm2daggerI yI zhellipI x DaggerI z iexcl I ydagger

iexcl3m2I x I z ŠDaggerm2I x I ypermilhK x L yK zi0hellip2I x iexcl2I y iexcl I zdagger

iexclhK yL x K zi0hellip2I y iexcl2I x iexcl I zdaggerŠ

DaggerhK x L zK yi0permilhellipx iexclm2daggerI x I zhellip2I y DaggerI z iexcl I xdagger iexcl3m2I x I yŠ

iexclhK yL zK x i0permilhellipx iexclm2daggerI y I zhellip2I x DaggerI z iexcl I ydagger iexcl3m2I x I y Šg

Dagger2hK 2x i0I yI z fx I zpermil16I x I yhellipI y iexcl I x dagger

iexcl2I yI zhellipI x Dagger9I ydagger Dagger4I xhellipI x iexcl I zdagger2

Dagger9xI yI zhellipI x Dagger3I ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I x I zhellipI z iexcl I xdagger2 DaggerI yI zhellip32I2x

iexcl7I xhellip8I y iexcl I zdagger Dagger45I yI zdagger

Dagger18hellipx iexclm2daggerI y I zhellip2I x I y iexcl I x I z iexcl3I yI zdagger

iexcl9m2I y I2zhellipI x Dagger3I ydaggerŠg

Dagger2hK 2yi0I x I z fx I zpermil16I x I yhellipI x iexcl I ydagger

iexcl2I x I zhellip9I x DaggerI ydagger Dagger4I yhellipI y iexcl I zdagger2

Dagger9xI x I zhellip3I x DaggerI ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I yI zhellipI z iexcl I ydagger2 DaggerI x I zhellip32I2y iexcl7I yhellip8I x iexcl I zdagger

Dagger45I x I zdagger Dagger18hellipx iexclm2daggerI x I zhellip2I x I y iexcl3I x I z iexcl I y I zdagger

iexcl9m2I x I 2zhellip3I x DaggerI ydaggerŠg

Dagger8hK 2zi0I x I y fhellipx iexclm2daggerI zpermilI xhellipI x iexcl I zdagger2

DaggerI yhellip I y iexcl I zdagger2 DaggerI x I yhellip4I z iexcl I x iexcl I ydaggerŠ

Daggerm2I x I ypermil9hellipx iexclm2daggerI zhellipI x DaggerI ydagger Dagger18m2I x I y iexcl8I2z Šg

R lm1 ˆ 16hL zK zi0I2

x I2ym

2 f2I2z iexcl 2hellipIx iexcl I ydagger2 iexcl 3xI zhellipIx Dagger I ydagger

Dagger3m2hellipIx I z Dagger I yI z iexcl 2I x I ydaggerg

Dagger4hLx K x i0I2yI

2z fhellipx iexcl m2daggerpermil4I2

x iexcl 4hellipI y iexcl Izdagger2

iexcl 6I zhellipIx iexcl Iydagger iexcl 3x IzhellipI x Dagger3IydaggerŠ

Dagger3m2permil3I zhellipIx iexcl I ydagger DaggerxhellipI zhellipI x Dagger3I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl Ix I z iexcl 3I yI zdaggerŠg

Dagger4hLyK yi0I2x I2

z fhellipx iexcl m2daggerpermil4I2y iexcl 4hellipI x iexcl I zdagger2

iexcl 6I zhellipIy iexcl I xdagger iexcl 3x Izhellip3Ix Dagger IydaggerŠ

Dagger3m2permil3I zhellipIy iexcl Ixdagger DaggerxhellipI zhellip3I x Dagger I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl 3I x I z iexcl I yI zdaggerŠg

R lm2 ˆ 15permil2Ix Iy iexcl IzhellipI x DaggerI ydaggerŠpermil8I2

x I2y iexcl 8I x I yI zhellipIx Dagger I ydagger

Dagger I2zhellip5I2

x iexcl 2Ix I y Dagger5I2ydaggerŠm6

Dagger15f16Ix I yhellipI x iexcl Iydagger2permilI2x I2

y iexcl I zhellipIx Dagger I ydaggerŠ

Dagger2I2zhellip3I4

x Dagger43I3x Iy iexcl 84I2

x I2y Dagger43I x I3

y Dagger3I4ydagger

Dagger3xI zpermil8I2x I2

yhellipI x Dagger I ydagger iexcl 4I x I yI zhellip3I2x Dagger2I x Iy Dagger3I2

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip5I2

x iexcl 2I x I y Dagger5I2ydaggerŠ

iexcl I3zhellipI x Dagger Iydaggerhellip47I2

x iexcl 78I x Iy Dagger47I2ydagger

Dagger2I4zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggergm4

Dagger f16I x IyhellipI x iexcl I ydagger2hellip3I2x iexcl 2I x I y Dagger3I2

ydagger

iexcl 8I zhellipIx Dagger IydaggerhellipIx iexcl I ydagger2hellip3I2x Dagger34I x I y Dagger3I2

ydagger

Dagger2I2zhellipI x iexcl Iydagger2hellip107I2

x Dagger602I x I y Dagger107I2ydagger

iexcl 32I3zhellipIx Dagger Iydaggerhellip25I2

x iexcl 41Ix I y Dagger25I2ydagger

Dagger2I4zhellip107I2

x Dagger98I x Iy Dagger107I2ydagger iexcl 24I5

zhellipI x DaggerI ydagger

Dagger30xI zpermil16Ix I yhellipI x Dagger IydaggerhellipIx iexcl Iydagger2

iexcl 4I zhellip3I4x Dagger19I3

x Iy iexcl 36I2x I2

y Dagger19I x I3y Dagger3I4

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip69I2

x iexcl 106I x Iy Dagger69I2ydagger

iexcl 4I3zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggerŠ

Dagger90I2zx

2permil2I x Iyhellip3I2x Dagger2Ix I y Dagger3I2

ydagger

Spectral moments of CFs of asymmetric top molecules 1917

iexcl I zhellipI x Dagger I ydaggerhellip5I 2x iexcl 2I x I y Dagger5I2

ydaggerŠgm2

Daggerx I z f4hellipI x iexcl I ydagger2permil6hellipI x Dagger I ydagger3

iexcl I zhellip31I2x Dagger106I x I y Dagger 31I2

ydaggerŠ

Dagger64I2zhellipI x Dagger I ydaggerhellip5I2

x iexcl 7I x I y Dagger5I2ydagger

iexcl 4I3zhellip31I2

x Dagger26I x I y Dagger31I2ydagger Dagger24I4

zhellipI x Dagger I ydagger

Dagger15xI zpermil2hellipI x iexcl I ydagger2hellip3I x Dagger I ydaggerhellip I x Dagger3I ydagger

iexcl 2I zhellipI x Dagger I ydaggerhellip11I2x iexcl 14I x I y Dagger11I2

ydagger

Dagger2I2zhellip3I 2

x Dagger2I x I y Dagger 3I 2ydagger

Daggerx I zhellipI x Dagger I ydaggerhellip5I2x iexcl 2I x I y Dagger5I2

ydaggerŠg

In this derivation we have assumed that

hK ii0 ˆ 0 hellip i ˆ x y zdaggerand

hK iK ji0 ˆ 0 hL iK ji0 ˆ 0 for i 6ˆ j hellip i j ˆ x y zdagger

References[1] BURHSTEIN A I and TEMKIN S I 1994 Spectroscopy

of M olecular Rotation in Gases and Liquids (CambridgeUniversity Press)

[2] ST PIERRE A G and STEELE W A 1981 M olecPhys 43 123

[3] GORDON R G 1963 J chem Phys 39 2788

[4] LEICKMAN JCL and GUISSANIY 1981 M olec Phys42 1105

[5] AGUADO-GOMEZM and LEICKMAN JCL 1986 PhysRev A 34 4195

[6] AGUADO-GOMEZM and LEICKMAN JCL 1987 PhysRev A 35 286

[7] BORISEVICH N A BLOKHINA PZALESSKAYAG ALASTOCHKINA V A and SHUKUROV T 1984 IzvAkad Nauk SSSR Ser Fiz 48 709

[8] KALMYKOV YUP 1985 Opt Spektrosk 58 804 [1985Opt Spectrosc 58 493]

[9] KALMYKOV YU P and TITOV S V 1999 J molecS truct 479 123

[10] GORDON R G 1963 J chem Phys 38 1724[11] GORDON R G 1964 J chem Phys 41 1819[12] GORDON R G 1964 J chem Phys 40 1973[13] BERNE B J and HARP G D 1970 Adv chem Phys

17 63[14] LEICKMAN JCL and GUISSANIY 1984 M olec Phys

53 761[15] VARSHALOVICH D A MOSKALEV A N and

KHERSONSKII V K 1988 Quantum Theory of AngularM omentum (Singapore World Scienti c)

[16] LANDAULD and LIFSHITZEM 1976 M echanics ACourse of theoretical Physics 3rd Edn Vol 1 (OxfordPergamon Press)

[17] BOiumlTTCHERC JF and BORDEWIJK P 1979 Theory ofElectric Polarization Vol 2 (Amsterdam Elsevier)

[18] RODRIGUEZ R and MCHALE J L 1988 J chemPhys 88 2264

[19] BURSHTEINA I and MCCONNELL J R 1989 PhysicaA 157 933

[20] KALMYKOV YU P and MCCONNELL J R 1993Physica A 193 394

1918 Spectral moments of CFs of asymmetric top molecules

pound hellip2 _Os1Os2

Dagger Os1_Os2

daggeri0C lnDaggers 01

ln1s 01C lnDaggers 0

1Daggers 02

lnDaggers 01 1 s 0

2C lmDaggers1

lm1s1

pound C lmDaggers1 Daggers2lmDaggers1 1s2

dnDaggers 01Daggers 0

2 mDaggers1Daggers2

Dagger l5=2hellip l Dagger1dagger5=2X1

s 01 s 0

2 s1 s2 s3ˆiexcl1

permilhhellip2 _Ocurrens 01Ocurren

s02

DaggerOcurrens 01

_Ocurrens 02daggerOs1

Os2Os3

i0C lnDaggers 01

ln1 s 01

pound C lnDaggers 01Daggers 0

2lnDaggers 0

1 1 s 02ClmDaggers1

lm1s1C lmDaggers1 Daggers2

lmDaggers1 1s2C lmDaggers1 Daggers2 Daggers3

lmDaggers1 Daggers2 1s3

pound dnDaggers 01 Daggers 0

2 mDaggers1 Daggers2 Daggers3

Dagger hhellip2 _Os 01Os 0

2Dagger Os 0

1

_Os 02daggerOcurren

s1Ocurren

s2Ocurren

s3i0C lnDaggers1

ln 1s1C lnDaggers1 Daggers2

lnDaggers1 1 s2

pound C lnDaggers1Daggers2Daggers3lnDaggers1Daggers2 1 s3

ClmDaggers 01

lm1 s 01C lmDaggers 0

1Daggers 02

lmDaggers 01 1s 0

2dnDaggers1 Daggers2 Daggers3 mDaggers 0

1 Daggers 02Š

iexcl l3hellip l Dagger1dagger3X1

s 01 s 0

2 s 03 s1 s2 s3ˆiexcl1

hOcurrens 01Ocurren

s 02Ocurren

s 03Os1

Os2Os3

i0

pound ClnDaggers 0

1

ln1 s 01C

lnDaggers 01 Daggers 0

2

lnDaggers 01 1s 0

2C

lnDaggers 01Daggers 0

2Daggers 03

lnDaggers 01Daggers 0

2 1 s 03C lmDaggers1

lm1s1

pound C lmDaggers1 Daggers2lmDaggers1 1s2

C lmDaggers1 Daggers2 Daggers3lmDaggers1 Daggers2 1s3

dnDaggers 01Daggers 0

2 Daggers 03 mDaggers1Daggers2 Daggers3

hellip32dagger

where macrik is Kronekerrsquo s symbol Here it has been takeninto account that [15]

Xl

rˆiexcll

Dlcurrenr mhellip0daggerDl

rm 0hellip0dagger ˆ macrmm 0 hellip33dagger

Equilibrium averages from combinations of the com-ponents of the angular velocity vector in equations(30)- (32) are easily calculated on noting that

hOiOji0 ˆ kTI i

dij hO2i O2

j i0 ˆ kTI i

hellip dagger2

hellip1 Dagger2dijdagger

hO3i O2

j i0 ˆ 15kTI i

hellip dagger3

dij hellip i j ˆ x y zdagger hellip34dagger

Equations for the time derivatives of the CFs C1hellip tdagger andC2hellip tdagger at t ˆ 0 have a de nite symmetry due to the factthat the values of R lhellip2kdagger

n m hellip0dagger (appearing in those equa-tions) depend on I x I y I z K x K y K z L x L y L z onlyand in the cyclic transformation of indices

fx y zg fy z xg fz x yg

transform into each other as follows

R 1 hellip2kdagger0 0 hellip0dagger R 1 hellip2kdagger

11 hellip0dagger DaggerR 1 hellip2kdagger1 iexcl1 hellip0dagger

R 1 hellip2kdagger11 hellip0dagger iexcl R 1 hellip2kdagger

1 iexcl1 hellip0dagger hellip35dagger

and

R 2 hellip2kdagger22 hellip0dagger iexcl R2 hellip2kdagger

2 iexcl2 hellip0dagger R 2 hellip2kdagger1 1 hellip0dagger iexcl R 2 hellip2kdagger

1iexcl1 hellip0dagger

R 2 hellip2kdagger1 1 hellip0dagger DaggerR 2 hellip2kdagger

1iexcl1 hellip0dagger hellip36dagger

and

R 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger fR 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger

Dagger3R 2 hellip2kdagger00 hellip0dagger iexcl

6

ppermilR 2 hellip2kdagger

20 hellip0dagger

DaggerR 2 hellip2kdagger02 hellip0daggerŠg=4

fR 2 hellip2kdagger2 2 hellip0dagger DaggerR 2 hellip2kdagger

2 iexcl2 hellip0dagger

Dagger3R 2 hellip2kdagger00 hellip0dagger Dagger

6

ppermilR 2 hellip2kdagger

20 hellip0dagger

DaggerR 2 hellip2kdagger02 hellip0daggerŠg=4 hellip37dagger

This circumstance allows us to evaluate in equations(30)- (32) only the time derivatives of R l

m mhellip tdagger and

R lmsect4 mhellip tdagger at t ˆ 0 Having determined R l hellip2kdagger

mm hellip0dagger and

R l hellip2kdaggermsect4 mhelliptdagger we can evaluate R 1 hellip2kdagger

0 0 hellip0dagger R 2 hellip2kdagger2 2 hellip0dagger iexcl

R 2 hellip2kdagger2iexcl2 hellip0dagger and R 2 hellip2kdagger

22 hellip0dagger DaggerR 2 hellip2kdagger2iexcl2 hellip0dagger and then the

time derivatives of all the other functions in equations(35)- (37) by changing the indices in I i K i and L i fromfx y zg to fy z xg and fz x yg respectively

Thus on using explicit equations for the Clebsch-Gordan coe cients [15] and equations (21)- (29) and(34) we can obtain from equations (30)- (32) all thequantities of interest (listed in the appendix)

3 Spectral moments for dipolar absorptionHaving determined R 1hellip2kdagger

00 hellip0dagger R 1 hellip2kdagger1 1 hellip0dagger DaggerR 1 hellip2kdagger

1 iexcl1 hellip0daggerand R 1 hellip2kdagger

1 1 hellip0dagger iexcl R 1 hellip2kdagger1 iexcl1 hellip0dagger for k ˆ 1 2 and 3 from equa-

tions (A 1) (A 3) and (A 5) of the appendix and taking

into account equations (2) and (6) we can obtain thesecond fourth and sixth spectral moments for thedipolar absorpt ion viz

M 12 ˆ kTpermilm2

xhellipI iexcl1y Dagger I iexcl1

z dagger Dagger m2yhellipI iexcl1

x Dagger I iexcl1z dagger

Daggerm2zhellipI iexcl1

x Dagger I iexcl1y daggerŠ hellip38dagger

Spectral moments of CFs of asymmetric top molecules 1911

M 14 ˆ m2

x fhellipkTdagger2permil6I iexcl1y I iexcl1

z iexcl I iexcl1x hellipI iexcl1

y Dagger I iexcl1z dagger

Dagger I iexcl1x hellipI zI

iexcl2y Dagger I yI

iexcl2z dagger

Dagger I x I iexcl1z I iexcl1

y hellipI iexcl1y Dagger I iexcl1

z dagger Dagger I iexcl2y Dagger I iexcl2

z Š

Dagger hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z g

Daggerm2y fhellipkTdagger2permil6I iexcl1

x I iexcl1z iexcl I iexcl1

y hellipI iexcl1x Dagger I iexcl1

z dagger

Dagger I iexcl1y hellipI zI

iexcl2x Dagger I x I iexcl2

z dagger

Dagger I y I iexcl1z I iexcl1

x hellipI iexcl1x Dagger I iexcl1

z dagger Dagger I iexcl2x Dagger I iexcl2

z Š

Dagger hK 2x i0I

iexcl2x Dagger hK 2

z i0I iexcl2z g

Daggerm2z fhellipkTdagger2permil6I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y Dagger I iexcl1

x dagger

Dagger I iexcl1z hellipI yI

iexcl2x Dagger I x I iexcl2

y dagger Dagger I zIiexcl1x I iexcl1

y hellipI iexcl1x Dagger I iexcl1

y dagger

Dagger I iexcl2x Dagger I iexcl2

y Š Dagger hK 2x i0I

iexcl2x Dagger hK 2

yi0I iexcl2y g hellip39dagger

M 16 ˆ kT

I 3x I3

yI3zhellipm2

x M x Dagger m2yM y Daggerm2

zM zdagger hellip40dagger

where

M x ˆ I x I yIzhellipI2xhellipIy DaggerI zdagger DaggerhellipI y DaggerI z iexcl2IxdaggerhellipIy iexcl I zdagger2daggerhK 2

x i0

DaggerI2x I zhellipI3

y iexcl2I2yhellipI x Dagger2Izdagger DaggerIyhellipIx Dagger2I zdagger2 Dagger9I x I2

zdaggerhK 2yi0

DaggerI2x I yhellipI3

z iexcl2I2zhellipIx Dagger2Iydagger DaggerI zhellipI x Dagger2Iydagger2 Dagger9Ix I2

ydaggerhK 2zi0

DaggerI2x I2

zhellipI x IzhhellipL yK ydagger2i0 DaggerIy IzhhellipL x K ydagger2i0

DaggerI x I yhhellipL zK ydagger2i0dagger DaggerI2x I2

yhellipIx IzhhellipL yK zdagger2i0

DaggerI yIzhhellipLx K zdagger2i0 DaggerIx I yhhellipLzK zdagger2i0dagger

Dagger2I2x I yI zpermilhK x L yK zi0IyhellipIx iexclI y DaggerI zdagger

iexclhK xL zK yi0I zhellipI x iexcl I z DaggerI ydagger

DaggerhK yL xK zi0I yhellipI x iexcl Iy Dagger2Izdagger

iexclhK zL xK yi0I zhellipI x iexcl I z Dagger2I ydaggerŠiexcl2hellipkTdaggerI2x

poundpermilhLyK yi0I2zhellipI2

x Dagger3Ix Iy iexcl I2y DaggerI2

z DaggerI x I zdagger

DaggerhLzK zi0I2yhellipI2

x DaggerI x Iy DaggerI2y iexcl I2

z Dagger3I x I zdaggerŠ

DaggerhellipkTdagger2I xpermil3I4xhellipIy DaggerIzdagger DaggerI3

xhellip7I2y Dagger2I yI z Dagger7I2

zdagger

iexclI2xhellipIy DaggerIzdaggerhellip5I2

y iexcl34I yIz Dagger5I2zdagger

DaggerI xhellipIy iexcl Izdagger2hellip7I2y Dagger22I yIz Dagger7I2

zdagger

Dagger3hellipIy iexcl Izdagger2hellipIy DaggerI zdagger3Š hellip41dagger

Equations for M y and M z are obtained by changing allthe indices in equation (41) by cyclic permutation of x y z

4 Moments of Raman spectraUsing the symmetry propert ies of the functions in

equations (36) and (37) we can also obtain from equa-tions (A 1)- (A 6) all the R 2hellip2kdagger

nm hellip0dagger which are needed forthe calculation of the spectral moments M 2

2M24 and M 2

6Thus on taking into account equations (2) (7) and(A 1)- (A 6) we can obtain

M 22 ˆ 2kTpermilhellipa2

xy iexcl axx ayydaggerhellipI iexcl1x Dagger I iexcl1

y Dagger 4I iexcl1z dagger

Daggerhellipa2yz iexcl ayyazzdaggerhellip4I iexcl1

x Dagger I iexcl1y Dagger I iexcl1

z dagger

Daggerhellipa2xz iexcl axx azzdaggerhellipI iexcl1

x Dagger 4I iexcl1y Dagger I iexcl1

z daggerŠ hellip42dagger

M 24 ˆ 2hellipkTdagger2 fhellipa2

xy iexcl axx ayydaggerpermil23I iexcl1z hellipI iexcl1

y Dagger I iexcl1x dagger

Dagger4I iexcl2z hellipI y I iexcl1

x Dagger I x I iexcl1y dagger Dagger I iexcl1

z hellipI yIiexcl2x Dagger I x I iexcl2

y dagger

Dagger I zIiexcl1x I iexcl1

y hellipI iexcl1x Dagger I iexcl1

y dagger DaggerhellipI iexcl1x Dagger I iexcl1

y dagger2

Dagger40I iexcl2z DaggerhellipkTdaggeriexcl2helliphK 2

x i0Iiexcl2x Dagger hK 2

yi0I iexcl2y

Dagger4hK 2z i0I iexcl2

z daggerŠ Daggerhellipa2yz iexcl ayyazzdaggerpermil23I iexcl1

x hellipI iexcl1y Dagger I iexcl1

z dagger

Dagger4I iexcl2x hellipI y I iexcl1

z Dagger I zIiexcl1y dagger Dagger I iexcl1

x hellipI y I iexcl2z Dagger I zI

iexcl2y dagger

Dagger I x I iexcl1y I iexcl1

z hellipI iexcl1y Dagger I iexcl1

z dagger DaggerhellipI iexcl1y Dagger I iexcl1

z dagger2 Dagger40I iexcl2x

DaggerhellipkTdaggeriexcl2hellip4hK 2x i0I iexcl2

x Dagger hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z daggerŠ

Daggerhellipa2xz iexcl axx azzdaggerpermil23I iexcl1

y hellipI iexcl1x Dagger I iexcl1

z dagger

Dagger4I iexcl2y hellipI x I iexcl1

z Dagger I zIiexcl1x dagger Dagger I iexcl1

y hellipI x I iexcl2z Dagger I zI

iexcl2x dagger

Dagger I y I iexcl1x I iexcl1

z hellipI iexcl1x Dagger I iexcl1

z dagger DaggerhellipI iexcl1x Dagger I iexcl1

z dagger2 Dagger40I iexcl2y

DaggerhellipkTdaggeriexcl2helliphK 2x i0I iexcl2

x Dagger 4hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z daggerŠ

iexcl 9axx ayyhellipI iexcl1x iexcl I iexcl1

y dagger2 iexcl 9ayyazzhellipI iexcl1y iexcl I iexcl1

z dagger2

iexcl 9axx azzhellipI iexcl1x iexcl I iexcl1

z dagger2g hellip43dagger

M 26 ˆ 2kT

I3x I3

y I3z

fhellipa2xy iexcl axx ayydaggerM xy Daggerhellipa2

yz iexcl ayyazzdaggerM yz

Daggerhellipa2xz iexcl axx azzdaggerM zx iexcl axx ayyN xy iexcl ayyazzN yz

iexcl axx azzN zxg hellip44dagger

where

1912 Y P Kalmykov and S V Titov

M xy ˆ Ix IzhellipI2x I2

zhhellipLyK ydagger2i0 DaggerI2yI

2zhhellipL yK xdagger2i0

Dagger4I2x I2

yhhellipL yK zdagger2i0dagger DaggerI yIzhellipI2x I2

zhhellipL xK ydagger2i0

DaggerI2yI

2zhhellipL xK xdagger2i0 Dagger4I2

x I2yhhellipL x K zdagger2i0dagger

DaggerI x I yhellipI2x I2

z hhellipL zK ydagger2i0 DaggerI2yI

2zhhellipLzK xdagger2i0

Dagger4I2x I2

yhhellipL zK zdagger2i0dagger DaggerIyI zhellipI3xhellip4I y DaggerI zdagger

iexcl2I2xhellip4I2

y iexcl2I yIz DaggerI2zdagger DaggerI xhellip4I3

y Dagger20I2yIz

Dagger20I yI2z DaggerI3

zdagger Dagger9I2yI

2zdaggerhK 2

xi0 DaggerIx IzhellipI3yhellip4Ix DaggerIzdagger

iexcl2I2yhellip4I2

x iexcl2I x Iz DaggerI2zdagger

DaggerI yhellip4I3x Dagger20I2

x Iz Dagger20Ix I2z DaggerI3

zdagger Dagger9I2x I2

zdaggerhK 2yi0

DaggerI x I yhellipI3zhellipIx DaggerI zdagger iexcl2I2

zhellipI2x Dagger10Ix Iy DaggerI2

ydagger

DaggerI zhellipI x DaggerIydaggerhellipI2x Dagger34Ix Iy DaggerI2

ydagger Dagger144I2x I2

ydaggerhK 2zi0

Dagger2I x I yIzpermil2I x I yhelliphK x LyK zi0hellip2I x iexcl2I y iexcl Izdagger

DaggerhK yLx K zi0hellip2I x iexcl2I y DaggerI zdaggerdagger

DaggerI yIzhelliphK yL zK xi0hellip4Ix DaggerIy iexcl Izdagger

DaggerhK zLyK x i0hellip5I x DaggerIy iexcl I zdaggerdagger

iexclI x I zhelliphK xL zK yi0hellipIx Dagger4Iy iexcl I zdagger

DaggerhK zLx K yi0hellipIx Dagger5Iy iexcl I zdaggerdaggerŠ

iexcl2kTpermilhL yK yi0I2x I2

zhellipI2z Dagger3I yIz iexcl I2

y DaggerI xhellipI x Dagger12I y DaggerIzdaggerdagger

DaggerhLx K xi0I2yI

2zhellipI2

z Dagger3I x I z iexcl I2x DaggerI yhellipIy Dagger12Ix DaggerIzdaggerdagger

Dagger4hLzK zi0I2x I2

yhellipI2x Dagger10Ix Iy DaggerI2

y DaggerIzhellip3Ix Dagger3I y iexcl IzdaggerdaggerŠ

DaggerhellipkTdagger2permil3I5zhellipI x DaggerI ydagger Dagger7I4

zhellipI2x Dagger10I x Iy DaggerI2

ydagger

iexcl5I3zhellipI x DaggerI ydaggerhellipI x iexcl Iydagger2

DaggerI2zhellip7I4

x Dagger68I3x Iy Dagger90I2

x I2y Dagger68Ix I3

y Dagger7I4ydagger

DaggerI zhellipI x DaggerIydagger

poundhellip3I4x Dagger64I3

x Iy Dagger586I2x I2

y Dagger64I x I3y Dagger3I4

ydagger

Dagger4I x I yhellip3I4x Dagger52I3

x Iy Dagger130I2x I2

y Dagger52Ix I3y Dagger3I4

ydaggerŠ hellip45dagger

N xy ˆ 9I2zhellipI x iexcl Iydaggerf3I zhellipI2

xhK 2yi0 iexcl I2

yhK 2x i0dagger

iexcl2kT IzhellipI2xhL yK yi0 iexcl I2

yhLx K xi0dagger

DaggerhellipkTdagger2permil5hellipIx iexcl IydaggerhellipI2x Dagger10Ix Iy DaggerI2

y

Dagger3I zhellipIx DaggerI ydagger DaggerI2zdaggerŠg hellip46dagger

Equations for M yz M zx and N yz N zx in equation (44)can be obtained readily by changing all the indices inequations (45) and (46) from fx y zg to fy z xg andfz x yg respectively

5 Linear spherical and symmetric top moleculesThe results we have obtained contain the particular

cases of classical ensembles of symmetric tops sphericaltops and linear molecules In these cases the theory isconsiderably simpli ed as the calculation of all the spec-tral moments M l

2k needs only that of R l hellip2kdaggermm hellip0dagger [2] Thus

for symmetric top molecules (I x ˆ I y ˆ I 6ˆ I zdagger equa-tions (A 1) (A 3) and (A 5) from the appendix yield

R lmmhellip0daggerˆiexclkT

Ipermilx Daggersup2m2Š hellip47dagger

R l hellip4daggermm hellip0daggerˆ kT

Ihellip dagger2

xhellip3x iexcl1daggerDaggersup2 6xm2 Daggerm2 iexclx1Daggersup2hellip daggerDagger3sup22m4

Dagger 12I2permilhellipx iexclm2daggerhK 2

i0 Dagger2hellip1Daggersup2dagger2m2hK 2zi0Š hellip48dagger

R l hellip6daggermm hellip0daggerˆiexcl kT

Ihellip dagger3 11Daggersup2

copy5xhellip3x2 iexcl3x Dagger1dagger

Dagger3sup2hellipxhellip5x2 iexcl10x Dagger8daggerDagger3m2hellip5x2 iexcl4daggerdaggerDagger15sup22m2hellipxhellip3x iexcl2daggerDaggerm2hellip3x Dagger1daggerdagger

Dagger15sup23m4hellip3x Daggerm2daggerDagger15sup24m6 iexcl3sup2hellipx iexclm2dagger1Daggersup2

Dagger 1

hellipkTdagger2permil12hK 2

i0hellipxhellip9x iexcl5daggeriexclm2hellip9x iexcl7dagger

Daggersup2permilxhellip9x iexcl5daggeriexclm2hellip9m2 iexcl1daggerŠDaggersup22permilx Daggerm2hellip9x iexcl9m2 iexcl7daggerŠdaggerDaggerhK 2

zi0hellip2x Daggerm2hellip9x iexcl11daggerDaggersup2permil2x Daggerm2hellip27x Dagger9m2 iexcl20daggerŠDagger9sup22m2hellip3x iexcl1Dagger3m2daggerDagger9sup23m2hellipx Dagger3m2daggerDagger9sup24m4daggerDagger1

2helliphhellipLx K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipLyK xdagger2

DaggerhellipL yK ydagger2i0 Daggerhellip1Daggersup2daggerhhellipLzK xdagger2

DaggerhellipL zK ydagger2i0daggerhellip1Daggersup2daggerhellipx iexclm2daggerDaggerhelliphhellipL xK zdagger2

DaggerhellipL yK zdagger2i0 Daggerhellip1Daggersup2daggerhhellipL zK zdagger2i0daggerhellip1Daggersup2dagger3m2

DaggerhK yLx K z iexclK x LyK zi0m2hellip1Daggersup2dagger2

DaggerhK yLzK x iexclK x LzK yi0permilm2hellip5Dagger4sup2daggeriexclxhellip2Daggersup2daggerŠpoundhellip1Daggersup2daggerŠ

Dagger 1kT

permilhLxK x DaggerL yK yi0hellipx iexclm2daggerhellip1iexcl3hellip1Daggersup2dagger

poundhellipx Daggersup2m2daggerDagger2sup2daggerDagger2hL zK zi0hellip1Daggersup2dagger2m2

poundhellip1iexcl3hellip1Daggersup2daggerhellipx Daggersup2m2daggerdaggerŠordf

hellip49dagger

Spectral moments of CFs of asymmetric top molecules 1913

where

x ˆ lhellip l Dagger1dagger sup2 ˆ I=I z iexcl 1 K 2 ˆ K 2

x DaggerK 2y

Here we have taken into account that [14]

L x L y iexcl L yL x ˆ L z

Equations (47) and (48) for the second and the fourthmoments coincide with the results of [2] which wereobtained from the consideration of an ensemble of hin-dered symmetric tops Equation (49) for freely hellipV ˆ 0daggerrotating symmetric tops was obtained in [2] However itwas presented there with several misprints

For spherical tops hellipI x ˆ I y ˆ I z ˆ Idagger we simplyput sup2 ˆ 0 in equations (47)- (49) Thus for m ˆ 0 wehave

R l00hellip0dagger ˆ iexcl

kTI

x hellip50dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 1dagger Dagger 12I2

hK 2i0 hellip51dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 15x Dagger 5 iexcl 3x iexcl 1kT

pound hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 5dagger Dagger4hK 2

z i0

Dagger4hK x L zK y iexcl K yL zK x i0

Dagger hhellipL x K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipL yK xdagger2

DaggerhellipL yK ydagger2 DaggerhellipL zK xdagger2 DaggerhellipL zK ydagger2i0Šordf

hellip52dagger

Equations (50)- (52) are in accordance with the results of[2]

For linear molecules hellipI x ˆ I y ˆ I I z ˆ 0dagger one has

R l00hellip0dagger ˆ iexcl kT

Ix hellip53dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 2dagger Dagger 12I2

hK 2i0 hellip54dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 30x Dagger 24

iexcl 3x iexcl 2kT

hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 1dagger Dagger hhellipL x K xdagger2

DaggerhellipL x K ydagger2 DaggerhellipL yK x dagger2 DaggerhellipL yK ydagger2i0Šordf

hellip55dagger

Equations (53)- (55) are also in agreement with theresults of [2]

6 Spectral moments and coe cients in the Taylorseries expansion of the memory functions

As has been shown on many occasions (eg [4- 6 8])the calculation of the CFs C lhellip tdagger for asymmetric topmolecules can be simpli ed considerably by makinguse of the memory function approach [13] Thisapproach allows one to express the absorpt ion andRaman spectra in terms of the memory functionsK l

nhellip tdagger of arbitrary order n [2 13] The coe cients inthe Taylor series expansions of the memory functionsK l

nhellip tdagger are also related to the spectral moments M ln as

the CFs C lhellip tdagger are connected to the correspondingmemory functions K l

nhellip tdagger by equations (see eg [13])

ddt

C lhellip tdagger ˆ iexclhellip t

0K l

1hellip t iexcl t 0daggerChellip t 0daggerdt 0 hellip56dagger

ddt

K lnhellip tdagger ˆ iexcl

hellip t

0K l

nDagger1hellip t iexcl t 0daggerK lnhellip t 0daggerdt 0

hellip57dagger

In the classical limit the CF C lhellip tdagger and the memory func-tions K l

nhellip tdagger which are both even functions of time t canbe expanded in Taylor series as [13]

C lhellip tdagger ˆX1

kˆ0

Chellip2kdaggerl hellip0dagger t2k

hellip2kdaggerˆ

X1

kˆ0

M l2k

hellip itdagger2k

hellip2kdagger hellip58dagger

K lnhellip tdagger ˆ

X1

kˆ0

K lhellip2kdaggern hellip0dagger t2k

hellip2kdagger hellip59dagger

Here we have taken into account equation (2)Substituting equations (58) and (59) in (56) and (57)one can obtain after some algebraic transformations

C lhellip0daggerK lhellip2ndagger1 hellip0dagger ˆ iexclChellip2nDagger2dagger

l hellip0dagger

iexclXn

sˆ1

Chellip2sdaggerl hellip0daggerK lhellip2hellipniexclsdaggerdagger

1 hellip0dagger hellip60dagger

K lihellip0daggerK lhellip2ndagger

iDagger1 hellip0dagger ˆ iexclK lhellip2nDagger2daggeri hellip0dagger

iexclXn

sˆ1

K lhellip2sdaggeri hellip0daggerK lhellip2hellipniexclsdaggerdagger

iDagger1 hellip0dagger

hellip61dagger

It is convenient to present equations (60) and (61) interms of determinants

1914 Y P Kalmykov and S V Titov

K lhellip2ndagger1 hellip0dagger ˆdet

M l2=Clhellip0dagger 1 0 centcent cent 0

M l4=Clhellip0dagger M l

2=Clhellip0dagger 1

0

M l6=Clhellip0dagger M l

4=Clhellip0dagger M l2=Clhellip0dagger

1

M l2hellipnDagger1dagger=Clhellip0dagger M l

2n=Clhellip0dagger cent centcent M l4=Clhellip0dagger M l

2=Clhellip0dagger

7777777777777777777

7777777777777777777

hellip62dagger

K lihellip0daggerK l hellip2ndagger

iDagger1 hellip0dagger ˆ i2n det

K lhellip2daggeri hellip0dagger 1 0 cent cent cent 0

K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger 1

1

K lhellip2hellipnDagger1daggerdaggeri hellip0dagger K lhellip2ndagger

i hellip0dagger cent cent cent K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger

777777777777777

777777777777777

hellip63dagger

where n ˆ 0 1 2 and

M l2n ˆClhellip0daggerdet

K l1hellip0dagger 1 0 centcentcent 0

K lhellip2dagger1 hellip0dagger K l

1hellip0dagger 1

1

K lhellip2niexcl2dagger1 hellip0dagger K lhellip2hellipniexcl3daggerdagger

1 hellip0dagger centcent cent K lhellip2dagger1 hellip0dagger K l

1hellip0dagger

777777777777777

777777777777777

hellip64dagger

Thus if the moments M l2n are known it is possible to

calculate all the memory functions and their derivativesat t ˆ 0 Conversely if the derivatives of the memoryfunctions K lhellip2ndagger

i hellip0dagger are known it is possible to calculateall the spectral moments M l

2n

7 DiscussionAs we have already mentioned in the analysis of

absorption and scattering spectra of molecular uidsin terms of the moments of the CFs the key problemis the evaluation of the short term behaviour of thecorrelation functions R l

nmhellip tdagger This problem is relatedin turn to the calculation of the coe cients in aTaylor series expansion of R l

n mhellip tdagger The results obtainedin the present work allow one to calculate these coe -cients and hence the spectral moments in a systematicmanner Equations for M l

n obtained in the present paperpermit one also to calculate from equation (64) the co-e cients in Taylor series expansions of the appropriatememory functions Here for simplicity we have con- ned ourselves to the evaluation of the second (M l

2)fourth (M l

4) and sixth (M l6) moments only However

the method used is quite general and it may be extendedreadily for evaluating higher spectral moments as thehigher time derivatives of R l

n mhellip tdagger at t ˆ 0 can be calcu-lated by a similar way For example according to equa-tion (11) on evaluating the fourth derivative of theWigner D function Dl

n mhellipsup3rsquoAacutedagger at t ˆ 0 one can calcu-late the eighth derivative of R lhellip8dagger

n mhellip0dagger etcThe second moment M 1

2 for the dipole absorption ofasymmetric top molecules has been obtained in many

works as within the framework of classical andquantum theory (eg [10 17]) The derivation of thefourth moment M 1

4 for molecules of C2v symmetry wasgiven in [17] The second and fourth moments for anarbitrary asymmetric top molecule have been calculatedby Gordon [11] on using quantum-mechanical methodswith the subsequent transition to the classical limithellip -h 0dagger Our results (equations (38) and (39)) are incomplete agreement with those of Gordon [11] Inmatrix notations of [11] equations (38) and (39) canbe written as

M 12 ˆ kTpermilm2Tr fBg iexcl m cent B cent mTŠ hellip65dagger

M 14 ˆ hellipkTdagger2 fm2permil2hellipTr fBgdagger2 DaggerTr fB2ghellipTr fBgTr fBiexcl1g iexcl 3dagger

iexcl Tr fB3gTr fBiexcl1gŠ

Daggerm cent B2 cent mThellip7 iexcl Tr fBgTr fBiexcl1gdagger

iexcl 6 m cent B cent mTTr fBg Daggerm cent B3 cent mTTr fBiexcl1gg

Daggerm2hK cent B2 cent KTi0 iexcl hhellipm cent B cent KTdagger2i0 hellip66dagger

where B ˆ Iiexcl1 and the upper index T denotes the trans-position

Equations (42) and (43) for the second and fourthRaman spectral moments M 2

2 and M 24 coincide with

the results of Gordon [12] and can be rewritten in thematrix form as follows

M 22 ˆ 2kTpermil2Tr fBgTr fa2g iexcl 3Tr fa cent B cent agŠ hellip67dagger

M 24 ˆ hellipkTdagger2permil27hellipTr fBgdagger2Tr fa2g iexcl 72Tr fa cent B cent agTr fBg

Dagger18Tr fa cent B cent a cent Bg

Dagger6Tr fa cent B2 cent aghellip1 iexcl Tr fBgTr fBiexcl1gdagger

DaggerTr fa2gTr fB2ghellip5 Dagger4Tr fBgTr fBiexcl1gdagger

iexcl 4Tr fa2gTr fB3gTr fBiexcl1g

Dagger6Tr fa cent B3 cent agTr fBiexcl1gŠ

Dagger4hK cent B2 cent KTTr fa2gi0 iexcl 6hK cent B cent a2 cent B cent KT i0

hellip68dagger

It should be noted that there was a misprint in theequation for the fourth moment M 2

4 in [12] equation(58) To our knowledge the sixth moments forthe dipole absorpt ion (equation (40) and Ramanscattering (equation (44)) spectra of an arbitrary asym-metric top molecule have been calculated for the rsttime

Spectral moments of CFs of asymmetric top molecules 1915

The spectral moments allow one to carry out thequantitative analysis of experimental data on IR absorp-tion and Raman spectra of uids with asymmetrictop molecules As is well known the spectral momentM l

2 does not depend on intermolecular interactions Thespectral moment M l

4 depends on the mean-squaretorque acting on the molecule The spectral momentM l

6 includes also averages of angular derivatives ofthe torque Thus the spectral moments allow one toobtain information both about molecular motions andintermolecular interactions Another important prop-erty of spectral moments is the possibility of evaluatingasymptotic properties of spectra For example ifspectral moments M l

2 M l4 and M l

6 exist the integral inthe right hand side of equation (1) exists for n ˆ 2 4and 6 As a result the spectral function I lhellipdagger mustdecrease faster than iexcl6 Moreover the spectralmoments play a rather important role in the theoreticalcalculation of the absorption and scattering spectraand in the modelling of molecular rotation in uidsAs the second spectral moment M l

2 does not dependon the torques acting on molecules (it is exclusivelydetermined by molecular parameters) this momentcalculated in the context of any particular modelshould not depend on parameters describing inter-actions of molecules In other words the secondmoment can be used as a criterion of the correctnessof the model assumptions As the higher momentsdepend on angular derivatives of the intermolecularpotential energy they can be used for the evaluationof model parameters in systems of interacting moleculesFor example the mean-square torque acting on themolecule can be determined from the fourth spectralmoment M l

4 [18] With the help of the spectral momentsM l

2 and M l4 one can evaluate the mean time of molecular

collisions [19 20] Examples of such evaluations forasymmetric top molecules like H 2O and D 2O weregiven in [9]

We thank Professor J L Dejardin for usefulcomments and suggestions The support of this workby the International Association for the Promotionof Co-operation with Scientists from the NewIndependent States of the Former Soviet Union(Project INTAS 96-1411) is gratefully acknowledged

Appendix

Equat ions for R l hellip2kdaggermm hellip0dagger and R l hellip2kdagger

mmsect4hellip0daggerEquations (21)- (32) and (34) yield

R l hellip2daggermm hellip0dagger ˆ iexcl

kT2

permilhellipx iexcl m2daggerhellipI iexcl1x Dagger I iexcl1

y dagger Dagger 2m2I iexcl1z Š hellipA 1dagger

R l hellip2daggermmsect4hellip0dagger ˆ 0 hellipA 2dagger

R l hellip4daggermmhellip0dagger ˆhellipkTdagger2 fhellipx iexcl 3m2daggerpermil2I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y DaggerI iexcl1

x daggerŠ

Daggerm2I iexcl2z hellipI yI

iexcl1x DaggerI x I iexcl1

y iexcl 2dagger

Dagger12hellipx Dagger6m2x iexcl6m4 iexcl5m2daggerI iexcl1

z hellipI iexcl1x DaggerI iexcl1

y dagger

Dagger12hellipx iexcl m2daggerpermilI iexcl1

z hellipI y I iexcl2x DaggerI x I iexcl2

y dagger

DaggerI zIiexcl1x I iexcl1

y hellipI iexcl1x DaggerI iexcl1

y dagger iexcl 2hellipI iexcl2x DaggerI iexcl2

y daggerŠ

Dagger18 permil3hellipx iexclm2dagger2 iexcl2x Dagger5m2Š

pound permil3I iexcl2x Dagger3I iexcl2

y Dagger2I iexcl1x I iexcl1

y Š Dagger3m4I iexcl2z g

Dagger12hellipx iexcl m2daggerhellip I iexcl2

x hK 2x i0 DaggerI iexcl2

y hK 2yi0dagger

Daggerm2I iexcl2z hK 2

z i0 hellipA 3dagger

R lhellip4daggermsect4 mhellip0dagger ˆ 3hellipkTdagger2

16hellipI iexcl1

x iexcl I iexcl1y dagger2

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 4dagger

R lhellip6daggermmhellip0dagger ˆ iexcl kT

16I3x I3

yI3zhellipR lm

0 DaggerkTR lm1 DaggerhellipkTdagger2R lm

2 dagger hellipA 5dagger

R lhellip6daggermsect4 m ˆ iexcl

3kThellipI x iexcl I ydagger32I3

x I 3yI z

f6I zhellipI2x hK 2

yi0 iexcl I2yhK 2

x i0dagger

Dagger4kT I zhellipI2yhL x K x i0 iexcl I2

x hL yK yi0dagger

Dagger5hellipkTdagger2hellipI x iexcl I ydaggerpermil3xhellipI x DaggerI ydaggerI z

Dagger3hellip2I x I y iexclhellipI x DaggerI ydaggerI zdagger

pound mhellipm sect 4dagger

Dagger2hellipI 2x Dagger22I x I y DaggerI 2

y iexcl12hellipI x DaggerI ydaggerI z DaggerI2zdaggerŠg

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 6dagger

where x ˆ lhellip l Dagger1dagger

1916 Y P Kalmykov and S V Titov

R lm0 ˆ 8f2m2I 2

x I2yhI x I zhellipL yK zdagger2 DaggerI y I zhellipL x K zdagger2

DaggerI x I yhellipL zK zdagger2i0 Daggerhellipx iexclm2daggerI2z permilI2

x hI x I zhellipL yK ydagger2

DaggerI yI zhellipL x K ydagger2 DaggerI x I yhellipL zK ydagger2i0 DaggerI2yhI x I zhellipL yK xdagger2

DaggerI yI zhellipL x K xdagger2 DaggerI x I yhellipL zK xdagger2i0Š

Dagger16I x I yI z fhK zL x K yi0permilhellipx iexclm2daggerI x I zhellipI y DaggerI z iexcl I x dagger

iexcl3m2I y I z Š iexclhK zL yK x i0permilhellipx iexclm2daggerI yI zhellipI x DaggerI z iexcl I ydagger

iexcl3m2I x I z ŠDaggerm2I x I ypermilhK x L yK zi0hellip2I x iexcl2I y iexcl I zdagger

iexclhK yL x K zi0hellip2I y iexcl2I x iexcl I zdaggerŠ

DaggerhK x L zK yi0permilhellipx iexclm2daggerI x I zhellip2I y DaggerI z iexcl I xdagger iexcl3m2I x I yŠ

iexclhK yL zK x i0permilhellipx iexclm2daggerI y I zhellip2I x DaggerI z iexcl I ydagger iexcl3m2I x I y Šg

Dagger2hK 2x i0I yI z fx I zpermil16I x I yhellipI y iexcl I x dagger

iexcl2I yI zhellipI x Dagger9I ydagger Dagger4I xhellipI x iexcl I zdagger2

Dagger9xI yI zhellipI x Dagger3I ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I x I zhellipI z iexcl I xdagger2 DaggerI yI zhellip32I2x

iexcl7I xhellip8I y iexcl I zdagger Dagger45I yI zdagger

Dagger18hellipx iexclm2daggerI y I zhellip2I x I y iexcl I x I z iexcl3I yI zdagger

iexcl9m2I y I2zhellipI x Dagger3I ydaggerŠg

Dagger2hK 2yi0I x I z fx I zpermil16I x I yhellipI x iexcl I ydagger

iexcl2I x I zhellip9I x DaggerI ydagger Dagger4I yhellipI y iexcl I zdagger2

Dagger9xI x I zhellip3I x DaggerI ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I yI zhellipI z iexcl I ydagger2 DaggerI x I zhellip32I2y iexcl7I yhellip8I x iexcl I zdagger

Dagger45I x I zdagger Dagger18hellipx iexclm2daggerI x I zhellip2I x I y iexcl3I x I z iexcl I y I zdagger

iexcl9m2I x I 2zhellip3I x DaggerI ydaggerŠg

Dagger8hK 2zi0I x I y fhellipx iexclm2daggerI zpermilI xhellipI x iexcl I zdagger2

DaggerI yhellip I y iexcl I zdagger2 DaggerI x I yhellip4I z iexcl I x iexcl I ydaggerŠ

Daggerm2I x I ypermil9hellipx iexclm2daggerI zhellipI x DaggerI ydagger Dagger18m2I x I y iexcl8I2z Šg

R lm1 ˆ 16hL zK zi0I2

x I2ym

2 f2I2z iexcl 2hellipIx iexcl I ydagger2 iexcl 3xI zhellipIx Dagger I ydagger

Dagger3m2hellipIx I z Dagger I yI z iexcl 2I x I ydaggerg

Dagger4hLx K x i0I2yI

2z fhellipx iexcl m2daggerpermil4I2

x iexcl 4hellipI y iexcl Izdagger2

iexcl 6I zhellipIx iexcl Iydagger iexcl 3x IzhellipI x Dagger3IydaggerŠ

Dagger3m2permil3I zhellipIx iexcl I ydagger DaggerxhellipI zhellipI x Dagger3I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl Ix I z iexcl 3I yI zdaggerŠg

Dagger4hLyK yi0I2x I2

z fhellipx iexcl m2daggerpermil4I2y iexcl 4hellipI x iexcl I zdagger2

iexcl 6I zhellipIy iexcl I xdagger iexcl 3x Izhellip3Ix Dagger IydaggerŠ

Dagger3m2permil3I zhellipIy iexcl Ixdagger DaggerxhellipI zhellip3I x Dagger I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl 3I x I z iexcl I yI zdaggerŠg

R lm2 ˆ 15permil2Ix Iy iexcl IzhellipI x DaggerI ydaggerŠpermil8I2

x I2y iexcl 8I x I yI zhellipIx Dagger I ydagger

Dagger I2zhellip5I2

x iexcl 2Ix I y Dagger5I2ydaggerŠm6

Dagger15f16Ix I yhellipI x iexcl Iydagger2permilI2x I2

y iexcl I zhellipIx Dagger I ydaggerŠ

Dagger2I2zhellip3I4

x Dagger43I3x Iy iexcl 84I2

x I2y Dagger43I x I3

y Dagger3I4ydagger

Dagger3xI zpermil8I2x I2

yhellipI x Dagger I ydagger iexcl 4I x I yI zhellip3I2x Dagger2I x Iy Dagger3I2

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip5I2

x iexcl 2I x I y Dagger5I2ydaggerŠ

iexcl I3zhellipI x Dagger Iydaggerhellip47I2

x iexcl 78I x Iy Dagger47I2ydagger

Dagger2I4zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggergm4

Dagger f16I x IyhellipI x iexcl I ydagger2hellip3I2x iexcl 2I x I y Dagger3I2

ydagger

iexcl 8I zhellipIx Dagger IydaggerhellipIx iexcl I ydagger2hellip3I2x Dagger34I x I y Dagger3I2

ydagger

Dagger2I2zhellipI x iexcl Iydagger2hellip107I2

x Dagger602I x I y Dagger107I2ydagger

iexcl 32I3zhellipIx Dagger Iydaggerhellip25I2

x iexcl 41Ix I y Dagger25I2ydagger

Dagger2I4zhellip107I2

x Dagger98I x Iy Dagger107I2ydagger iexcl 24I5

zhellipI x DaggerI ydagger

Dagger30xI zpermil16Ix I yhellipI x Dagger IydaggerhellipIx iexcl Iydagger2

iexcl 4I zhellip3I4x Dagger19I3

x Iy iexcl 36I2x I2

y Dagger19I x I3y Dagger3I4

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip69I2

x iexcl 106I x Iy Dagger69I2ydagger

iexcl 4I3zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggerŠ

Dagger90I2zx

2permil2I x Iyhellip3I2x Dagger2Ix I y Dagger3I2

ydagger

Spectral moments of CFs of asymmetric top molecules 1917

iexcl I zhellipI x Dagger I ydaggerhellip5I 2x iexcl 2I x I y Dagger5I2

ydaggerŠgm2

Daggerx I z f4hellipI x iexcl I ydagger2permil6hellipI x Dagger I ydagger3

iexcl I zhellip31I2x Dagger106I x I y Dagger 31I2

ydaggerŠ

Dagger64I2zhellipI x Dagger I ydaggerhellip5I2

x iexcl 7I x I y Dagger5I2ydagger

iexcl 4I3zhellip31I2

x Dagger26I x I y Dagger31I2ydagger Dagger24I4

zhellipI x Dagger I ydagger

Dagger15xI zpermil2hellipI x iexcl I ydagger2hellip3I x Dagger I ydaggerhellip I x Dagger3I ydagger

iexcl 2I zhellipI x Dagger I ydaggerhellip11I2x iexcl 14I x I y Dagger11I2

ydagger

Dagger2I2zhellip3I 2

x Dagger2I x I y Dagger 3I 2ydagger

Daggerx I zhellipI x Dagger I ydaggerhellip5I2x iexcl 2I x I y Dagger5I2

ydaggerŠg

In this derivation we have assumed that

hK ii0 ˆ 0 hellip i ˆ x y zdaggerand

hK iK ji0 ˆ 0 hL iK ji0 ˆ 0 for i 6ˆ j hellip i j ˆ x y zdagger

References[1] BURHSTEIN A I and TEMKIN S I 1994 Spectroscopy

of M olecular Rotation in Gases and Liquids (CambridgeUniversity Press)

[2] ST PIERRE A G and STEELE W A 1981 M olecPhys 43 123

[3] GORDON R G 1963 J chem Phys 39 2788

[4] LEICKMAN JCL and GUISSANIY 1981 M olec Phys42 1105

[5] AGUADO-GOMEZM and LEICKMAN JCL 1986 PhysRev A 34 4195

[6] AGUADO-GOMEZM and LEICKMAN JCL 1987 PhysRev A 35 286

[7] BORISEVICH N A BLOKHINA PZALESSKAYAG ALASTOCHKINA V A and SHUKUROV T 1984 IzvAkad Nauk SSSR Ser Fiz 48 709

[8] KALMYKOV YUP 1985 Opt Spektrosk 58 804 [1985Opt Spectrosc 58 493]

[9] KALMYKOV YU P and TITOV S V 1999 J molecS truct 479 123

[10] GORDON R G 1963 J chem Phys 38 1724[11] GORDON R G 1964 J chem Phys 41 1819[12] GORDON R G 1964 J chem Phys 40 1973[13] BERNE B J and HARP G D 1970 Adv chem Phys

17 63[14] LEICKMAN JCL and GUISSANIY 1984 M olec Phys

53 761[15] VARSHALOVICH D A MOSKALEV A N and

KHERSONSKII V K 1988 Quantum Theory of AngularM omentum (Singapore World Scienti c)

[16] LANDAULD and LIFSHITZEM 1976 M echanics ACourse of theoretical Physics 3rd Edn Vol 1 (OxfordPergamon Press)

[17] BOiumlTTCHERC JF and BORDEWIJK P 1979 Theory ofElectric Polarization Vol 2 (Amsterdam Elsevier)

[18] RODRIGUEZ R and MCHALE J L 1988 J chemPhys 88 2264

[19] BURSHTEINA I and MCCONNELL J R 1989 PhysicaA 157 933

[20] KALMYKOV YU P and MCCONNELL J R 1993Physica A 193 394

1918 Spectral moments of CFs of asymmetric top molecules

M 14 ˆ m2

x fhellipkTdagger2permil6I iexcl1y I iexcl1

z iexcl I iexcl1x hellipI iexcl1

y Dagger I iexcl1z dagger

Dagger I iexcl1x hellipI zI

iexcl2y Dagger I yI

iexcl2z dagger

Dagger I x I iexcl1z I iexcl1

y hellipI iexcl1y Dagger I iexcl1

z dagger Dagger I iexcl2y Dagger I iexcl2

z Š

Dagger hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z g

Daggerm2y fhellipkTdagger2permil6I iexcl1

x I iexcl1z iexcl I iexcl1

y hellipI iexcl1x Dagger I iexcl1

z dagger

Dagger I iexcl1y hellipI zI

iexcl2x Dagger I x I iexcl2

z dagger

Dagger I y I iexcl1z I iexcl1

x hellipI iexcl1x Dagger I iexcl1

z dagger Dagger I iexcl2x Dagger I iexcl2

z Š

Dagger hK 2x i0I

iexcl2x Dagger hK 2

z i0I iexcl2z g

Daggerm2z fhellipkTdagger2permil6I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y Dagger I iexcl1

x dagger

Dagger I iexcl1z hellipI yI

iexcl2x Dagger I x I iexcl2

y dagger Dagger I zIiexcl1x I iexcl1

y hellipI iexcl1x Dagger I iexcl1

y dagger

Dagger I iexcl2x Dagger I iexcl2

y Š Dagger hK 2x i0I

iexcl2x Dagger hK 2

yi0I iexcl2y g hellip39dagger

M 16 ˆ kT

I 3x I3

yI3zhellipm2

x M x Dagger m2yM y Daggerm2

zM zdagger hellip40dagger

where

M x ˆ I x I yIzhellipI2xhellipIy DaggerI zdagger DaggerhellipI y DaggerI z iexcl2IxdaggerhellipIy iexcl I zdagger2daggerhK 2

x i0

DaggerI2x I zhellipI3

y iexcl2I2yhellipI x Dagger2Izdagger DaggerIyhellipIx Dagger2I zdagger2 Dagger9I x I2

zdaggerhK 2yi0

DaggerI2x I yhellipI3

z iexcl2I2zhellipIx Dagger2Iydagger DaggerI zhellipI x Dagger2Iydagger2 Dagger9Ix I2

ydaggerhK 2zi0

DaggerI2x I2

zhellipI x IzhhellipL yK ydagger2i0 DaggerIy IzhhellipL x K ydagger2i0

DaggerI x I yhhellipL zK ydagger2i0dagger DaggerI2x I2

yhellipIx IzhhellipL yK zdagger2i0

DaggerI yIzhhellipLx K zdagger2i0 DaggerIx I yhhellipLzK zdagger2i0dagger

Dagger2I2x I yI zpermilhK x L yK zi0IyhellipIx iexclI y DaggerI zdagger

iexclhK xL zK yi0I zhellipI x iexcl I z DaggerI ydagger

DaggerhK yL xK zi0I yhellipI x iexcl Iy Dagger2Izdagger

iexclhK zL xK yi0I zhellipI x iexcl I z Dagger2I ydaggerŠiexcl2hellipkTdaggerI2x

poundpermilhLyK yi0I2zhellipI2

x Dagger3Ix Iy iexcl I2y DaggerI2

z DaggerI x I zdagger

DaggerhLzK zi0I2yhellipI2

x DaggerI x Iy DaggerI2y iexcl I2

z Dagger3I x I zdaggerŠ

DaggerhellipkTdagger2I xpermil3I4xhellipIy DaggerIzdagger DaggerI3

xhellip7I2y Dagger2I yI z Dagger7I2

zdagger

iexclI2xhellipIy DaggerIzdaggerhellip5I2

y iexcl34I yIz Dagger5I2zdagger

DaggerI xhellipIy iexcl Izdagger2hellip7I2y Dagger22I yIz Dagger7I2

zdagger

Dagger3hellipIy iexcl Izdagger2hellipIy DaggerI zdagger3Š hellip41dagger

Equations for M y and M z are obtained by changing allthe indices in equation (41) by cyclic permutation of x y z

4 Moments of Raman spectraUsing the symmetry propert ies of the functions in

equations (36) and (37) we can also obtain from equa-tions (A 1)- (A 6) all the R 2hellip2kdagger

nm hellip0dagger which are needed forthe calculation of the spectral moments M 2

2M24 and M 2

6Thus on taking into account equations (2) (7) and(A 1)- (A 6) we can obtain

M 22 ˆ 2kTpermilhellipa2

xy iexcl axx ayydaggerhellipI iexcl1x Dagger I iexcl1

y Dagger 4I iexcl1z dagger

Daggerhellipa2yz iexcl ayyazzdaggerhellip4I iexcl1

x Dagger I iexcl1y Dagger I iexcl1

z dagger

Daggerhellipa2xz iexcl axx azzdaggerhellipI iexcl1

x Dagger 4I iexcl1y Dagger I iexcl1

z daggerŠ hellip42dagger

M 24 ˆ 2hellipkTdagger2 fhellipa2

xy iexcl axx ayydaggerpermil23I iexcl1z hellipI iexcl1

y Dagger I iexcl1x dagger

Dagger4I iexcl2z hellipI y I iexcl1

x Dagger I x I iexcl1y dagger Dagger I iexcl1

z hellipI yIiexcl2x Dagger I x I iexcl2

y dagger

Dagger I zIiexcl1x I iexcl1

y hellipI iexcl1x Dagger I iexcl1

y dagger DaggerhellipI iexcl1x Dagger I iexcl1

y dagger2

Dagger40I iexcl2z DaggerhellipkTdaggeriexcl2helliphK 2

x i0Iiexcl2x Dagger hK 2

yi0I iexcl2y

Dagger4hK 2z i0I iexcl2

z daggerŠ Daggerhellipa2yz iexcl ayyazzdaggerpermil23I iexcl1

x hellipI iexcl1y Dagger I iexcl1

z dagger

Dagger4I iexcl2x hellipI y I iexcl1

z Dagger I zIiexcl1y dagger Dagger I iexcl1

x hellipI y I iexcl2z Dagger I zI

iexcl2y dagger

Dagger I x I iexcl1y I iexcl1

z hellipI iexcl1y Dagger I iexcl1

z dagger DaggerhellipI iexcl1y Dagger I iexcl1

z dagger2 Dagger40I iexcl2x

DaggerhellipkTdaggeriexcl2hellip4hK 2x i0I iexcl2

x Dagger hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z daggerŠ

Daggerhellipa2xz iexcl axx azzdaggerpermil23I iexcl1

y hellipI iexcl1x Dagger I iexcl1

z dagger

Dagger4I iexcl2y hellipI x I iexcl1

z Dagger I zIiexcl1x dagger Dagger I iexcl1

y hellipI x I iexcl2z Dagger I zI

iexcl2x dagger

Dagger I y I iexcl1x I iexcl1

z hellipI iexcl1x Dagger I iexcl1

z dagger DaggerhellipI iexcl1x Dagger I iexcl1

z dagger2 Dagger40I iexcl2y

DaggerhellipkTdaggeriexcl2helliphK 2x i0I iexcl2

x Dagger 4hK 2yi0I iexcl2

y Dagger hK 2z i0I

iexcl2z daggerŠ

iexcl 9axx ayyhellipI iexcl1x iexcl I iexcl1

y dagger2 iexcl 9ayyazzhellipI iexcl1y iexcl I iexcl1

z dagger2

iexcl 9axx azzhellipI iexcl1x iexcl I iexcl1

z dagger2g hellip43dagger

M 26 ˆ 2kT

I3x I3

y I3z

fhellipa2xy iexcl axx ayydaggerM xy Daggerhellipa2

yz iexcl ayyazzdaggerM yz

Daggerhellipa2xz iexcl axx azzdaggerM zx iexcl axx ayyN xy iexcl ayyazzN yz

iexcl axx azzN zxg hellip44dagger

where

1912 Y P Kalmykov and S V Titov

M xy ˆ Ix IzhellipI2x I2

zhhellipLyK ydagger2i0 DaggerI2yI

2zhhellipL yK xdagger2i0

Dagger4I2x I2

yhhellipL yK zdagger2i0dagger DaggerI yIzhellipI2x I2

zhhellipL xK ydagger2i0

DaggerI2yI

2zhhellipL xK xdagger2i0 Dagger4I2

x I2yhhellipL x K zdagger2i0dagger

DaggerI x I yhellipI2x I2

z hhellipL zK ydagger2i0 DaggerI2yI

2zhhellipLzK xdagger2i0

Dagger4I2x I2

yhhellipL zK zdagger2i0dagger DaggerIyI zhellipI3xhellip4I y DaggerI zdagger

iexcl2I2xhellip4I2

y iexcl2I yIz DaggerI2zdagger DaggerI xhellip4I3

y Dagger20I2yIz

Dagger20I yI2z DaggerI3

zdagger Dagger9I2yI

2zdaggerhK 2

xi0 DaggerIx IzhellipI3yhellip4Ix DaggerIzdagger

iexcl2I2yhellip4I2

x iexcl2I x Iz DaggerI2zdagger

DaggerI yhellip4I3x Dagger20I2

x Iz Dagger20Ix I2z DaggerI3

zdagger Dagger9I2x I2

zdaggerhK 2yi0

DaggerI x I yhellipI3zhellipIx DaggerI zdagger iexcl2I2

zhellipI2x Dagger10Ix Iy DaggerI2

ydagger

DaggerI zhellipI x DaggerIydaggerhellipI2x Dagger34Ix Iy DaggerI2

ydagger Dagger144I2x I2

ydaggerhK 2zi0

Dagger2I x I yIzpermil2I x I yhelliphK x LyK zi0hellip2I x iexcl2I y iexcl Izdagger

DaggerhK yLx K zi0hellip2I x iexcl2I y DaggerI zdaggerdagger

DaggerI yIzhelliphK yL zK xi0hellip4Ix DaggerIy iexcl Izdagger

DaggerhK zLyK x i0hellip5I x DaggerIy iexcl I zdaggerdagger

iexclI x I zhelliphK xL zK yi0hellipIx Dagger4Iy iexcl I zdagger

DaggerhK zLx K yi0hellipIx Dagger5Iy iexcl I zdaggerdaggerŠ

iexcl2kTpermilhL yK yi0I2x I2

zhellipI2z Dagger3I yIz iexcl I2

y DaggerI xhellipI x Dagger12I y DaggerIzdaggerdagger

DaggerhLx K xi0I2yI

2zhellipI2

z Dagger3I x I z iexcl I2x DaggerI yhellipIy Dagger12Ix DaggerIzdaggerdagger

Dagger4hLzK zi0I2x I2

yhellipI2x Dagger10Ix Iy DaggerI2

y DaggerIzhellip3Ix Dagger3I y iexcl IzdaggerdaggerŠ

DaggerhellipkTdagger2permil3I5zhellipI x DaggerI ydagger Dagger7I4

zhellipI2x Dagger10I x Iy DaggerI2

ydagger

iexcl5I3zhellipI x DaggerI ydaggerhellipI x iexcl Iydagger2

DaggerI2zhellip7I4

x Dagger68I3x Iy Dagger90I2

x I2y Dagger68Ix I3

y Dagger7I4ydagger

DaggerI zhellipI x DaggerIydagger

poundhellip3I4x Dagger64I3

x Iy Dagger586I2x I2

y Dagger64I x I3y Dagger3I4

ydagger

Dagger4I x I yhellip3I4x Dagger52I3

x Iy Dagger130I2x I2

y Dagger52Ix I3y Dagger3I4

ydaggerŠ hellip45dagger

N xy ˆ 9I2zhellipI x iexcl Iydaggerf3I zhellipI2

xhK 2yi0 iexcl I2

yhK 2x i0dagger

iexcl2kT IzhellipI2xhL yK yi0 iexcl I2

yhLx K xi0dagger

DaggerhellipkTdagger2permil5hellipIx iexcl IydaggerhellipI2x Dagger10Ix Iy DaggerI2

y

Dagger3I zhellipIx DaggerI ydagger DaggerI2zdaggerŠg hellip46dagger

Equations for M yz M zx and N yz N zx in equation (44)can be obtained readily by changing all the indices inequations (45) and (46) from fx y zg to fy z xg andfz x yg respectively

5 Linear spherical and symmetric top moleculesThe results we have obtained contain the particular

cases of classical ensembles of symmetric tops sphericaltops and linear molecules In these cases the theory isconsiderably simpli ed as the calculation of all the spec-tral moments M l

2k needs only that of R l hellip2kdaggermm hellip0dagger [2] Thus

for symmetric top molecules (I x ˆ I y ˆ I 6ˆ I zdagger equa-tions (A 1) (A 3) and (A 5) from the appendix yield

R lmmhellip0daggerˆiexclkT

Ipermilx Daggersup2m2Š hellip47dagger

R l hellip4daggermm hellip0daggerˆ kT

Ihellip dagger2

xhellip3x iexcl1daggerDaggersup2 6xm2 Daggerm2 iexclx1Daggersup2hellip daggerDagger3sup22m4

Dagger 12I2permilhellipx iexclm2daggerhK 2

i0 Dagger2hellip1Daggersup2dagger2m2hK 2zi0Š hellip48dagger

R l hellip6daggermm hellip0daggerˆiexcl kT

Ihellip dagger3 11Daggersup2

copy5xhellip3x2 iexcl3x Dagger1dagger

Dagger3sup2hellipxhellip5x2 iexcl10x Dagger8daggerDagger3m2hellip5x2 iexcl4daggerdaggerDagger15sup22m2hellipxhellip3x iexcl2daggerDaggerm2hellip3x Dagger1daggerdagger

Dagger15sup23m4hellip3x Daggerm2daggerDagger15sup24m6 iexcl3sup2hellipx iexclm2dagger1Daggersup2

Dagger 1

hellipkTdagger2permil12hK 2

i0hellipxhellip9x iexcl5daggeriexclm2hellip9x iexcl7dagger

Daggersup2permilxhellip9x iexcl5daggeriexclm2hellip9m2 iexcl1daggerŠDaggersup22permilx Daggerm2hellip9x iexcl9m2 iexcl7daggerŠdaggerDaggerhK 2

zi0hellip2x Daggerm2hellip9x iexcl11daggerDaggersup2permil2x Daggerm2hellip27x Dagger9m2 iexcl20daggerŠDagger9sup22m2hellip3x iexcl1Dagger3m2daggerDagger9sup23m2hellipx Dagger3m2daggerDagger9sup24m4daggerDagger1

2helliphhellipLx K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipLyK xdagger2

DaggerhellipL yK ydagger2i0 Daggerhellip1Daggersup2daggerhhellipLzK xdagger2

DaggerhellipL zK ydagger2i0daggerhellip1Daggersup2daggerhellipx iexclm2daggerDaggerhelliphhellipL xK zdagger2

DaggerhellipL yK zdagger2i0 Daggerhellip1Daggersup2daggerhhellipL zK zdagger2i0daggerhellip1Daggersup2dagger3m2

DaggerhK yLx K z iexclK x LyK zi0m2hellip1Daggersup2dagger2

DaggerhK yLzK x iexclK x LzK yi0permilm2hellip5Dagger4sup2daggeriexclxhellip2Daggersup2daggerŠpoundhellip1Daggersup2daggerŠ

Dagger 1kT

permilhLxK x DaggerL yK yi0hellipx iexclm2daggerhellip1iexcl3hellip1Daggersup2dagger

poundhellipx Daggersup2m2daggerDagger2sup2daggerDagger2hL zK zi0hellip1Daggersup2dagger2m2

poundhellip1iexcl3hellip1Daggersup2daggerhellipx Daggersup2m2daggerdaggerŠordf

hellip49dagger

Spectral moments of CFs of asymmetric top molecules 1913

where

x ˆ lhellip l Dagger1dagger sup2 ˆ I=I z iexcl 1 K 2 ˆ K 2

x DaggerK 2y

Here we have taken into account that [14]

L x L y iexcl L yL x ˆ L z

Equations (47) and (48) for the second and the fourthmoments coincide with the results of [2] which wereobtained from the consideration of an ensemble of hin-dered symmetric tops Equation (49) for freely hellipV ˆ 0daggerrotating symmetric tops was obtained in [2] However itwas presented there with several misprints

For spherical tops hellipI x ˆ I y ˆ I z ˆ Idagger we simplyput sup2 ˆ 0 in equations (47)- (49) Thus for m ˆ 0 wehave

R l00hellip0dagger ˆ iexcl

kTI

x hellip50dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 1dagger Dagger 12I2

hK 2i0 hellip51dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 15x Dagger 5 iexcl 3x iexcl 1kT

pound hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 5dagger Dagger4hK 2

z i0

Dagger4hK x L zK y iexcl K yL zK x i0

Dagger hhellipL x K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipL yK xdagger2

DaggerhellipL yK ydagger2 DaggerhellipL zK xdagger2 DaggerhellipL zK ydagger2i0Šordf

hellip52dagger

Equations (50)- (52) are in accordance with the results of[2]

For linear molecules hellipI x ˆ I y ˆ I I z ˆ 0dagger one has

R l00hellip0dagger ˆ iexcl kT

Ix hellip53dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 2dagger Dagger 12I2

hK 2i0 hellip54dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 30x Dagger 24

iexcl 3x iexcl 2kT

hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 1dagger Dagger hhellipL x K xdagger2

DaggerhellipL x K ydagger2 DaggerhellipL yK x dagger2 DaggerhellipL yK ydagger2i0Šordf

hellip55dagger

Equations (53)- (55) are also in agreement with theresults of [2]

6 Spectral moments and coe cients in the Taylorseries expansion of the memory functions

As has been shown on many occasions (eg [4- 6 8])the calculation of the CFs C lhellip tdagger for asymmetric topmolecules can be simpli ed considerably by makinguse of the memory function approach [13] Thisapproach allows one to express the absorpt ion andRaman spectra in terms of the memory functionsK l

nhellip tdagger of arbitrary order n [2 13] The coe cients inthe Taylor series expansions of the memory functionsK l

nhellip tdagger are also related to the spectral moments M ln as

the CFs C lhellip tdagger are connected to the correspondingmemory functions K l

nhellip tdagger by equations (see eg [13])

ddt

C lhellip tdagger ˆ iexclhellip t

0K l

1hellip t iexcl t 0daggerChellip t 0daggerdt 0 hellip56dagger

ddt

K lnhellip tdagger ˆ iexcl

hellip t

0K l

nDagger1hellip t iexcl t 0daggerK lnhellip t 0daggerdt 0

hellip57dagger

In the classical limit the CF C lhellip tdagger and the memory func-tions K l

nhellip tdagger which are both even functions of time t canbe expanded in Taylor series as [13]

C lhellip tdagger ˆX1

kˆ0

Chellip2kdaggerl hellip0dagger t2k

hellip2kdaggerˆ

X1

kˆ0

M l2k

hellip itdagger2k

hellip2kdagger hellip58dagger

K lnhellip tdagger ˆ

X1

kˆ0

K lhellip2kdaggern hellip0dagger t2k

hellip2kdagger hellip59dagger

Here we have taken into account equation (2)Substituting equations (58) and (59) in (56) and (57)one can obtain after some algebraic transformations

C lhellip0daggerK lhellip2ndagger1 hellip0dagger ˆ iexclChellip2nDagger2dagger

l hellip0dagger

iexclXn

sˆ1

Chellip2sdaggerl hellip0daggerK lhellip2hellipniexclsdaggerdagger

1 hellip0dagger hellip60dagger

K lihellip0daggerK lhellip2ndagger

iDagger1 hellip0dagger ˆ iexclK lhellip2nDagger2daggeri hellip0dagger

iexclXn

sˆ1

K lhellip2sdaggeri hellip0daggerK lhellip2hellipniexclsdaggerdagger

iDagger1 hellip0dagger

hellip61dagger

It is convenient to present equations (60) and (61) interms of determinants

1914 Y P Kalmykov and S V Titov

K lhellip2ndagger1 hellip0dagger ˆdet

M l2=Clhellip0dagger 1 0 centcent cent 0

M l4=Clhellip0dagger M l

2=Clhellip0dagger 1

0

M l6=Clhellip0dagger M l

4=Clhellip0dagger M l2=Clhellip0dagger

1

M l2hellipnDagger1dagger=Clhellip0dagger M l

2n=Clhellip0dagger cent centcent M l4=Clhellip0dagger M l

2=Clhellip0dagger

7777777777777777777

7777777777777777777

hellip62dagger

K lihellip0daggerK l hellip2ndagger

iDagger1 hellip0dagger ˆ i2n det

K lhellip2daggeri hellip0dagger 1 0 cent cent cent 0

K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger 1

1

K lhellip2hellipnDagger1daggerdaggeri hellip0dagger K lhellip2ndagger

i hellip0dagger cent cent cent K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger

777777777777777

777777777777777

hellip63dagger

where n ˆ 0 1 2 and

M l2n ˆClhellip0daggerdet

K l1hellip0dagger 1 0 centcentcent 0

K lhellip2dagger1 hellip0dagger K l

1hellip0dagger 1

1

K lhellip2niexcl2dagger1 hellip0dagger K lhellip2hellipniexcl3daggerdagger

1 hellip0dagger centcent cent K lhellip2dagger1 hellip0dagger K l

1hellip0dagger

777777777777777

777777777777777

hellip64dagger

Thus if the moments M l2n are known it is possible to

calculate all the memory functions and their derivativesat t ˆ 0 Conversely if the derivatives of the memoryfunctions K lhellip2ndagger

i hellip0dagger are known it is possible to calculateall the spectral moments M l

2n

7 DiscussionAs we have already mentioned in the analysis of

absorption and scattering spectra of molecular uidsin terms of the moments of the CFs the key problemis the evaluation of the short term behaviour of thecorrelation functions R l

nmhellip tdagger This problem is relatedin turn to the calculation of the coe cients in aTaylor series expansion of R l

n mhellip tdagger The results obtainedin the present work allow one to calculate these coe -cients and hence the spectral moments in a systematicmanner Equations for M l

n obtained in the present paperpermit one also to calculate from equation (64) the co-e cients in Taylor series expansions of the appropriatememory functions Here for simplicity we have con- ned ourselves to the evaluation of the second (M l

2)fourth (M l

4) and sixth (M l6) moments only However

the method used is quite general and it may be extendedreadily for evaluating higher spectral moments as thehigher time derivatives of R l

n mhellip tdagger at t ˆ 0 can be calcu-lated by a similar way For example according to equa-tion (11) on evaluating the fourth derivative of theWigner D function Dl

n mhellipsup3rsquoAacutedagger at t ˆ 0 one can calcu-late the eighth derivative of R lhellip8dagger

n mhellip0dagger etcThe second moment M 1

2 for the dipole absorption ofasymmetric top molecules has been obtained in many

works as within the framework of classical andquantum theory (eg [10 17]) The derivation of thefourth moment M 1

4 for molecules of C2v symmetry wasgiven in [17] The second and fourth moments for anarbitrary asymmetric top molecule have been calculatedby Gordon [11] on using quantum-mechanical methodswith the subsequent transition to the classical limithellip -h 0dagger Our results (equations (38) and (39)) are incomplete agreement with those of Gordon [11] Inmatrix notations of [11] equations (38) and (39) canbe written as

M 12 ˆ kTpermilm2Tr fBg iexcl m cent B cent mTŠ hellip65dagger

M 14 ˆ hellipkTdagger2 fm2permil2hellipTr fBgdagger2 DaggerTr fB2ghellipTr fBgTr fBiexcl1g iexcl 3dagger

iexcl Tr fB3gTr fBiexcl1gŠ

Daggerm cent B2 cent mThellip7 iexcl Tr fBgTr fBiexcl1gdagger

iexcl 6 m cent B cent mTTr fBg Daggerm cent B3 cent mTTr fBiexcl1gg

Daggerm2hK cent B2 cent KTi0 iexcl hhellipm cent B cent KTdagger2i0 hellip66dagger

where B ˆ Iiexcl1 and the upper index T denotes the trans-position

Equations (42) and (43) for the second and fourthRaman spectral moments M 2

2 and M 24 coincide with

the results of Gordon [12] and can be rewritten in thematrix form as follows

M 22 ˆ 2kTpermil2Tr fBgTr fa2g iexcl 3Tr fa cent B cent agŠ hellip67dagger

M 24 ˆ hellipkTdagger2permil27hellipTr fBgdagger2Tr fa2g iexcl 72Tr fa cent B cent agTr fBg

Dagger18Tr fa cent B cent a cent Bg

Dagger6Tr fa cent B2 cent aghellip1 iexcl Tr fBgTr fBiexcl1gdagger

DaggerTr fa2gTr fB2ghellip5 Dagger4Tr fBgTr fBiexcl1gdagger

iexcl 4Tr fa2gTr fB3gTr fBiexcl1g

Dagger6Tr fa cent B3 cent agTr fBiexcl1gŠ

Dagger4hK cent B2 cent KTTr fa2gi0 iexcl 6hK cent B cent a2 cent B cent KT i0

hellip68dagger

It should be noted that there was a misprint in theequation for the fourth moment M 2

4 in [12] equation(58) To our knowledge the sixth moments forthe dipole absorpt ion (equation (40) and Ramanscattering (equation (44)) spectra of an arbitrary asym-metric top molecule have been calculated for the rsttime

Spectral moments of CFs of asymmetric top molecules 1915

The spectral moments allow one to carry out thequantitative analysis of experimental data on IR absorp-tion and Raman spectra of uids with asymmetrictop molecules As is well known the spectral momentM l

2 does not depend on intermolecular interactions Thespectral moment M l

4 depends on the mean-squaretorque acting on the molecule The spectral momentM l

6 includes also averages of angular derivatives ofthe torque Thus the spectral moments allow one toobtain information both about molecular motions andintermolecular interactions Another important prop-erty of spectral moments is the possibility of evaluatingasymptotic properties of spectra For example ifspectral moments M l

2 M l4 and M l

6 exist the integral inthe right hand side of equation (1) exists for n ˆ 2 4and 6 As a result the spectral function I lhellipdagger mustdecrease faster than iexcl6 Moreover the spectralmoments play a rather important role in the theoreticalcalculation of the absorption and scattering spectraand in the modelling of molecular rotation in uidsAs the second spectral moment M l

2 does not dependon the torques acting on molecules (it is exclusivelydetermined by molecular parameters) this momentcalculated in the context of any particular modelshould not depend on parameters describing inter-actions of molecules In other words the secondmoment can be used as a criterion of the correctnessof the model assumptions As the higher momentsdepend on angular derivatives of the intermolecularpotential energy they can be used for the evaluationof model parameters in systems of interacting moleculesFor example the mean-square torque acting on themolecule can be determined from the fourth spectralmoment M l

4 [18] With the help of the spectral momentsM l

2 and M l4 one can evaluate the mean time of molecular

collisions [19 20] Examples of such evaluations forasymmetric top molecules like H 2O and D 2O weregiven in [9]

We thank Professor J L Dejardin for usefulcomments and suggestions The support of this workby the International Association for the Promotionof Co-operation with Scientists from the NewIndependent States of the Former Soviet Union(Project INTAS 96-1411) is gratefully acknowledged

Appendix

Equat ions for R l hellip2kdaggermm hellip0dagger and R l hellip2kdagger

mmsect4hellip0daggerEquations (21)- (32) and (34) yield

R l hellip2daggermm hellip0dagger ˆ iexcl

kT2

permilhellipx iexcl m2daggerhellipI iexcl1x Dagger I iexcl1

y dagger Dagger 2m2I iexcl1z Š hellipA 1dagger

R l hellip2daggermmsect4hellip0dagger ˆ 0 hellipA 2dagger

R l hellip4daggermmhellip0dagger ˆhellipkTdagger2 fhellipx iexcl 3m2daggerpermil2I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y DaggerI iexcl1

x daggerŠ

Daggerm2I iexcl2z hellipI yI

iexcl1x DaggerI x I iexcl1

y iexcl 2dagger

Dagger12hellipx Dagger6m2x iexcl6m4 iexcl5m2daggerI iexcl1

z hellipI iexcl1x DaggerI iexcl1

y dagger

Dagger12hellipx iexcl m2daggerpermilI iexcl1

z hellipI y I iexcl2x DaggerI x I iexcl2

y dagger

DaggerI zIiexcl1x I iexcl1

y hellipI iexcl1x DaggerI iexcl1

y dagger iexcl 2hellipI iexcl2x DaggerI iexcl2

y daggerŠ

Dagger18 permil3hellipx iexclm2dagger2 iexcl2x Dagger5m2Š

pound permil3I iexcl2x Dagger3I iexcl2

y Dagger2I iexcl1x I iexcl1

y Š Dagger3m4I iexcl2z g

Dagger12hellipx iexcl m2daggerhellip I iexcl2

x hK 2x i0 DaggerI iexcl2

y hK 2yi0dagger

Daggerm2I iexcl2z hK 2

z i0 hellipA 3dagger

R lhellip4daggermsect4 mhellip0dagger ˆ 3hellipkTdagger2

16hellipI iexcl1

x iexcl I iexcl1y dagger2

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 4dagger

R lhellip6daggermmhellip0dagger ˆ iexcl kT

16I3x I3

yI3zhellipR lm

0 DaggerkTR lm1 DaggerhellipkTdagger2R lm

2 dagger hellipA 5dagger

R lhellip6daggermsect4 m ˆ iexcl

3kThellipI x iexcl I ydagger32I3

x I 3yI z

f6I zhellipI2x hK 2

yi0 iexcl I2yhK 2

x i0dagger

Dagger4kT I zhellipI2yhL x K x i0 iexcl I2

x hL yK yi0dagger

Dagger5hellipkTdagger2hellipI x iexcl I ydaggerpermil3xhellipI x DaggerI ydaggerI z

Dagger3hellip2I x I y iexclhellipI x DaggerI ydaggerI zdagger

pound mhellipm sect 4dagger

Dagger2hellipI 2x Dagger22I x I y DaggerI 2

y iexcl12hellipI x DaggerI ydaggerI z DaggerI2zdaggerŠg

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 6dagger

where x ˆ lhellip l Dagger1dagger

1916 Y P Kalmykov and S V Titov

R lm0 ˆ 8f2m2I 2

x I2yhI x I zhellipL yK zdagger2 DaggerI y I zhellipL x K zdagger2

DaggerI x I yhellipL zK zdagger2i0 Daggerhellipx iexclm2daggerI2z permilI2

x hI x I zhellipL yK ydagger2

DaggerI yI zhellipL x K ydagger2 DaggerI x I yhellipL zK ydagger2i0 DaggerI2yhI x I zhellipL yK xdagger2

DaggerI yI zhellipL x K xdagger2 DaggerI x I yhellipL zK xdagger2i0Š

Dagger16I x I yI z fhK zL x K yi0permilhellipx iexclm2daggerI x I zhellipI y DaggerI z iexcl I x dagger

iexcl3m2I y I z Š iexclhK zL yK x i0permilhellipx iexclm2daggerI yI zhellipI x DaggerI z iexcl I ydagger

iexcl3m2I x I z ŠDaggerm2I x I ypermilhK x L yK zi0hellip2I x iexcl2I y iexcl I zdagger

iexclhK yL x K zi0hellip2I y iexcl2I x iexcl I zdaggerŠ

DaggerhK x L zK yi0permilhellipx iexclm2daggerI x I zhellip2I y DaggerI z iexcl I xdagger iexcl3m2I x I yŠ

iexclhK yL zK x i0permilhellipx iexclm2daggerI y I zhellip2I x DaggerI z iexcl I ydagger iexcl3m2I x I y Šg

Dagger2hK 2x i0I yI z fx I zpermil16I x I yhellipI y iexcl I x dagger

iexcl2I yI zhellipI x Dagger9I ydagger Dagger4I xhellipI x iexcl I zdagger2

Dagger9xI yI zhellipI x Dagger3I ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I x I zhellipI z iexcl I xdagger2 DaggerI yI zhellip32I2x

iexcl7I xhellip8I y iexcl I zdagger Dagger45I yI zdagger

Dagger18hellipx iexclm2daggerI y I zhellip2I x I y iexcl I x I z iexcl3I yI zdagger

iexcl9m2I y I2zhellipI x Dagger3I ydaggerŠg

Dagger2hK 2yi0I x I z fx I zpermil16I x I yhellipI x iexcl I ydagger

iexcl2I x I zhellip9I x DaggerI ydagger Dagger4I yhellipI y iexcl I zdagger2

Dagger9xI x I zhellip3I x DaggerI ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I yI zhellipI z iexcl I ydagger2 DaggerI x I zhellip32I2y iexcl7I yhellip8I x iexcl I zdagger

Dagger45I x I zdagger Dagger18hellipx iexclm2daggerI x I zhellip2I x I y iexcl3I x I z iexcl I y I zdagger

iexcl9m2I x I 2zhellip3I x DaggerI ydaggerŠg

Dagger8hK 2zi0I x I y fhellipx iexclm2daggerI zpermilI xhellipI x iexcl I zdagger2

DaggerI yhellip I y iexcl I zdagger2 DaggerI x I yhellip4I z iexcl I x iexcl I ydaggerŠ

Daggerm2I x I ypermil9hellipx iexclm2daggerI zhellipI x DaggerI ydagger Dagger18m2I x I y iexcl8I2z Šg

R lm1 ˆ 16hL zK zi0I2

x I2ym

2 f2I2z iexcl 2hellipIx iexcl I ydagger2 iexcl 3xI zhellipIx Dagger I ydagger

Dagger3m2hellipIx I z Dagger I yI z iexcl 2I x I ydaggerg

Dagger4hLx K x i0I2yI

2z fhellipx iexcl m2daggerpermil4I2

x iexcl 4hellipI y iexcl Izdagger2

iexcl 6I zhellipIx iexcl Iydagger iexcl 3x IzhellipI x Dagger3IydaggerŠ

Dagger3m2permil3I zhellipIx iexcl I ydagger DaggerxhellipI zhellipI x Dagger3I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl Ix I z iexcl 3I yI zdaggerŠg

Dagger4hLyK yi0I2x I2

z fhellipx iexcl m2daggerpermil4I2y iexcl 4hellipI x iexcl I zdagger2

iexcl 6I zhellipIy iexcl I xdagger iexcl 3x Izhellip3Ix Dagger IydaggerŠ

Dagger3m2permil3I zhellipIy iexcl Ixdagger DaggerxhellipI zhellip3I x Dagger I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl 3I x I z iexcl I yI zdaggerŠg

R lm2 ˆ 15permil2Ix Iy iexcl IzhellipI x DaggerI ydaggerŠpermil8I2

x I2y iexcl 8I x I yI zhellipIx Dagger I ydagger

Dagger I2zhellip5I2

x iexcl 2Ix I y Dagger5I2ydaggerŠm6

Dagger15f16Ix I yhellipI x iexcl Iydagger2permilI2x I2

y iexcl I zhellipIx Dagger I ydaggerŠ

Dagger2I2zhellip3I4

x Dagger43I3x Iy iexcl 84I2

x I2y Dagger43I x I3

y Dagger3I4ydagger

Dagger3xI zpermil8I2x I2

yhellipI x Dagger I ydagger iexcl 4I x I yI zhellip3I2x Dagger2I x Iy Dagger3I2

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip5I2

x iexcl 2I x I y Dagger5I2ydaggerŠ

iexcl I3zhellipI x Dagger Iydaggerhellip47I2

x iexcl 78I x Iy Dagger47I2ydagger

Dagger2I4zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggergm4

Dagger f16I x IyhellipI x iexcl I ydagger2hellip3I2x iexcl 2I x I y Dagger3I2

ydagger

iexcl 8I zhellipIx Dagger IydaggerhellipIx iexcl I ydagger2hellip3I2x Dagger34I x I y Dagger3I2

ydagger

Dagger2I2zhellipI x iexcl Iydagger2hellip107I2

x Dagger602I x I y Dagger107I2ydagger

iexcl 32I3zhellipIx Dagger Iydaggerhellip25I2

x iexcl 41Ix I y Dagger25I2ydagger

Dagger2I4zhellip107I2

x Dagger98I x Iy Dagger107I2ydagger iexcl 24I5

zhellipI x DaggerI ydagger

Dagger30xI zpermil16Ix I yhellipI x Dagger IydaggerhellipIx iexcl Iydagger2

iexcl 4I zhellip3I4x Dagger19I3

x Iy iexcl 36I2x I2

y Dagger19I x I3y Dagger3I4

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip69I2

x iexcl 106I x Iy Dagger69I2ydagger

iexcl 4I3zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggerŠ

Dagger90I2zx

2permil2I x Iyhellip3I2x Dagger2Ix I y Dagger3I2

ydagger

Spectral moments of CFs of asymmetric top molecules 1917

iexcl I zhellipI x Dagger I ydaggerhellip5I 2x iexcl 2I x I y Dagger5I2

ydaggerŠgm2

Daggerx I z f4hellipI x iexcl I ydagger2permil6hellipI x Dagger I ydagger3

iexcl I zhellip31I2x Dagger106I x I y Dagger 31I2

ydaggerŠ

Dagger64I2zhellipI x Dagger I ydaggerhellip5I2

x iexcl 7I x I y Dagger5I2ydagger

iexcl 4I3zhellip31I2

x Dagger26I x I y Dagger31I2ydagger Dagger24I4

zhellipI x Dagger I ydagger

Dagger15xI zpermil2hellipI x iexcl I ydagger2hellip3I x Dagger I ydaggerhellip I x Dagger3I ydagger

iexcl 2I zhellipI x Dagger I ydaggerhellip11I2x iexcl 14I x I y Dagger11I2

ydagger

Dagger2I2zhellip3I 2

x Dagger2I x I y Dagger 3I 2ydagger

Daggerx I zhellipI x Dagger I ydaggerhellip5I2x iexcl 2I x I y Dagger5I2

ydaggerŠg

In this derivation we have assumed that

hK ii0 ˆ 0 hellip i ˆ x y zdaggerand

hK iK ji0 ˆ 0 hL iK ji0 ˆ 0 for i 6ˆ j hellip i j ˆ x y zdagger

References[1] BURHSTEIN A I and TEMKIN S I 1994 Spectroscopy

of M olecular Rotation in Gases and Liquids (CambridgeUniversity Press)

[2] ST PIERRE A G and STEELE W A 1981 M olecPhys 43 123

[3] GORDON R G 1963 J chem Phys 39 2788

[4] LEICKMAN JCL and GUISSANIY 1981 M olec Phys42 1105

[5] AGUADO-GOMEZM and LEICKMAN JCL 1986 PhysRev A 34 4195

[6] AGUADO-GOMEZM and LEICKMAN JCL 1987 PhysRev A 35 286

[7] BORISEVICH N A BLOKHINA PZALESSKAYAG ALASTOCHKINA V A and SHUKUROV T 1984 IzvAkad Nauk SSSR Ser Fiz 48 709

[8] KALMYKOV YUP 1985 Opt Spektrosk 58 804 [1985Opt Spectrosc 58 493]

[9] KALMYKOV YU P and TITOV S V 1999 J molecS truct 479 123

[10] GORDON R G 1963 J chem Phys 38 1724[11] GORDON R G 1964 J chem Phys 41 1819[12] GORDON R G 1964 J chem Phys 40 1973[13] BERNE B J and HARP G D 1970 Adv chem Phys

17 63[14] LEICKMAN JCL and GUISSANIY 1984 M olec Phys

53 761[15] VARSHALOVICH D A MOSKALEV A N and

KHERSONSKII V K 1988 Quantum Theory of AngularM omentum (Singapore World Scienti c)

[16] LANDAULD and LIFSHITZEM 1976 M echanics ACourse of theoretical Physics 3rd Edn Vol 1 (OxfordPergamon Press)

[17] BOiumlTTCHERC JF and BORDEWIJK P 1979 Theory ofElectric Polarization Vol 2 (Amsterdam Elsevier)

[18] RODRIGUEZ R and MCHALE J L 1988 J chemPhys 88 2264

[19] BURSHTEINA I and MCCONNELL J R 1989 PhysicaA 157 933

[20] KALMYKOV YU P and MCCONNELL J R 1993Physica A 193 394

1918 Spectral moments of CFs of asymmetric top molecules

M xy ˆ Ix IzhellipI2x I2

zhhellipLyK ydagger2i0 DaggerI2yI

2zhhellipL yK xdagger2i0

Dagger4I2x I2

yhhellipL yK zdagger2i0dagger DaggerI yIzhellipI2x I2

zhhellipL xK ydagger2i0

DaggerI2yI

2zhhellipL xK xdagger2i0 Dagger4I2

x I2yhhellipL x K zdagger2i0dagger

DaggerI x I yhellipI2x I2

z hhellipL zK ydagger2i0 DaggerI2yI

2zhhellipLzK xdagger2i0

Dagger4I2x I2

yhhellipL zK zdagger2i0dagger DaggerIyI zhellipI3xhellip4I y DaggerI zdagger

iexcl2I2xhellip4I2

y iexcl2I yIz DaggerI2zdagger DaggerI xhellip4I3

y Dagger20I2yIz

Dagger20I yI2z DaggerI3

zdagger Dagger9I2yI

2zdaggerhK 2

xi0 DaggerIx IzhellipI3yhellip4Ix DaggerIzdagger

iexcl2I2yhellip4I2

x iexcl2I x Iz DaggerI2zdagger

DaggerI yhellip4I3x Dagger20I2

x Iz Dagger20Ix I2z DaggerI3

zdagger Dagger9I2x I2

zdaggerhK 2yi0

DaggerI x I yhellipI3zhellipIx DaggerI zdagger iexcl2I2

zhellipI2x Dagger10Ix Iy DaggerI2

ydagger

DaggerI zhellipI x DaggerIydaggerhellipI2x Dagger34Ix Iy DaggerI2

ydagger Dagger144I2x I2

ydaggerhK 2zi0

Dagger2I x I yIzpermil2I x I yhelliphK x LyK zi0hellip2I x iexcl2I y iexcl Izdagger

DaggerhK yLx K zi0hellip2I x iexcl2I y DaggerI zdaggerdagger

DaggerI yIzhelliphK yL zK xi0hellip4Ix DaggerIy iexcl Izdagger

DaggerhK zLyK x i0hellip5I x DaggerIy iexcl I zdaggerdagger

iexclI x I zhelliphK xL zK yi0hellipIx Dagger4Iy iexcl I zdagger

DaggerhK zLx K yi0hellipIx Dagger5Iy iexcl I zdaggerdaggerŠ

iexcl2kTpermilhL yK yi0I2x I2

zhellipI2z Dagger3I yIz iexcl I2

y DaggerI xhellipI x Dagger12I y DaggerIzdaggerdagger

DaggerhLx K xi0I2yI

2zhellipI2

z Dagger3I x I z iexcl I2x DaggerI yhellipIy Dagger12Ix DaggerIzdaggerdagger

Dagger4hLzK zi0I2x I2

yhellipI2x Dagger10Ix Iy DaggerI2

y DaggerIzhellip3Ix Dagger3I y iexcl IzdaggerdaggerŠ

DaggerhellipkTdagger2permil3I5zhellipI x DaggerI ydagger Dagger7I4

zhellipI2x Dagger10I x Iy DaggerI2

ydagger

iexcl5I3zhellipI x DaggerI ydaggerhellipI x iexcl Iydagger2

DaggerI2zhellip7I4

x Dagger68I3x Iy Dagger90I2

x I2y Dagger68Ix I3

y Dagger7I4ydagger

DaggerI zhellipI x DaggerIydagger

poundhellip3I4x Dagger64I3

x Iy Dagger586I2x I2

y Dagger64I x I3y Dagger3I4

ydagger

Dagger4I x I yhellip3I4x Dagger52I3

x Iy Dagger130I2x I2

y Dagger52Ix I3y Dagger3I4

ydaggerŠ hellip45dagger

N xy ˆ 9I2zhellipI x iexcl Iydaggerf3I zhellipI2

xhK 2yi0 iexcl I2

yhK 2x i0dagger

iexcl2kT IzhellipI2xhL yK yi0 iexcl I2

yhLx K xi0dagger

DaggerhellipkTdagger2permil5hellipIx iexcl IydaggerhellipI2x Dagger10Ix Iy DaggerI2

y

Dagger3I zhellipIx DaggerI ydagger DaggerI2zdaggerŠg hellip46dagger

Equations for M yz M zx and N yz N zx in equation (44)can be obtained readily by changing all the indices inequations (45) and (46) from fx y zg to fy z xg andfz x yg respectively

5 Linear spherical and symmetric top moleculesThe results we have obtained contain the particular

cases of classical ensembles of symmetric tops sphericaltops and linear molecules In these cases the theory isconsiderably simpli ed as the calculation of all the spec-tral moments M l

2k needs only that of R l hellip2kdaggermm hellip0dagger [2] Thus

for symmetric top molecules (I x ˆ I y ˆ I 6ˆ I zdagger equa-tions (A 1) (A 3) and (A 5) from the appendix yield

R lmmhellip0daggerˆiexclkT

Ipermilx Daggersup2m2Š hellip47dagger

R l hellip4daggermm hellip0daggerˆ kT

Ihellip dagger2

xhellip3x iexcl1daggerDaggersup2 6xm2 Daggerm2 iexclx1Daggersup2hellip daggerDagger3sup22m4

Dagger 12I2permilhellipx iexclm2daggerhK 2

i0 Dagger2hellip1Daggersup2dagger2m2hK 2zi0Š hellip48dagger

R l hellip6daggermm hellip0daggerˆiexcl kT

Ihellip dagger3 11Daggersup2

copy5xhellip3x2 iexcl3x Dagger1dagger

Dagger3sup2hellipxhellip5x2 iexcl10x Dagger8daggerDagger3m2hellip5x2 iexcl4daggerdaggerDagger15sup22m2hellipxhellip3x iexcl2daggerDaggerm2hellip3x Dagger1daggerdagger

Dagger15sup23m4hellip3x Daggerm2daggerDagger15sup24m6 iexcl3sup2hellipx iexclm2dagger1Daggersup2

Dagger 1

hellipkTdagger2permil12hK 2

i0hellipxhellip9x iexcl5daggeriexclm2hellip9x iexcl7dagger

Daggersup2permilxhellip9x iexcl5daggeriexclm2hellip9m2 iexcl1daggerŠDaggersup22permilx Daggerm2hellip9x iexcl9m2 iexcl7daggerŠdaggerDaggerhK 2

zi0hellip2x Daggerm2hellip9x iexcl11daggerDaggersup2permil2x Daggerm2hellip27x Dagger9m2 iexcl20daggerŠDagger9sup22m2hellip3x iexcl1Dagger3m2daggerDagger9sup23m2hellipx Dagger3m2daggerDagger9sup24m4daggerDagger1

2helliphhellipLx K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipLyK xdagger2

DaggerhellipL yK ydagger2i0 Daggerhellip1Daggersup2daggerhhellipLzK xdagger2

DaggerhellipL zK ydagger2i0daggerhellip1Daggersup2daggerhellipx iexclm2daggerDaggerhelliphhellipL xK zdagger2

DaggerhellipL yK zdagger2i0 Daggerhellip1Daggersup2daggerhhellipL zK zdagger2i0daggerhellip1Daggersup2dagger3m2

DaggerhK yLx K z iexclK x LyK zi0m2hellip1Daggersup2dagger2

DaggerhK yLzK x iexclK x LzK yi0permilm2hellip5Dagger4sup2daggeriexclxhellip2Daggersup2daggerŠpoundhellip1Daggersup2daggerŠ

Dagger 1kT

permilhLxK x DaggerL yK yi0hellipx iexclm2daggerhellip1iexcl3hellip1Daggersup2dagger

poundhellipx Daggersup2m2daggerDagger2sup2daggerDagger2hL zK zi0hellip1Daggersup2dagger2m2

poundhellip1iexcl3hellip1Daggersup2daggerhellipx Daggersup2m2daggerdaggerŠordf

hellip49dagger

Spectral moments of CFs of asymmetric top molecules 1913

where

x ˆ lhellip l Dagger1dagger sup2 ˆ I=I z iexcl 1 K 2 ˆ K 2

x DaggerK 2y

Here we have taken into account that [14]

L x L y iexcl L yL x ˆ L z

Equations (47) and (48) for the second and the fourthmoments coincide with the results of [2] which wereobtained from the consideration of an ensemble of hin-dered symmetric tops Equation (49) for freely hellipV ˆ 0daggerrotating symmetric tops was obtained in [2] However itwas presented there with several misprints

For spherical tops hellipI x ˆ I y ˆ I z ˆ Idagger we simplyput sup2 ˆ 0 in equations (47)- (49) Thus for m ˆ 0 wehave

R l00hellip0dagger ˆ iexcl

kTI

x hellip50dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 1dagger Dagger 12I2

hK 2i0 hellip51dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 15x Dagger 5 iexcl 3x iexcl 1kT

pound hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 5dagger Dagger4hK 2

z i0

Dagger4hK x L zK y iexcl K yL zK x i0

Dagger hhellipL x K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipL yK xdagger2

DaggerhellipL yK ydagger2 DaggerhellipL zK xdagger2 DaggerhellipL zK ydagger2i0Šordf

hellip52dagger

Equations (50)- (52) are in accordance with the results of[2]

For linear molecules hellipI x ˆ I y ˆ I I z ˆ 0dagger one has

R l00hellip0dagger ˆ iexcl kT

Ix hellip53dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 2dagger Dagger 12I2

hK 2i0 hellip54dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 30x Dagger 24

iexcl 3x iexcl 2kT

hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 1dagger Dagger hhellipL x K xdagger2

DaggerhellipL x K ydagger2 DaggerhellipL yK x dagger2 DaggerhellipL yK ydagger2i0Šordf

hellip55dagger

Equations (53)- (55) are also in agreement with theresults of [2]

6 Spectral moments and coe cients in the Taylorseries expansion of the memory functions

As has been shown on many occasions (eg [4- 6 8])the calculation of the CFs C lhellip tdagger for asymmetric topmolecules can be simpli ed considerably by makinguse of the memory function approach [13] Thisapproach allows one to express the absorpt ion andRaman spectra in terms of the memory functionsK l

nhellip tdagger of arbitrary order n [2 13] The coe cients inthe Taylor series expansions of the memory functionsK l

nhellip tdagger are also related to the spectral moments M ln as

the CFs C lhellip tdagger are connected to the correspondingmemory functions K l

nhellip tdagger by equations (see eg [13])

ddt

C lhellip tdagger ˆ iexclhellip t

0K l

1hellip t iexcl t 0daggerChellip t 0daggerdt 0 hellip56dagger

ddt

K lnhellip tdagger ˆ iexcl

hellip t

0K l

nDagger1hellip t iexcl t 0daggerK lnhellip t 0daggerdt 0

hellip57dagger

In the classical limit the CF C lhellip tdagger and the memory func-tions K l

nhellip tdagger which are both even functions of time t canbe expanded in Taylor series as [13]

C lhellip tdagger ˆX1

kˆ0

Chellip2kdaggerl hellip0dagger t2k

hellip2kdaggerˆ

X1

kˆ0

M l2k

hellip itdagger2k

hellip2kdagger hellip58dagger

K lnhellip tdagger ˆ

X1

kˆ0

K lhellip2kdaggern hellip0dagger t2k

hellip2kdagger hellip59dagger

Here we have taken into account equation (2)Substituting equations (58) and (59) in (56) and (57)one can obtain after some algebraic transformations

C lhellip0daggerK lhellip2ndagger1 hellip0dagger ˆ iexclChellip2nDagger2dagger

l hellip0dagger

iexclXn

sˆ1

Chellip2sdaggerl hellip0daggerK lhellip2hellipniexclsdaggerdagger

1 hellip0dagger hellip60dagger

K lihellip0daggerK lhellip2ndagger

iDagger1 hellip0dagger ˆ iexclK lhellip2nDagger2daggeri hellip0dagger

iexclXn

sˆ1

K lhellip2sdaggeri hellip0daggerK lhellip2hellipniexclsdaggerdagger

iDagger1 hellip0dagger

hellip61dagger

It is convenient to present equations (60) and (61) interms of determinants

1914 Y P Kalmykov and S V Titov

K lhellip2ndagger1 hellip0dagger ˆdet

M l2=Clhellip0dagger 1 0 centcent cent 0

M l4=Clhellip0dagger M l

2=Clhellip0dagger 1

0

M l6=Clhellip0dagger M l

4=Clhellip0dagger M l2=Clhellip0dagger

1

M l2hellipnDagger1dagger=Clhellip0dagger M l

2n=Clhellip0dagger cent centcent M l4=Clhellip0dagger M l

2=Clhellip0dagger

7777777777777777777

7777777777777777777

hellip62dagger

K lihellip0daggerK l hellip2ndagger

iDagger1 hellip0dagger ˆ i2n det

K lhellip2daggeri hellip0dagger 1 0 cent cent cent 0

K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger 1

1

K lhellip2hellipnDagger1daggerdaggeri hellip0dagger K lhellip2ndagger

i hellip0dagger cent cent cent K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger

777777777777777

777777777777777

hellip63dagger

where n ˆ 0 1 2 and

M l2n ˆClhellip0daggerdet

K l1hellip0dagger 1 0 centcentcent 0

K lhellip2dagger1 hellip0dagger K l

1hellip0dagger 1

1

K lhellip2niexcl2dagger1 hellip0dagger K lhellip2hellipniexcl3daggerdagger

1 hellip0dagger centcent cent K lhellip2dagger1 hellip0dagger K l

1hellip0dagger

777777777777777

777777777777777

hellip64dagger

Thus if the moments M l2n are known it is possible to

calculate all the memory functions and their derivativesat t ˆ 0 Conversely if the derivatives of the memoryfunctions K lhellip2ndagger

i hellip0dagger are known it is possible to calculateall the spectral moments M l

2n

7 DiscussionAs we have already mentioned in the analysis of

absorption and scattering spectra of molecular uidsin terms of the moments of the CFs the key problemis the evaluation of the short term behaviour of thecorrelation functions R l

nmhellip tdagger This problem is relatedin turn to the calculation of the coe cients in aTaylor series expansion of R l

n mhellip tdagger The results obtainedin the present work allow one to calculate these coe -cients and hence the spectral moments in a systematicmanner Equations for M l

n obtained in the present paperpermit one also to calculate from equation (64) the co-e cients in Taylor series expansions of the appropriatememory functions Here for simplicity we have con- ned ourselves to the evaluation of the second (M l

2)fourth (M l

4) and sixth (M l6) moments only However

the method used is quite general and it may be extendedreadily for evaluating higher spectral moments as thehigher time derivatives of R l

n mhellip tdagger at t ˆ 0 can be calcu-lated by a similar way For example according to equa-tion (11) on evaluating the fourth derivative of theWigner D function Dl

n mhellipsup3rsquoAacutedagger at t ˆ 0 one can calcu-late the eighth derivative of R lhellip8dagger

n mhellip0dagger etcThe second moment M 1

2 for the dipole absorption ofasymmetric top molecules has been obtained in many

works as within the framework of classical andquantum theory (eg [10 17]) The derivation of thefourth moment M 1

4 for molecules of C2v symmetry wasgiven in [17] The second and fourth moments for anarbitrary asymmetric top molecule have been calculatedby Gordon [11] on using quantum-mechanical methodswith the subsequent transition to the classical limithellip -h 0dagger Our results (equations (38) and (39)) are incomplete agreement with those of Gordon [11] Inmatrix notations of [11] equations (38) and (39) canbe written as

M 12 ˆ kTpermilm2Tr fBg iexcl m cent B cent mTŠ hellip65dagger

M 14 ˆ hellipkTdagger2 fm2permil2hellipTr fBgdagger2 DaggerTr fB2ghellipTr fBgTr fBiexcl1g iexcl 3dagger

iexcl Tr fB3gTr fBiexcl1gŠ

Daggerm cent B2 cent mThellip7 iexcl Tr fBgTr fBiexcl1gdagger

iexcl 6 m cent B cent mTTr fBg Daggerm cent B3 cent mTTr fBiexcl1gg

Daggerm2hK cent B2 cent KTi0 iexcl hhellipm cent B cent KTdagger2i0 hellip66dagger

where B ˆ Iiexcl1 and the upper index T denotes the trans-position

Equations (42) and (43) for the second and fourthRaman spectral moments M 2

2 and M 24 coincide with

the results of Gordon [12] and can be rewritten in thematrix form as follows

M 22 ˆ 2kTpermil2Tr fBgTr fa2g iexcl 3Tr fa cent B cent agŠ hellip67dagger

M 24 ˆ hellipkTdagger2permil27hellipTr fBgdagger2Tr fa2g iexcl 72Tr fa cent B cent agTr fBg

Dagger18Tr fa cent B cent a cent Bg

Dagger6Tr fa cent B2 cent aghellip1 iexcl Tr fBgTr fBiexcl1gdagger

DaggerTr fa2gTr fB2ghellip5 Dagger4Tr fBgTr fBiexcl1gdagger

iexcl 4Tr fa2gTr fB3gTr fBiexcl1g

Dagger6Tr fa cent B3 cent agTr fBiexcl1gŠ

Dagger4hK cent B2 cent KTTr fa2gi0 iexcl 6hK cent B cent a2 cent B cent KT i0

hellip68dagger

It should be noted that there was a misprint in theequation for the fourth moment M 2

4 in [12] equation(58) To our knowledge the sixth moments forthe dipole absorpt ion (equation (40) and Ramanscattering (equation (44)) spectra of an arbitrary asym-metric top molecule have been calculated for the rsttime

Spectral moments of CFs of asymmetric top molecules 1915

The spectral moments allow one to carry out thequantitative analysis of experimental data on IR absorp-tion and Raman spectra of uids with asymmetrictop molecules As is well known the spectral momentM l

2 does not depend on intermolecular interactions Thespectral moment M l

4 depends on the mean-squaretorque acting on the molecule The spectral momentM l

6 includes also averages of angular derivatives ofthe torque Thus the spectral moments allow one toobtain information both about molecular motions andintermolecular interactions Another important prop-erty of spectral moments is the possibility of evaluatingasymptotic properties of spectra For example ifspectral moments M l

2 M l4 and M l

6 exist the integral inthe right hand side of equation (1) exists for n ˆ 2 4and 6 As a result the spectral function I lhellipdagger mustdecrease faster than iexcl6 Moreover the spectralmoments play a rather important role in the theoreticalcalculation of the absorption and scattering spectraand in the modelling of molecular rotation in uidsAs the second spectral moment M l

2 does not dependon the torques acting on molecules (it is exclusivelydetermined by molecular parameters) this momentcalculated in the context of any particular modelshould not depend on parameters describing inter-actions of molecules In other words the secondmoment can be used as a criterion of the correctnessof the model assumptions As the higher momentsdepend on angular derivatives of the intermolecularpotential energy they can be used for the evaluationof model parameters in systems of interacting moleculesFor example the mean-square torque acting on themolecule can be determined from the fourth spectralmoment M l

4 [18] With the help of the spectral momentsM l

2 and M l4 one can evaluate the mean time of molecular

collisions [19 20] Examples of such evaluations forasymmetric top molecules like H 2O and D 2O weregiven in [9]

We thank Professor J L Dejardin for usefulcomments and suggestions The support of this workby the International Association for the Promotionof Co-operation with Scientists from the NewIndependent States of the Former Soviet Union(Project INTAS 96-1411) is gratefully acknowledged

Appendix

Equat ions for R l hellip2kdaggermm hellip0dagger and R l hellip2kdagger

mmsect4hellip0daggerEquations (21)- (32) and (34) yield

R l hellip2daggermm hellip0dagger ˆ iexcl

kT2

permilhellipx iexcl m2daggerhellipI iexcl1x Dagger I iexcl1

y dagger Dagger 2m2I iexcl1z Š hellipA 1dagger

R l hellip2daggermmsect4hellip0dagger ˆ 0 hellipA 2dagger

R l hellip4daggermmhellip0dagger ˆhellipkTdagger2 fhellipx iexcl 3m2daggerpermil2I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y DaggerI iexcl1

x daggerŠ

Daggerm2I iexcl2z hellipI yI

iexcl1x DaggerI x I iexcl1

y iexcl 2dagger

Dagger12hellipx Dagger6m2x iexcl6m4 iexcl5m2daggerI iexcl1

z hellipI iexcl1x DaggerI iexcl1

y dagger

Dagger12hellipx iexcl m2daggerpermilI iexcl1

z hellipI y I iexcl2x DaggerI x I iexcl2

y dagger

DaggerI zIiexcl1x I iexcl1

y hellipI iexcl1x DaggerI iexcl1

y dagger iexcl 2hellipI iexcl2x DaggerI iexcl2

y daggerŠ

Dagger18 permil3hellipx iexclm2dagger2 iexcl2x Dagger5m2Š

pound permil3I iexcl2x Dagger3I iexcl2

y Dagger2I iexcl1x I iexcl1

y Š Dagger3m4I iexcl2z g

Dagger12hellipx iexcl m2daggerhellip I iexcl2

x hK 2x i0 DaggerI iexcl2

y hK 2yi0dagger

Daggerm2I iexcl2z hK 2

z i0 hellipA 3dagger

R lhellip4daggermsect4 mhellip0dagger ˆ 3hellipkTdagger2

16hellipI iexcl1

x iexcl I iexcl1y dagger2

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 4dagger

R lhellip6daggermmhellip0dagger ˆ iexcl kT

16I3x I3

yI3zhellipR lm

0 DaggerkTR lm1 DaggerhellipkTdagger2R lm

2 dagger hellipA 5dagger

R lhellip6daggermsect4 m ˆ iexcl

3kThellipI x iexcl I ydagger32I3

x I 3yI z

f6I zhellipI2x hK 2

yi0 iexcl I2yhK 2

x i0dagger

Dagger4kT I zhellipI2yhL x K x i0 iexcl I2

x hL yK yi0dagger

Dagger5hellipkTdagger2hellipI x iexcl I ydaggerpermil3xhellipI x DaggerI ydaggerI z

Dagger3hellip2I x I y iexclhellipI x DaggerI ydaggerI zdagger

pound mhellipm sect 4dagger

Dagger2hellipI 2x Dagger22I x I y DaggerI 2

y iexcl12hellipI x DaggerI ydaggerI z DaggerI2zdaggerŠg

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 6dagger

where x ˆ lhellip l Dagger1dagger

1916 Y P Kalmykov and S V Titov

R lm0 ˆ 8f2m2I 2

x I2yhI x I zhellipL yK zdagger2 DaggerI y I zhellipL x K zdagger2

DaggerI x I yhellipL zK zdagger2i0 Daggerhellipx iexclm2daggerI2z permilI2

x hI x I zhellipL yK ydagger2

DaggerI yI zhellipL x K ydagger2 DaggerI x I yhellipL zK ydagger2i0 DaggerI2yhI x I zhellipL yK xdagger2

DaggerI yI zhellipL x K xdagger2 DaggerI x I yhellipL zK xdagger2i0Š

Dagger16I x I yI z fhK zL x K yi0permilhellipx iexclm2daggerI x I zhellipI y DaggerI z iexcl I x dagger

iexcl3m2I y I z Š iexclhK zL yK x i0permilhellipx iexclm2daggerI yI zhellipI x DaggerI z iexcl I ydagger

iexcl3m2I x I z ŠDaggerm2I x I ypermilhK x L yK zi0hellip2I x iexcl2I y iexcl I zdagger

iexclhK yL x K zi0hellip2I y iexcl2I x iexcl I zdaggerŠ

DaggerhK x L zK yi0permilhellipx iexclm2daggerI x I zhellip2I y DaggerI z iexcl I xdagger iexcl3m2I x I yŠ

iexclhK yL zK x i0permilhellipx iexclm2daggerI y I zhellip2I x DaggerI z iexcl I ydagger iexcl3m2I x I y Šg

Dagger2hK 2x i0I yI z fx I zpermil16I x I yhellipI y iexcl I x dagger

iexcl2I yI zhellipI x Dagger9I ydagger Dagger4I xhellipI x iexcl I zdagger2

Dagger9xI yI zhellipI x Dagger3I ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I x I zhellipI z iexcl I xdagger2 DaggerI yI zhellip32I2x

iexcl7I xhellip8I y iexcl I zdagger Dagger45I yI zdagger

Dagger18hellipx iexclm2daggerI y I zhellip2I x I y iexcl I x I z iexcl3I yI zdagger

iexcl9m2I y I2zhellipI x Dagger3I ydaggerŠg

Dagger2hK 2yi0I x I z fx I zpermil16I x I yhellipI x iexcl I ydagger

iexcl2I x I zhellip9I x DaggerI ydagger Dagger4I yhellipI y iexcl I zdagger2

Dagger9xI x I zhellip3I x DaggerI ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I yI zhellipI z iexcl I ydagger2 DaggerI x I zhellip32I2y iexcl7I yhellip8I x iexcl I zdagger

Dagger45I x I zdagger Dagger18hellipx iexclm2daggerI x I zhellip2I x I y iexcl3I x I z iexcl I y I zdagger

iexcl9m2I x I 2zhellip3I x DaggerI ydaggerŠg

Dagger8hK 2zi0I x I y fhellipx iexclm2daggerI zpermilI xhellipI x iexcl I zdagger2

DaggerI yhellip I y iexcl I zdagger2 DaggerI x I yhellip4I z iexcl I x iexcl I ydaggerŠ

Daggerm2I x I ypermil9hellipx iexclm2daggerI zhellipI x DaggerI ydagger Dagger18m2I x I y iexcl8I2z Šg

R lm1 ˆ 16hL zK zi0I2

x I2ym

2 f2I2z iexcl 2hellipIx iexcl I ydagger2 iexcl 3xI zhellipIx Dagger I ydagger

Dagger3m2hellipIx I z Dagger I yI z iexcl 2I x I ydaggerg

Dagger4hLx K x i0I2yI

2z fhellipx iexcl m2daggerpermil4I2

x iexcl 4hellipI y iexcl Izdagger2

iexcl 6I zhellipIx iexcl Iydagger iexcl 3x IzhellipI x Dagger3IydaggerŠ

Dagger3m2permil3I zhellipIx iexcl I ydagger DaggerxhellipI zhellipI x Dagger3I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl Ix I z iexcl 3I yI zdaggerŠg

Dagger4hLyK yi0I2x I2

z fhellipx iexcl m2daggerpermil4I2y iexcl 4hellipI x iexcl I zdagger2

iexcl 6I zhellipIy iexcl I xdagger iexcl 3x Izhellip3Ix Dagger IydaggerŠ

Dagger3m2permil3I zhellipIy iexcl Ixdagger DaggerxhellipI zhellip3I x Dagger I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl 3I x I z iexcl I yI zdaggerŠg

R lm2 ˆ 15permil2Ix Iy iexcl IzhellipI x DaggerI ydaggerŠpermil8I2

x I2y iexcl 8I x I yI zhellipIx Dagger I ydagger

Dagger I2zhellip5I2

x iexcl 2Ix I y Dagger5I2ydaggerŠm6

Dagger15f16Ix I yhellipI x iexcl Iydagger2permilI2x I2

y iexcl I zhellipIx Dagger I ydaggerŠ

Dagger2I2zhellip3I4

x Dagger43I3x Iy iexcl 84I2

x I2y Dagger43I x I3

y Dagger3I4ydagger

Dagger3xI zpermil8I2x I2

yhellipI x Dagger I ydagger iexcl 4I x I yI zhellip3I2x Dagger2I x Iy Dagger3I2

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip5I2

x iexcl 2I x I y Dagger5I2ydaggerŠ

iexcl I3zhellipI x Dagger Iydaggerhellip47I2

x iexcl 78I x Iy Dagger47I2ydagger

Dagger2I4zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggergm4

Dagger f16I x IyhellipI x iexcl I ydagger2hellip3I2x iexcl 2I x I y Dagger3I2

ydagger

iexcl 8I zhellipIx Dagger IydaggerhellipIx iexcl I ydagger2hellip3I2x Dagger34I x I y Dagger3I2

ydagger

Dagger2I2zhellipI x iexcl Iydagger2hellip107I2

x Dagger602I x I y Dagger107I2ydagger

iexcl 32I3zhellipIx Dagger Iydaggerhellip25I2

x iexcl 41Ix I y Dagger25I2ydagger

Dagger2I4zhellip107I2

x Dagger98I x Iy Dagger107I2ydagger iexcl 24I5

zhellipI x DaggerI ydagger

Dagger30xI zpermil16Ix I yhellipI x Dagger IydaggerhellipIx iexcl Iydagger2

iexcl 4I zhellip3I4x Dagger19I3

x Iy iexcl 36I2x I2

y Dagger19I x I3y Dagger3I4

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip69I2

x iexcl 106I x Iy Dagger69I2ydagger

iexcl 4I3zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggerŠ

Dagger90I2zx

2permil2I x Iyhellip3I2x Dagger2Ix I y Dagger3I2

ydagger

Spectral moments of CFs of asymmetric top molecules 1917

iexcl I zhellipI x Dagger I ydaggerhellip5I 2x iexcl 2I x I y Dagger5I2

ydaggerŠgm2

Daggerx I z f4hellipI x iexcl I ydagger2permil6hellipI x Dagger I ydagger3

iexcl I zhellip31I2x Dagger106I x I y Dagger 31I2

ydaggerŠ

Dagger64I2zhellipI x Dagger I ydaggerhellip5I2

x iexcl 7I x I y Dagger5I2ydagger

iexcl 4I3zhellip31I2

x Dagger26I x I y Dagger31I2ydagger Dagger24I4

zhellipI x Dagger I ydagger

Dagger15xI zpermil2hellipI x iexcl I ydagger2hellip3I x Dagger I ydaggerhellip I x Dagger3I ydagger

iexcl 2I zhellipI x Dagger I ydaggerhellip11I2x iexcl 14I x I y Dagger11I2

ydagger

Dagger2I2zhellip3I 2

x Dagger2I x I y Dagger 3I 2ydagger

Daggerx I zhellipI x Dagger I ydaggerhellip5I2x iexcl 2I x I y Dagger5I2

ydaggerŠg

In this derivation we have assumed that

hK ii0 ˆ 0 hellip i ˆ x y zdaggerand

hK iK ji0 ˆ 0 hL iK ji0 ˆ 0 for i 6ˆ j hellip i j ˆ x y zdagger

References[1] BURHSTEIN A I and TEMKIN S I 1994 Spectroscopy

of M olecular Rotation in Gases and Liquids (CambridgeUniversity Press)

[2] ST PIERRE A G and STEELE W A 1981 M olecPhys 43 123

[3] GORDON R G 1963 J chem Phys 39 2788

[4] LEICKMAN JCL and GUISSANIY 1981 M olec Phys42 1105

[5] AGUADO-GOMEZM and LEICKMAN JCL 1986 PhysRev A 34 4195

[6] AGUADO-GOMEZM and LEICKMAN JCL 1987 PhysRev A 35 286

[7] BORISEVICH N A BLOKHINA PZALESSKAYAG ALASTOCHKINA V A and SHUKUROV T 1984 IzvAkad Nauk SSSR Ser Fiz 48 709

[8] KALMYKOV YUP 1985 Opt Spektrosk 58 804 [1985Opt Spectrosc 58 493]

[9] KALMYKOV YU P and TITOV S V 1999 J molecS truct 479 123

[10] GORDON R G 1963 J chem Phys 38 1724[11] GORDON R G 1964 J chem Phys 41 1819[12] GORDON R G 1964 J chem Phys 40 1973[13] BERNE B J and HARP G D 1970 Adv chem Phys

17 63[14] LEICKMAN JCL and GUISSANIY 1984 M olec Phys

53 761[15] VARSHALOVICH D A MOSKALEV A N and

KHERSONSKII V K 1988 Quantum Theory of AngularM omentum (Singapore World Scienti c)

[16] LANDAULD and LIFSHITZEM 1976 M echanics ACourse of theoretical Physics 3rd Edn Vol 1 (OxfordPergamon Press)

[17] BOiumlTTCHERC JF and BORDEWIJK P 1979 Theory ofElectric Polarization Vol 2 (Amsterdam Elsevier)

[18] RODRIGUEZ R and MCHALE J L 1988 J chemPhys 88 2264

[19] BURSHTEINA I and MCCONNELL J R 1989 PhysicaA 157 933

[20] KALMYKOV YU P and MCCONNELL J R 1993Physica A 193 394

1918 Spectral moments of CFs of asymmetric top molecules

where

x ˆ lhellip l Dagger1dagger sup2 ˆ I=I z iexcl 1 K 2 ˆ K 2

x DaggerK 2y

Here we have taken into account that [14]

L x L y iexcl L yL x ˆ L z

Equations (47) and (48) for the second and the fourthmoments coincide with the results of [2] which wereobtained from the consideration of an ensemble of hin-dered symmetric tops Equation (49) for freely hellipV ˆ 0daggerrotating symmetric tops was obtained in [2] However itwas presented there with several misprints

For spherical tops hellipI x ˆ I y ˆ I z ˆ Idagger we simplyput sup2 ˆ 0 in equations (47)- (49) Thus for m ˆ 0 wehave

R l00hellip0dagger ˆ iexcl

kTI

x hellip50dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 1dagger Dagger 12I2

hK 2i0 hellip51dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 15x Dagger 5 iexcl 3x iexcl 1kT

pound hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 5dagger Dagger4hK 2

z i0

Dagger4hK x L zK y iexcl K yL zK x i0

Dagger hhellipL x K xdagger2 DaggerhellipL x K ydagger2 DaggerhellipL yK xdagger2

DaggerhellipL yK ydagger2 DaggerhellipL zK xdagger2 DaggerhellipL zK ydagger2i0Šordf

hellip52dagger

Equations (50)- (52) are in accordance with the results of[2]

For linear molecules hellipI x ˆ I y ˆ I I z ˆ 0dagger one has

R l00hellip0dagger ˆ iexcl kT

Ix hellip53dagger

R l hellip4dagger0 0 hellip0dagger ˆ kT

Ihellip dagger2

xhellip3x iexcl 2dagger Dagger 12I2

hK 2i0 hellip54dagger

R l hellip6dagger0 0 hellip0dagger ˆ iexcl kT

Ihellip dagger3

xcopy

15x 2 iexcl 30x Dagger 24

iexcl 3x iexcl 2kT

hL x K x Dagger L yK yi0

Dagger 1

2hellipkTdagger2 permilhK 2i0hellip9x iexcl 1dagger Dagger hhellipL x K xdagger2

DaggerhellipL x K ydagger2 DaggerhellipL yK x dagger2 DaggerhellipL yK ydagger2i0Šordf

hellip55dagger

Equations (53)- (55) are also in agreement with theresults of [2]

6 Spectral moments and coe cients in the Taylorseries expansion of the memory functions

As has been shown on many occasions (eg [4- 6 8])the calculation of the CFs C lhellip tdagger for asymmetric topmolecules can be simpli ed considerably by makinguse of the memory function approach [13] Thisapproach allows one to express the absorpt ion andRaman spectra in terms of the memory functionsK l

nhellip tdagger of arbitrary order n [2 13] The coe cients inthe Taylor series expansions of the memory functionsK l

nhellip tdagger are also related to the spectral moments M ln as

the CFs C lhellip tdagger are connected to the correspondingmemory functions K l

nhellip tdagger by equations (see eg [13])

ddt

C lhellip tdagger ˆ iexclhellip t

0K l

1hellip t iexcl t 0daggerChellip t 0daggerdt 0 hellip56dagger

ddt

K lnhellip tdagger ˆ iexcl

hellip t

0K l

nDagger1hellip t iexcl t 0daggerK lnhellip t 0daggerdt 0

hellip57dagger

In the classical limit the CF C lhellip tdagger and the memory func-tions K l

nhellip tdagger which are both even functions of time t canbe expanded in Taylor series as [13]

C lhellip tdagger ˆX1

kˆ0

Chellip2kdaggerl hellip0dagger t2k

hellip2kdaggerˆ

X1

kˆ0

M l2k

hellip itdagger2k

hellip2kdagger hellip58dagger

K lnhellip tdagger ˆ

X1

kˆ0

K lhellip2kdaggern hellip0dagger t2k

hellip2kdagger hellip59dagger

Here we have taken into account equation (2)Substituting equations (58) and (59) in (56) and (57)one can obtain after some algebraic transformations

C lhellip0daggerK lhellip2ndagger1 hellip0dagger ˆ iexclChellip2nDagger2dagger

l hellip0dagger

iexclXn

sˆ1

Chellip2sdaggerl hellip0daggerK lhellip2hellipniexclsdaggerdagger

1 hellip0dagger hellip60dagger

K lihellip0daggerK lhellip2ndagger

iDagger1 hellip0dagger ˆ iexclK lhellip2nDagger2daggeri hellip0dagger

iexclXn

sˆ1

K lhellip2sdaggeri hellip0daggerK lhellip2hellipniexclsdaggerdagger

iDagger1 hellip0dagger

hellip61dagger

It is convenient to present equations (60) and (61) interms of determinants

1914 Y P Kalmykov and S V Titov

K lhellip2ndagger1 hellip0dagger ˆdet

M l2=Clhellip0dagger 1 0 centcent cent 0

M l4=Clhellip0dagger M l

2=Clhellip0dagger 1

0

M l6=Clhellip0dagger M l

4=Clhellip0dagger M l2=Clhellip0dagger

1

M l2hellipnDagger1dagger=Clhellip0dagger M l

2n=Clhellip0dagger cent centcent M l4=Clhellip0dagger M l

2=Clhellip0dagger

7777777777777777777

7777777777777777777

hellip62dagger

K lihellip0daggerK l hellip2ndagger

iDagger1 hellip0dagger ˆ i2n det

K lhellip2daggeri hellip0dagger 1 0 cent cent cent 0

K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger 1

1

K lhellip2hellipnDagger1daggerdaggeri hellip0dagger K lhellip2ndagger

i hellip0dagger cent cent cent K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger

777777777777777

777777777777777

hellip63dagger

where n ˆ 0 1 2 and

M l2n ˆClhellip0daggerdet

K l1hellip0dagger 1 0 centcentcent 0

K lhellip2dagger1 hellip0dagger K l

1hellip0dagger 1

1

K lhellip2niexcl2dagger1 hellip0dagger K lhellip2hellipniexcl3daggerdagger

1 hellip0dagger centcent cent K lhellip2dagger1 hellip0dagger K l

1hellip0dagger

777777777777777

777777777777777

hellip64dagger

Thus if the moments M l2n are known it is possible to

calculate all the memory functions and their derivativesat t ˆ 0 Conversely if the derivatives of the memoryfunctions K lhellip2ndagger

i hellip0dagger are known it is possible to calculateall the spectral moments M l

2n

7 DiscussionAs we have already mentioned in the analysis of

absorption and scattering spectra of molecular uidsin terms of the moments of the CFs the key problemis the evaluation of the short term behaviour of thecorrelation functions R l

nmhellip tdagger This problem is relatedin turn to the calculation of the coe cients in aTaylor series expansion of R l

n mhellip tdagger The results obtainedin the present work allow one to calculate these coe -cients and hence the spectral moments in a systematicmanner Equations for M l

n obtained in the present paperpermit one also to calculate from equation (64) the co-e cients in Taylor series expansions of the appropriatememory functions Here for simplicity we have con- ned ourselves to the evaluation of the second (M l

2)fourth (M l

4) and sixth (M l6) moments only However

the method used is quite general and it may be extendedreadily for evaluating higher spectral moments as thehigher time derivatives of R l

n mhellip tdagger at t ˆ 0 can be calcu-lated by a similar way For example according to equa-tion (11) on evaluating the fourth derivative of theWigner D function Dl

n mhellipsup3rsquoAacutedagger at t ˆ 0 one can calcu-late the eighth derivative of R lhellip8dagger

n mhellip0dagger etcThe second moment M 1

2 for the dipole absorption ofasymmetric top molecules has been obtained in many

works as within the framework of classical andquantum theory (eg [10 17]) The derivation of thefourth moment M 1

4 for molecules of C2v symmetry wasgiven in [17] The second and fourth moments for anarbitrary asymmetric top molecule have been calculatedby Gordon [11] on using quantum-mechanical methodswith the subsequent transition to the classical limithellip -h 0dagger Our results (equations (38) and (39)) are incomplete agreement with those of Gordon [11] Inmatrix notations of [11] equations (38) and (39) canbe written as

M 12 ˆ kTpermilm2Tr fBg iexcl m cent B cent mTŠ hellip65dagger

M 14 ˆ hellipkTdagger2 fm2permil2hellipTr fBgdagger2 DaggerTr fB2ghellipTr fBgTr fBiexcl1g iexcl 3dagger

iexcl Tr fB3gTr fBiexcl1gŠ

Daggerm cent B2 cent mThellip7 iexcl Tr fBgTr fBiexcl1gdagger

iexcl 6 m cent B cent mTTr fBg Daggerm cent B3 cent mTTr fBiexcl1gg

Daggerm2hK cent B2 cent KTi0 iexcl hhellipm cent B cent KTdagger2i0 hellip66dagger

where B ˆ Iiexcl1 and the upper index T denotes the trans-position

Equations (42) and (43) for the second and fourthRaman spectral moments M 2

2 and M 24 coincide with

the results of Gordon [12] and can be rewritten in thematrix form as follows

M 22 ˆ 2kTpermil2Tr fBgTr fa2g iexcl 3Tr fa cent B cent agŠ hellip67dagger

M 24 ˆ hellipkTdagger2permil27hellipTr fBgdagger2Tr fa2g iexcl 72Tr fa cent B cent agTr fBg

Dagger18Tr fa cent B cent a cent Bg

Dagger6Tr fa cent B2 cent aghellip1 iexcl Tr fBgTr fBiexcl1gdagger

DaggerTr fa2gTr fB2ghellip5 Dagger4Tr fBgTr fBiexcl1gdagger

iexcl 4Tr fa2gTr fB3gTr fBiexcl1g

Dagger6Tr fa cent B3 cent agTr fBiexcl1gŠ

Dagger4hK cent B2 cent KTTr fa2gi0 iexcl 6hK cent B cent a2 cent B cent KT i0

hellip68dagger

It should be noted that there was a misprint in theequation for the fourth moment M 2

4 in [12] equation(58) To our knowledge the sixth moments forthe dipole absorpt ion (equation (40) and Ramanscattering (equation (44)) spectra of an arbitrary asym-metric top molecule have been calculated for the rsttime

Spectral moments of CFs of asymmetric top molecules 1915

The spectral moments allow one to carry out thequantitative analysis of experimental data on IR absorp-tion and Raman spectra of uids with asymmetrictop molecules As is well known the spectral momentM l

2 does not depend on intermolecular interactions Thespectral moment M l

4 depends on the mean-squaretorque acting on the molecule The spectral momentM l

6 includes also averages of angular derivatives ofthe torque Thus the spectral moments allow one toobtain information both about molecular motions andintermolecular interactions Another important prop-erty of spectral moments is the possibility of evaluatingasymptotic properties of spectra For example ifspectral moments M l

2 M l4 and M l

6 exist the integral inthe right hand side of equation (1) exists for n ˆ 2 4and 6 As a result the spectral function I lhellipdagger mustdecrease faster than iexcl6 Moreover the spectralmoments play a rather important role in the theoreticalcalculation of the absorption and scattering spectraand in the modelling of molecular rotation in uidsAs the second spectral moment M l

2 does not dependon the torques acting on molecules (it is exclusivelydetermined by molecular parameters) this momentcalculated in the context of any particular modelshould not depend on parameters describing inter-actions of molecules In other words the secondmoment can be used as a criterion of the correctnessof the model assumptions As the higher momentsdepend on angular derivatives of the intermolecularpotential energy they can be used for the evaluationof model parameters in systems of interacting moleculesFor example the mean-square torque acting on themolecule can be determined from the fourth spectralmoment M l

4 [18] With the help of the spectral momentsM l

2 and M l4 one can evaluate the mean time of molecular

collisions [19 20] Examples of such evaluations forasymmetric top molecules like H 2O and D 2O weregiven in [9]

We thank Professor J L Dejardin for usefulcomments and suggestions The support of this workby the International Association for the Promotionof Co-operation with Scientists from the NewIndependent States of the Former Soviet Union(Project INTAS 96-1411) is gratefully acknowledged

Appendix

Equat ions for R l hellip2kdaggermm hellip0dagger and R l hellip2kdagger

mmsect4hellip0daggerEquations (21)- (32) and (34) yield

R l hellip2daggermm hellip0dagger ˆ iexcl

kT2

permilhellipx iexcl m2daggerhellipI iexcl1x Dagger I iexcl1

y dagger Dagger 2m2I iexcl1z Š hellipA 1dagger

R l hellip2daggermmsect4hellip0dagger ˆ 0 hellipA 2dagger

R l hellip4daggermmhellip0dagger ˆhellipkTdagger2 fhellipx iexcl 3m2daggerpermil2I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y DaggerI iexcl1

x daggerŠ

Daggerm2I iexcl2z hellipI yI

iexcl1x DaggerI x I iexcl1

y iexcl 2dagger

Dagger12hellipx Dagger6m2x iexcl6m4 iexcl5m2daggerI iexcl1

z hellipI iexcl1x DaggerI iexcl1

y dagger

Dagger12hellipx iexcl m2daggerpermilI iexcl1

z hellipI y I iexcl2x DaggerI x I iexcl2

y dagger

DaggerI zIiexcl1x I iexcl1

y hellipI iexcl1x DaggerI iexcl1

y dagger iexcl 2hellipI iexcl2x DaggerI iexcl2

y daggerŠ

Dagger18 permil3hellipx iexclm2dagger2 iexcl2x Dagger5m2Š

pound permil3I iexcl2x Dagger3I iexcl2

y Dagger2I iexcl1x I iexcl1

y Š Dagger3m4I iexcl2z g

Dagger12hellipx iexcl m2daggerhellip I iexcl2

x hK 2x i0 DaggerI iexcl2

y hK 2yi0dagger

Daggerm2I iexcl2z hK 2

z i0 hellipA 3dagger

R lhellip4daggermsect4 mhellip0dagger ˆ 3hellipkTdagger2

16hellipI iexcl1

x iexcl I iexcl1y dagger2

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 4dagger

R lhellip6daggermmhellip0dagger ˆ iexcl kT

16I3x I3

yI3zhellipR lm

0 DaggerkTR lm1 DaggerhellipkTdagger2R lm

2 dagger hellipA 5dagger

R lhellip6daggermsect4 m ˆ iexcl

3kThellipI x iexcl I ydagger32I3

x I 3yI z

f6I zhellipI2x hK 2

yi0 iexcl I2yhK 2

x i0dagger

Dagger4kT I zhellipI2yhL x K x i0 iexcl I2

x hL yK yi0dagger

Dagger5hellipkTdagger2hellipI x iexcl I ydaggerpermil3xhellipI x DaggerI ydaggerI z

Dagger3hellip2I x I y iexclhellipI x DaggerI ydaggerI zdagger

pound mhellipm sect 4dagger

Dagger2hellipI 2x Dagger22I x I y DaggerI 2

y iexcl12hellipI x DaggerI ydaggerI z DaggerI2zdaggerŠg

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 6dagger

where x ˆ lhellip l Dagger1dagger

1916 Y P Kalmykov and S V Titov

R lm0 ˆ 8f2m2I 2

x I2yhI x I zhellipL yK zdagger2 DaggerI y I zhellipL x K zdagger2

DaggerI x I yhellipL zK zdagger2i0 Daggerhellipx iexclm2daggerI2z permilI2

x hI x I zhellipL yK ydagger2

DaggerI yI zhellipL x K ydagger2 DaggerI x I yhellipL zK ydagger2i0 DaggerI2yhI x I zhellipL yK xdagger2

DaggerI yI zhellipL x K xdagger2 DaggerI x I yhellipL zK xdagger2i0Š

Dagger16I x I yI z fhK zL x K yi0permilhellipx iexclm2daggerI x I zhellipI y DaggerI z iexcl I x dagger

iexcl3m2I y I z Š iexclhK zL yK x i0permilhellipx iexclm2daggerI yI zhellipI x DaggerI z iexcl I ydagger

iexcl3m2I x I z ŠDaggerm2I x I ypermilhK x L yK zi0hellip2I x iexcl2I y iexcl I zdagger

iexclhK yL x K zi0hellip2I y iexcl2I x iexcl I zdaggerŠ

DaggerhK x L zK yi0permilhellipx iexclm2daggerI x I zhellip2I y DaggerI z iexcl I xdagger iexcl3m2I x I yŠ

iexclhK yL zK x i0permilhellipx iexclm2daggerI y I zhellip2I x DaggerI z iexcl I ydagger iexcl3m2I x I y Šg

Dagger2hK 2x i0I yI z fx I zpermil16I x I yhellipI y iexcl I x dagger

iexcl2I yI zhellipI x Dagger9I ydagger Dagger4I xhellipI x iexcl I zdagger2

Dagger9xI yI zhellipI x Dagger3I ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I x I zhellipI z iexcl I xdagger2 DaggerI yI zhellip32I2x

iexcl7I xhellip8I y iexcl I zdagger Dagger45I yI zdagger

Dagger18hellipx iexclm2daggerI y I zhellip2I x I y iexcl I x I z iexcl3I yI zdagger

iexcl9m2I y I2zhellipI x Dagger3I ydaggerŠg

Dagger2hK 2yi0I x I z fx I zpermil16I x I yhellipI x iexcl I ydagger

iexcl2I x I zhellip9I x DaggerI ydagger Dagger4I yhellipI y iexcl I zdagger2

Dagger9xI x I zhellip3I x DaggerI ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I yI zhellipI z iexcl I ydagger2 DaggerI x I zhellip32I2y iexcl7I yhellip8I x iexcl I zdagger

Dagger45I x I zdagger Dagger18hellipx iexclm2daggerI x I zhellip2I x I y iexcl3I x I z iexcl I y I zdagger

iexcl9m2I x I 2zhellip3I x DaggerI ydaggerŠg

Dagger8hK 2zi0I x I y fhellipx iexclm2daggerI zpermilI xhellipI x iexcl I zdagger2

DaggerI yhellip I y iexcl I zdagger2 DaggerI x I yhellip4I z iexcl I x iexcl I ydaggerŠ

Daggerm2I x I ypermil9hellipx iexclm2daggerI zhellipI x DaggerI ydagger Dagger18m2I x I y iexcl8I2z Šg

R lm1 ˆ 16hL zK zi0I2

x I2ym

2 f2I2z iexcl 2hellipIx iexcl I ydagger2 iexcl 3xI zhellipIx Dagger I ydagger

Dagger3m2hellipIx I z Dagger I yI z iexcl 2I x I ydaggerg

Dagger4hLx K x i0I2yI

2z fhellipx iexcl m2daggerpermil4I2

x iexcl 4hellipI y iexcl Izdagger2

iexcl 6I zhellipIx iexcl Iydagger iexcl 3x IzhellipI x Dagger3IydaggerŠ

Dagger3m2permil3I zhellipIx iexcl I ydagger DaggerxhellipI zhellipI x Dagger3I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl Ix I z iexcl 3I yI zdaggerŠg

Dagger4hLyK yi0I2x I2

z fhellipx iexcl m2daggerpermil4I2y iexcl 4hellipI x iexcl I zdagger2

iexcl 6I zhellipIy iexcl I xdagger iexcl 3x Izhellip3Ix Dagger IydaggerŠ

Dagger3m2permil3I zhellipIy iexcl Ixdagger DaggerxhellipI zhellip3I x Dagger I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl 3I x I z iexcl I yI zdaggerŠg

R lm2 ˆ 15permil2Ix Iy iexcl IzhellipI x DaggerI ydaggerŠpermil8I2

x I2y iexcl 8I x I yI zhellipIx Dagger I ydagger

Dagger I2zhellip5I2

x iexcl 2Ix I y Dagger5I2ydaggerŠm6

Dagger15f16Ix I yhellipI x iexcl Iydagger2permilI2x I2

y iexcl I zhellipIx Dagger I ydaggerŠ

Dagger2I2zhellip3I4

x Dagger43I3x Iy iexcl 84I2

x I2y Dagger43I x I3

y Dagger3I4ydagger

Dagger3xI zpermil8I2x I2

yhellipI x Dagger I ydagger iexcl 4I x I yI zhellip3I2x Dagger2I x Iy Dagger3I2

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip5I2

x iexcl 2I x I y Dagger5I2ydaggerŠ

iexcl I3zhellipI x Dagger Iydaggerhellip47I2

x iexcl 78I x Iy Dagger47I2ydagger

Dagger2I4zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggergm4

Dagger f16I x IyhellipI x iexcl I ydagger2hellip3I2x iexcl 2I x I y Dagger3I2

ydagger

iexcl 8I zhellipIx Dagger IydaggerhellipIx iexcl I ydagger2hellip3I2x Dagger34I x I y Dagger3I2

ydagger

Dagger2I2zhellipI x iexcl Iydagger2hellip107I2

x Dagger602I x I y Dagger107I2ydagger

iexcl 32I3zhellipIx Dagger Iydaggerhellip25I2

x iexcl 41Ix I y Dagger25I2ydagger

Dagger2I4zhellip107I2

x Dagger98I x Iy Dagger107I2ydagger iexcl 24I5

zhellipI x DaggerI ydagger

Dagger30xI zpermil16Ix I yhellipI x Dagger IydaggerhellipIx iexcl Iydagger2

iexcl 4I zhellip3I4x Dagger19I3

x Iy iexcl 36I2x I2

y Dagger19I x I3y Dagger3I4

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip69I2

x iexcl 106I x Iy Dagger69I2ydagger

iexcl 4I3zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggerŠ

Dagger90I2zx

2permil2I x Iyhellip3I2x Dagger2Ix I y Dagger3I2

ydagger

Spectral moments of CFs of asymmetric top molecules 1917

iexcl I zhellipI x Dagger I ydaggerhellip5I 2x iexcl 2I x I y Dagger5I2

ydaggerŠgm2

Daggerx I z f4hellipI x iexcl I ydagger2permil6hellipI x Dagger I ydagger3

iexcl I zhellip31I2x Dagger106I x I y Dagger 31I2

ydaggerŠ

Dagger64I2zhellipI x Dagger I ydaggerhellip5I2

x iexcl 7I x I y Dagger5I2ydagger

iexcl 4I3zhellip31I2

x Dagger26I x I y Dagger31I2ydagger Dagger24I4

zhellipI x Dagger I ydagger

Dagger15xI zpermil2hellipI x iexcl I ydagger2hellip3I x Dagger I ydaggerhellip I x Dagger3I ydagger

iexcl 2I zhellipI x Dagger I ydaggerhellip11I2x iexcl 14I x I y Dagger11I2

ydagger

Dagger2I2zhellip3I 2

x Dagger2I x I y Dagger 3I 2ydagger

Daggerx I zhellipI x Dagger I ydaggerhellip5I2x iexcl 2I x I y Dagger5I2

ydaggerŠg

In this derivation we have assumed that

hK ii0 ˆ 0 hellip i ˆ x y zdaggerand

hK iK ji0 ˆ 0 hL iK ji0 ˆ 0 for i 6ˆ j hellip i j ˆ x y zdagger

References[1] BURHSTEIN A I and TEMKIN S I 1994 Spectroscopy

of M olecular Rotation in Gases and Liquids (CambridgeUniversity Press)

[2] ST PIERRE A G and STEELE W A 1981 M olecPhys 43 123

[3] GORDON R G 1963 J chem Phys 39 2788

[4] LEICKMAN JCL and GUISSANIY 1981 M olec Phys42 1105

[5] AGUADO-GOMEZM and LEICKMAN JCL 1986 PhysRev A 34 4195

[6] AGUADO-GOMEZM and LEICKMAN JCL 1987 PhysRev A 35 286

[7] BORISEVICH N A BLOKHINA PZALESSKAYAG ALASTOCHKINA V A and SHUKUROV T 1984 IzvAkad Nauk SSSR Ser Fiz 48 709

[8] KALMYKOV YUP 1985 Opt Spektrosk 58 804 [1985Opt Spectrosc 58 493]

[9] KALMYKOV YU P and TITOV S V 1999 J molecS truct 479 123

[10] GORDON R G 1963 J chem Phys 38 1724[11] GORDON R G 1964 J chem Phys 41 1819[12] GORDON R G 1964 J chem Phys 40 1973[13] BERNE B J and HARP G D 1970 Adv chem Phys

17 63[14] LEICKMAN JCL and GUISSANIY 1984 M olec Phys

53 761[15] VARSHALOVICH D A MOSKALEV A N and

KHERSONSKII V K 1988 Quantum Theory of AngularM omentum (Singapore World Scienti c)

[16] LANDAULD and LIFSHITZEM 1976 M echanics ACourse of theoretical Physics 3rd Edn Vol 1 (OxfordPergamon Press)

[17] BOiumlTTCHERC JF and BORDEWIJK P 1979 Theory ofElectric Polarization Vol 2 (Amsterdam Elsevier)

[18] RODRIGUEZ R and MCHALE J L 1988 J chemPhys 88 2264

[19] BURSHTEINA I and MCCONNELL J R 1989 PhysicaA 157 933

[20] KALMYKOV YU P and MCCONNELL J R 1993Physica A 193 394

1918 Spectral moments of CFs of asymmetric top molecules

K lhellip2ndagger1 hellip0dagger ˆdet

M l2=Clhellip0dagger 1 0 centcent cent 0

M l4=Clhellip0dagger M l

2=Clhellip0dagger 1

0

M l6=Clhellip0dagger M l

4=Clhellip0dagger M l2=Clhellip0dagger

1

M l2hellipnDagger1dagger=Clhellip0dagger M l

2n=Clhellip0dagger cent centcent M l4=Clhellip0dagger M l

2=Clhellip0dagger

7777777777777777777

7777777777777777777

hellip62dagger

K lihellip0daggerK l hellip2ndagger

iDagger1 hellip0dagger ˆ i2n det

K lhellip2daggeri hellip0dagger 1 0 cent cent cent 0

K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger 1

1

K lhellip2hellipnDagger1daggerdaggeri hellip0dagger K lhellip2ndagger

i hellip0dagger cent cent cent K lhellip4daggeri hellip0dagger K lhellip2dagger

i hellip0dagger

777777777777777

777777777777777

hellip63dagger

where n ˆ 0 1 2 and

M l2n ˆClhellip0daggerdet

K l1hellip0dagger 1 0 centcentcent 0

K lhellip2dagger1 hellip0dagger K l

1hellip0dagger 1

1

K lhellip2niexcl2dagger1 hellip0dagger K lhellip2hellipniexcl3daggerdagger

1 hellip0dagger centcent cent K lhellip2dagger1 hellip0dagger K l

1hellip0dagger

777777777777777

777777777777777

hellip64dagger

Thus if the moments M l2n are known it is possible to

calculate all the memory functions and their derivativesat t ˆ 0 Conversely if the derivatives of the memoryfunctions K lhellip2ndagger

i hellip0dagger are known it is possible to calculateall the spectral moments M l

2n

7 DiscussionAs we have already mentioned in the analysis of

absorption and scattering spectra of molecular uidsin terms of the moments of the CFs the key problemis the evaluation of the short term behaviour of thecorrelation functions R l

nmhellip tdagger This problem is relatedin turn to the calculation of the coe cients in aTaylor series expansion of R l

n mhellip tdagger The results obtainedin the present work allow one to calculate these coe -cients and hence the spectral moments in a systematicmanner Equations for M l

n obtained in the present paperpermit one also to calculate from equation (64) the co-e cients in Taylor series expansions of the appropriatememory functions Here for simplicity we have con- ned ourselves to the evaluation of the second (M l

2)fourth (M l

4) and sixth (M l6) moments only However

the method used is quite general and it may be extendedreadily for evaluating higher spectral moments as thehigher time derivatives of R l

n mhellip tdagger at t ˆ 0 can be calcu-lated by a similar way For example according to equa-tion (11) on evaluating the fourth derivative of theWigner D function Dl

n mhellipsup3rsquoAacutedagger at t ˆ 0 one can calcu-late the eighth derivative of R lhellip8dagger

n mhellip0dagger etcThe second moment M 1

2 for the dipole absorption ofasymmetric top molecules has been obtained in many

works as within the framework of classical andquantum theory (eg [10 17]) The derivation of thefourth moment M 1

4 for molecules of C2v symmetry wasgiven in [17] The second and fourth moments for anarbitrary asymmetric top molecule have been calculatedby Gordon [11] on using quantum-mechanical methodswith the subsequent transition to the classical limithellip -h 0dagger Our results (equations (38) and (39)) are incomplete agreement with those of Gordon [11] Inmatrix notations of [11] equations (38) and (39) canbe written as

M 12 ˆ kTpermilm2Tr fBg iexcl m cent B cent mTŠ hellip65dagger

M 14 ˆ hellipkTdagger2 fm2permil2hellipTr fBgdagger2 DaggerTr fB2ghellipTr fBgTr fBiexcl1g iexcl 3dagger

iexcl Tr fB3gTr fBiexcl1gŠ

Daggerm cent B2 cent mThellip7 iexcl Tr fBgTr fBiexcl1gdagger

iexcl 6 m cent B cent mTTr fBg Daggerm cent B3 cent mTTr fBiexcl1gg

Daggerm2hK cent B2 cent KTi0 iexcl hhellipm cent B cent KTdagger2i0 hellip66dagger

where B ˆ Iiexcl1 and the upper index T denotes the trans-position

Equations (42) and (43) for the second and fourthRaman spectral moments M 2

2 and M 24 coincide with

the results of Gordon [12] and can be rewritten in thematrix form as follows

M 22 ˆ 2kTpermil2Tr fBgTr fa2g iexcl 3Tr fa cent B cent agŠ hellip67dagger

M 24 ˆ hellipkTdagger2permil27hellipTr fBgdagger2Tr fa2g iexcl 72Tr fa cent B cent agTr fBg

Dagger18Tr fa cent B cent a cent Bg

Dagger6Tr fa cent B2 cent aghellip1 iexcl Tr fBgTr fBiexcl1gdagger

DaggerTr fa2gTr fB2ghellip5 Dagger4Tr fBgTr fBiexcl1gdagger

iexcl 4Tr fa2gTr fB3gTr fBiexcl1g

Dagger6Tr fa cent B3 cent agTr fBiexcl1gŠ

Dagger4hK cent B2 cent KTTr fa2gi0 iexcl 6hK cent B cent a2 cent B cent KT i0

hellip68dagger

It should be noted that there was a misprint in theequation for the fourth moment M 2

4 in [12] equation(58) To our knowledge the sixth moments forthe dipole absorpt ion (equation (40) and Ramanscattering (equation (44)) spectra of an arbitrary asym-metric top molecule have been calculated for the rsttime

Spectral moments of CFs of asymmetric top molecules 1915

The spectral moments allow one to carry out thequantitative analysis of experimental data on IR absorp-tion and Raman spectra of uids with asymmetrictop molecules As is well known the spectral momentM l

2 does not depend on intermolecular interactions Thespectral moment M l

4 depends on the mean-squaretorque acting on the molecule The spectral momentM l

6 includes also averages of angular derivatives ofthe torque Thus the spectral moments allow one toobtain information both about molecular motions andintermolecular interactions Another important prop-erty of spectral moments is the possibility of evaluatingasymptotic properties of spectra For example ifspectral moments M l

2 M l4 and M l

6 exist the integral inthe right hand side of equation (1) exists for n ˆ 2 4and 6 As a result the spectral function I lhellipdagger mustdecrease faster than iexcl6 Moreover the spectralmoments play a rather important role in the theoreticalcalculation of the absorption and scattering spectraand in the modelling of molecular rotation in uidsAs the second spectral moment M l

2 does not dependon the torques acting on molecules (it is exclusivelydetermined by molecular parameters) this momentcalculated in the context of any particular modelshould not depend on parameters describing inter-actions of molecules In other words the secondmoment can be used as a criterion of the correctnessof the model assumptions As the higher momentsdepend on angular derivatives of the intermolecularpotential energy they can be used for the evaluationof model parameters in systems of interacting moleculesFor example the mean-square torque acting on themolecule can be determined from the fourth spectralmoment M l

4 [18] With the help of the spectral momentsM l

2 and M l4 one can evaluate the mean time of molecular

collisions [19 20] Examples of such evaluations forasymmetric top molecules like H 2O and D 2O weregiven in [9]

We thank Professor J L Dejardin for usefulcomments and suggestions The support of this workby the International Association for the Promotionof Co-operation with Scientists from the NewIndependent States of the Former Soviet Union(Project INTAS 96-1411) is gratefully acknowledged

Appendix

Equat ions for R l hellip2kdaggermm hellip0dagger and R l hellip2kdagger

mmsect4hellip0daggerEquations (21)- (32) and (34) yield

R l hellip2daggermm hellip0dagger ˆ iexcl

kT2

permilhellipx iexcl m2daggerhellipI iexcl1x Dagger I iexcl1

y dagger Dagger 2m2I iexcl1z Š hellipA 1dagger

R l hellip2daggermmsect4hellip0dagger ˆ 0 hellipA 2dagger

R l hellip4daggermmhellip0dagger ˆhellipkTdagger2 fhellipx iexcl 3m2daggerpermil2I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y DaggerI iexcl1

x daggerŠ

Daggerm2I iexcl2z hellipI yI

iexcl1x DaggerI x I iexcl1

y iexcl 2dagger

Dagger12hellipx Dagger6m2x iexcl6m4 iexcl5m2daggerI iexcl1

z hellipI iexcl1x DaggerI iexcl1

y dagger

Dagger12hellipx iexcl m2daggerpermilI iexcl1

z hellipI y I iexcl2x DaggerI x I iexcl2

y dagger

DaggerI zIiexcl1x I iexcl1

y hellipI iexcl1x DaggerI iexcl1

y dagger iexcl 2hellipI iexcl2x DaggerI iexcl2

y daggerŠ

Dagger18 permil3hellipx iexclm2dagger2 iexcl2x Dagger5m2Š

pound permil3I iexcl2x Dagger3I iexcl2

y Dagger2I iexcl1x I iexcl1

y Š Dagger3m4I iexcl2z g

Dagger12hellipx iexcl m2daggerhellip I iexcl2

x hK 2x i0 DaggerI iexcl2

y hK 2yi0dagger

Daggerm2I iexcl2z hK 2

z i0 hellipA 3dagger

R lhellip4daggermsect4 mhellip0dagger ˆ 3hellipkTdagger2

16hellipI iexcl1

x iexcl I iexcl1y dagger2

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 4dagger

R lhellip6daggermmhellip0dagger ˆ iexcl kT

16I3x I3

yI3zhellipR lm

0 DaggerkTR lm1 DaggerhellipkTdagger2R lm

2 dagger hellipA 5dagger

R lhellip6daggermsect4 m ˆ iexcl

3kThellipI x iexcl I ydagger32I3

x I 3yI z

f6I zhellipI2x hK 2

yi0 iexcl I2yhK 2

x i0dagger

Dagger4kT I zhellipI2yhL x K x i0 iexcl I2

x hL yK yi0dagger

Dagger5hellipkTdagger2hellipI x iexcl I ydaggerpermil3xhellipI x DaggerI ydaggerI z

Dagger3hellip2I x I y iexclhellipI x DaggerI ydaggerI zdagger

pound mhellipm sect 4dagger

Dagger2hellipI 2x Dagger22I x I y DaggerI 2

y iexcl12hellipI x DaggerI ydaggerI z DaggerI2zdaggerŠg

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 6dagger

where x ˆ lhellip l Dagger1dagger

1916 Y P Kalmykov and S V Titov

R lm0 ˆ 8f2m2I 2

x I2yhI x I zhellipL yK zdagger2 DaggerI y I zhellipL x K zdagger2

DaggerI x I yhellipL zK zdagger2i0 Daggerhellipx iexclm2daggerI2z permilI2

x hI x I zhellipL yK ydagger2

DaggerI yI zhellipL x K ydagger2 DaggerI x I yhellipL zK ydagger2i0 DaggerI2yhI x I zhellipL yK xdagger2

DaggerI yI zhellipL x K xdagger2 DaggerI x I yhellipL zK xdagger2i0Š

Dagger16I x I yI z fhK zL x K yi0permilhellipx iexclm2daggerI x I zhellipI y DaggerI z iexcl I x dagger

iexcl3m2I y I z Š iexclhK zL yK x i0permilhellipx iexclm2daggerI yI zhellipI x DaggerI z iexcl I ydagger

iexcl3m2I x I z ŠDaggerm2I x I ypermilhK x L yK zi0hellip2I x iexcl2I y iexcl I zdagger

iexclhK yL x K zi0hellip2I y iexcl2I x iexcl I zdaggerŠ

DaggerhK x L zK yi0permilhellipx iexclm2daggerI x I zhellip2I y DaggerI z iexcl I xdagger iexcl3m2I x I yŠ

iexclhK yL zK x i0permilhellipx iexclm2daggerI y I zhellip2I x DaggerI z iexcl I ydagger iexcl3m2I x I y Šg

Dagger2hK 2x i0I yI z fx I zpermil16I x I yhellipI y iexcl I x dagger

iexcl2I yI zhellipI x Dagger9I ydagger Dagger4I xhellipI x iexcl I zdagger2

Dagger9xI yI zhellipI x Dagger3I ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I x I zhellipI z iexcl I xdagger2 DaggerI yI zhellip32I2x

iexcl7I xhellip8I y iexcl I zdagger Dagger45I yI zdagger

Dagger18hellipx iexclm2daggerI y I zhellip2I x I y iexcl I x I z iexcl3I yI zdagger

iexcl9m2I y I2zhellipI x Dagger3I ydaggerŠg

Dagger2hK 2yi0I x I z fx I zpermil16I x I yhellipI x iexcl I ydagger

iexcl2I x I zhellip9I x DaggerI ydagger Dagger4I yhellipI y iexcl I zdagger2

Dagger9xI x I zhellip3I x DaggerI ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I yI zhellipI z iexcl I ydagger2 DaggerI x I zhellip32I2y iexcl7I yhellip8I x iexcl I zdagger

Dagger45I x I zdagger Dagger18hellipx iexclm2daggerI x I zhellip2I x I y iexcl3I x I z iexcl I y I zdagger

iexcl9m2I x I 2zhellip3I x DaggerI ydaggerŠg

Dagger8hK 2zi0I x I y fhellipx iexclm2daggerI zpermilI xhellipI x iexcl I zdagger2

DaggerI yhellip I y iexcl I zdagger2 DaggerI x I yhellip4I z iexcl I x iexcl I ydaggerŠ

Daggerm2I x I ypermil9hellipx iexclm2daggerI zhellipI x DaggerI ydagger Dagger18m2I x I y iexcl8I2z Šg

R lm1 ˆ 16hL zK zi0I2

x I2ym

2 f2I2z iexcl 2hellipIx iexcl I ydagger2 iexcl 3xI zhellipIx Dagger I ydagger

Dagger3m2hellipIx I z Dagger I yI z iexcl 2I x I ydaggerg

Dagger4hLx K x i0I2yI

2z fhellipx iexcl m2daggerpermil4I2

x iexcl 4hellipI y iexcl Izdagger2

iexcl 6I zhellipIx iexcl Iydagger iexcl 3x IzhellipI x Dagger3IydaggerŠ

Dagger3m2permil3I zhellipIx iexcl I ydagger DaggerxhellipI zhellipI x Dagger3I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl Ix I z iexcl 3I yI zdaggerŠg

Dagger4hLyK yi0I2x I2

z fhellipx iexcl m2daggerpermil4I2y iexcl 4hellipI x iexcl I zdagger2

iexcl 6I zhellipIy iexcl I xdagger iexcl 3x Izhellip3Ix Dagger IydaggerŠ

Dagger3m2permil3I zhellipIy iexcl Ixdagger DaggerxhellipI zhellip3I x Dagger I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl 3I x I z iexcl I yI zdaggerŠg

R lm2 ˆ 15permil2Ix Iy iexcl IzhellipI x DaggerI ydaggerŠpermil8I2

x I2y iexcl 8I x I yI zhellipIx Dagger I ydagger

Dagger I2zhellip5I2

x iexcl 2Ix I y Dagger5I2ydaggerŠm6

Dagger15f16Ix I yhellipI x iexcl Iydagger2permilI2x I2

y iexcl I zhellipIx Dagger I ydaggerŠ

Dagger2I2zhellip3I4

x Dagger43I3x Iy iexcl 84I2

x I2y Dagger43I x I3

y Dagger3I4ydagger

Dagger3xI zpermil8I2x I2

yhellipI x Dagger I ydagger iexcl 4I x I yI zhellip3I2x Dagger2I x Iy Dagger3I2

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip5I2

x iexcl 2I x I y Dagger5I2ydaggerŠ

iexcl I3zhellipI x Dagger Iydaggerhellip47I2

x iexcl 78I x Iy Dagger47I2ydagger

Dagger2I4zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggergm4

Dagger f16I x IyhellipI x iexcl I ydagger2hellip3I2x iexcl 2I x I y Dagger3I2

ydagger

iexcl 8I zhellipIx Dagger IydaggerhellipIx iexcl I ydagger2hellip3I2x Dagger34I x I y Dagger3I2

ydagger

Dagger2I2zhellipI x iexcl Iydagger2hellip107I2

x Dagger602I x I y Dagger107I2ydagger

iexcl 32I3zhellipIx Dagger Iydaggerhellip25I2

x iexcl 41Ix I y Dagger25I2ydagger

Dagger2I4zhellip107I2

x Dagger98I x Iy Dagger107I2ydagger iexcl 24I5

zhellipI x DaggerI ydagger

Dagger30xI zpermil16Ix I yhellipI x Dagger IydaggerhellipIx iexcl Iydagger2

iexcl 4I zhellip3I4x Dagger19I3

x Iy iexcl 36I2x I2

y Dagger19I x I3y Dagger3I4

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip69I2

x iexcl 106I x Iy Dagger69I2ydagger

iexcl 4I3zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggerŠ

Dagger90I2zx

2permil2I x Iyhellip3I2x Dagger2Ix I y Dagger3I2

ydagger

Spectral moments of CFs of asymmetric top molecules 1917

iexcl I zhellipI x Dagger I ydaggerhellip5I 2x iexcl 2I x I y Dagger5I2

ydaggerŠgm2

Daggerx I z f4hellipI x iexcl I ydagger2permil6hellipI x Dagger I ydagger3

iexcl I zhellip31I2x Dagger106I x I y Dagger 31I2

ydaggerŠ

Dagger64I2zhellipI x Dagger I ydaggerhellip5I2

x iexcl 7I x I y Dagger5I2ydagger

iexcl 4I3zhellip31I2

x Dagger26I x I y Dagger31I2ydagger Dagger24I4

zhellipI x Dagger I ydagger

Dagger15xI zpermil2hellipI x iexcl I ydagger2hellip3I x Dagger I ydaggerhellip I x Dagger3I ydagger

iexcl 2I zhellipI x Dagger I ydaggerhellip11I2x iexcl 14I x I y Dagger11I2

ydagger

Dagger2I2zhellip3I 2

x Dagger2I x I y Dagger 3I 2ydagger

Daggerx I zhellipI x Dagger I ydaggerhellip5I2x iexcl 2I x I y Dagger5I2

ydaggerŠg

In this derivation we have assumed that

hK ii0 ˆ 0 hellip i ˆ x y zdaggerand

hK iK ji0 ˆ 0 hL iK ji0 ˆ 0 for i 6ˆ j hellip i j ˆ x y zdagger

References[1] BURHSTEIN A I and TEMKIN S I 1994 Spectroscopy

of M olecular Rotation in Gases and Liquids (CambridgeUniversity Press)

[2] ST PIERRE A G and STEELE W A 1981 M olecPhys 43 123

[3] GORDON R G 1963 J chem Phys 39 2788

[4] LEICKMAN JCL and GUISSANIY 1981 M olec Phys42 1105

[5] AGUADO-GOMEZM and LEICKMAN JCL 1986 PhysRev A 34 4195

[6] AGUADO-GOMEZM and LEICKMAN JCL 1987 PhysRev A 35 286

[7] BORISEVICH N A BLOKHINA PZALESSKAYAG ALASTOCHKINA V A and SHUKUROV T 1984 IzvAkad Nauk SSSR Ser Fiz 48 709

[8] KALMYKOV YUP 1985 Opt Spektrosk 58 804 [1985Opt Spectrosc 58 493]

[9] KALMYKOV YU P and TITOV S V 1999 J molecS truct 479 123

[10] GORDON R G 1963 J chem Phys 38 1724[11] GORDON R G 1964 J chem Phys 41 1819[12] GORDON R G 1964 J chem Phys 40 1973[13] BERNE B J and HARP G D 1970 Adv chem Phys

17 63[14] LEICKMAN JCL and GUISSANIY 1984 M olec Phys

53 761[15] VARSHALOVICH D A MOSKALEV A N and

KHERSONSKII V K 1988 Quantum Theory of AngularM omentum (Singapore World Scienti c)

[16] LANDAULD and LIFSHITZEM 1976 M echanics ACourse of theoretical Physics 3rd Edn Vol 1 (OxfordPergamon Press)

[17] BOiumlTTCHERC JF and BORDEWIJK P 1979 Theory ofElectric Polarization Vol 2 (Amsterdam Elsevier)

[18] RODRIGUEZ R and MCHALE J L 1988 J chemPhys 88 2264

[19] BURSHTEINA I and MCCONNELL J R 1989 PhysicaA 157 933

[20] KALMYKOV YU P and MCCONNELL J R 1993Physica A 193 394

1918 Spectral moments of CFs of asymmetric top molecules

The spectral moments allow one to carry out thequantitative analysis of experimental data on IR absorp-tion and Raman spectra of uids with asymmetrictop molecules As is well known the spectral momentM l

2 does not depend on intermolecular interactions Thespectral moment M l

4 depends on the mean-squaretorque acting on the molecule The spectral momentM l

6 includes also averages of angular derivatives ofthe torque Thus the spectral moments allow one toobtain information both about molecular motions andintermolecular interactions Another important prop-erty of spectral moments is the possibility of evaluatingasymptotic properties of spectra For example ifspectral moments M l

2 M l4 and M l

6 exist the integral inthe right hand side of equation (1) exists for n ˆ 2 4and 6 As a result the spectral function I lhellipdagger mustdecrease faster than iexcl6 Moreover the spectralmoments play a rather important role in the theoreticalcalculation of the absorption and scattering spectraand in the modelling of molecular rotation in uidsAs the second spectral moment M l

2 does not dependon the torques acting on molecules (it is exclusivelydetermined by molecular parameters) this momentcalculated in the context of any particular modelshould not depend on parameters describing inter-actions of molecules In other words the secondmoment can be used as a criterion of the correctnessof the model assumptions As the higher momentsdepend on angular derivatives of the intermolecularpotential energy they can be used for the evaluationof model parameters in systems of interacting moleculesFor example the mean-square torque acting on themolecule can be determined from the fourth spectralmoment M l

4 [18] With the help of the spectral momentsM l

2 and M l4 one can evaluate the mean time of molecular

collisions [19 20] Examples of such evaluations forasymmetric top molecules like H 2O and D 2O weregiven in [9]

We thank Professor J L Dejardin for usefulcomments and suggestions The support of this workby the International Association for the Promotionof Co-operation with Scientists from the NewIndependent States of the Former Soviet Union(Project INTAS 96-1411) is gratefully acknowledged

Appendix

Equat ions for R l hellip2kdaggermm hellip0dagger and R l hellip2kdagger

mmsect4hellip0daggerEquations (21)- (32) and (34) yield

R l hellip2daggermm hellip0dagger ˆ iexcl

kT2

permilhellipx iexcl m2daggerhellipI iexcl1x Dagger I iexcl1

y dagger Dagger 2m2I iexcl1z Š hellipA 1dagger

R l hellip2daggermmsect4hellip0dagger ˆ 0 hellipA 2dagger

R l hellip4daggermmhellip0dagger ˆhellipkTdagger2 fhellipx iexcl 3m2daggerpermil2I iexcl1

x I iexcl1y iexcl I iexcl1

z hellipI iexcl1y DaggerI iexcl1

x daggerŠ

Daggerm2I iexcl2z hellipI yI

iexcl1x DaggerI x I iexcl1

y iexcl 2dagger

Dagger12hellipx Dagger6m2x iexcl6m4 iexcl5m2daggerI iexcl1

z hellipI iexcl1x DaggerI iexcl1

y dagger

Dagger12hellipx iexcl m2daggerpermilI iexcl1

z hellipI y I iexcl2x DaggerI x I iexcl2

y dagger

DaggerI zIiexcl1x I iexcl1

y hellipI iexcl1x DaggerI iexcl1

y dagger iexcl 2hellipI iexcl2x DaggerI iexcl2

y daggerŠ

Dagger18 permil3hellipx iexclm2dagger2 iexcl2x Dagger5m2Š

pound permil3I iexcl2x Dagger3I iexcl2

y Dagger2I iexcl1x I iexcl1

y Š Dagger3m4I iexcl2z g

Dagger12hellipx iexcl m2daggerhellip I iexcl2

x hK 2x i0 DaggerI iexcl2

y hK 2yi0dagger

Daggerm2I iexcl2z hK 2

z i0 hellipA 3dagger

R lhellip4daggermsect4 mhellip0dagger ˆ 3hellipkTdagger2

16hellipI iexcl1

x iexcl I iexcl1y dagger2

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 4dagger

R lhellip6daggermmhellip0dagger ˆ iexcl kT

16I3x I3

yI3zhellipR lm

0 DaggerkTR lm1 DaggerhellipkTdagger2R lm

2 dagger hellipA 5dagger

R lhellip6daggermsect4 m ˆ iexcl

3kThellipI x iexcl I ydagger32I3

x I 3yI z

f6I zhellipI2x hK 2

yi0 iexcl I2yhK 2

x i0dagger

Dagger4kT I zhellipI2yhL x K x i0 iexcl I2

x hL yK yi0dagger

Dagger5hellipkTdagger2hellipI x iexcl I ydaggerpermil3xhellipI x DaggerI ydaggerI z

Dagger3hellip2I x I y iexclhellipI x DaggerI ydaggerI zdagger

pound mhellipm sect 4dagger

Dagger2hellipI 2x Dagger22I x I y DaggerI 2

y iexcl12hellipI x DaggerI ydaggerI z DaggerI2zdaggerŠg

poundpermilx iexclmhellipm sect 1daggerŠpermilx iexclhellipm sect 1daggerhellipm sect 2daggerŠ

p

poundpermilx iexclhellipm sect 2daggerhellipm sect 3daggerŠpermilx iexclhellipm sect 3daggerhellipm sect 4daggerŠ

p

hellipA 6dagger

where x ˆ lhellip l Dagger1dagger

1916 Y P Kalmykov and S V Titov

R lm0 ˆ 8f2m2I 2

x I2yhI x I zhellipL yK zdagger2 DaggerI y I zhellipL x K zdagger2

DaggerI x I yhellipL zK zdagger2i0 Daggerhellipx iexclm2daggerI2z permilI2

x hI x I zhellipL yK ydagger2

DaggerI yI zhellipL x K ydagger2 DaggerI x I yhellipL zK ydagger2i0 DaggerI2yhI x I zhellipL yK xdagger2

DaggerI yI zhellipL x K xdagger2 DaggerI x I yhellipL zK xdagger2i0Š

Dagger16I x I yI z fhK zL x K yi0permilhellipx iexclm2daggerI x I zhellipI y DaggerI z iexcl I x dagger

iexcl3m2I y I z Š iexclhK zL yK x i0permilhellipx iexclm2daggerI yI zhellipI x DaggerI z iexcl I ydagger

iexcl3m2I x I z ŠDaggerm2I x I ypermilhK x L yK zi0hellip2I x iexcl2I y iexcl I zdagger

iexclhK yL x K zi0hellip2I y iexcl2I x iexcl I zdaggerŠ

DaggerhK x L zK yi0permilhellipx iexclm2daggerI x I zhellip2I y DaggerI z iexcl I xdagger iexcl3m2I x I yŠ

iexclhK yL zK x i0permilhellipx iexclm2daggerI y I zhellip2I x DaggerI z iexcl I ydagger iexcl3m2I x I y Šg

Dagger2hK 2x i0I yI z fx I zpermil16I x I yhellipI y iexcl I x dagger

iexcl2I yI zhellipI x Dagger9I ydagger Dagger4I xhellipI x iexcl I zdagger2

Dagger9xI yI zhellipI x Dagger3I ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I x I zhellipI z iexcl I xdagger2 DaggerI yI zhellip32I2x

iexcl7I xhellip8I y iexcl I zdagger Dagger45I yI zdagger

Dagger18hellipx iexclm2daggerI y I zhellip2I x I y iexcl I x I z iexcl3I yI zdagger

iexcl9m2I y I2zhellipI x Dagger3I ydaggerŠg

Dagger2hK 2yi0I x I z fx I zpermil16I x I yhellipI x iexcl I ydagger

iexcl2I x I zhellip9I x DaggerI ydagger Dagger4I yhellipI y iexcl I zdagger2

Dagger9xI x I zhellip3I x DaggerI ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I yI zhellipI z iexcl I ydagger2 DaggerI x I zhellip32I2y iexcl7I yhellip8I x iexcl I zdagger

Dagger45I x I zdagger Dagger18hellipx iexclm2daggerI x I zhellip2I x I y iexcl3I x I z iexcl I y I zdagger

iexcl9m2I x I 2zhellip3I x DaggerI ydaggerŠg

Dagger8hK 2zi0I x I y fhellipx iexclm2daggerI zpermilI xhellipI x iexcl I zdagger2

DaggerI yhellip I y iexcl I zdagger2 DaggerI x I yhellip4I z iexcl I x iexcl I ydaggerŠ

Daggerm2I x I ypermil9hellipx iexclm2daggerI zhellipI x DaggerI ydagger Dagger18m2I x I y iexcl8I2z Šg

R lm1 ˆ 16hL zK zi0I2

x I2ym

2 f2I2z iexcl 2hellipIx iexcl I ydagger2 iexcl 3xI zhellipIx Dagger I ydagger

Dagger3m2hellipIx I z Dagger I yI z iexcl 2I x I ydaggerg

Dagger4hLx K x i0I2yI

2z fhellipx iexcl m2daggerpermil4I2

x iexcl 4hellipI y iexcl Izdagger2

iexcl 6I zhellipIx iexcl Iydagger iexcl 3x IzhellipI x Dagger3IydaggerŠ

Dagger3m2permil3I zhellipIx iexcl I ydagger DaggerxhellipI zhellipI x Dagger3I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl Ix I z iexcl 3I yI zdaggerŠg

Dagger4hLyK yi0I2x I2

z fhellipx iexcl m2daggerpermil4I2y iexcl 4hellipI x iexcl I zdagger2

iexcl 6I zhellipIy iexcl I xdagger iexcl 3x Izhellip3Ix Dagger IydaggerŠ

Dagger3m2permil3I zhellipIy iexcl Ixdagger DaggerxhellipI zhellip3I x Dagger I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl 3I x I z iexcl I yI zdaggerŠg

R lm2 ˆ 15permil2Ix Iy iexcl IzhellipI x DaggerI ydaggerŠpermil8I2

x I2y iexcl 8I x I yI zhellipIx Dagger I ydagger

Dagger I2zhellip5I2

x iexcl 2Ix I y Dagger5I2ydaggerŠm6

Dagger15f16Ix I yhellipI x iexcl Iydagger2permilI2x I2

y iexcl I zhellipIx Dagger I ydaggerŠ

Dagger2I2zhellip3I4

x Dagger43I3x Iy iexcl 84I2

x I2y Dagger43I x I3

y Dagger3I4ydagger

Dagger3xI zpermil8I2x I2

yhellipI x Dagger I ydagger iexcl 4I x I yI zhellip3I2x Dagger2I x Iy Dagger3I2

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip5I2

x iexcl 2I x I y Dagger5I2ydaggerŠ

iexcl I3zhellipI x Dagger Iydaggerhellip47I2

x iexcl 78I x Iy Dagger47I2ydagger

Dagger2I4zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggergm4

Dagger f16I x IyhellipI x iexcl I ydagger2hellip3I2x iexcl 2I x I y Dagger3I2

ydagger

iexcl 8I zhellipIx Dagger IydaggerhellipIx iexcl I ydagger2hellip3I2x Dagger34I x I y Dagger3I2

ydagger

Dagger2I2zhellipI x iexcl Iydagger2hellip107I2

x Dagger602I x I y Dagger107I2ydagger

iexcl 32I3zhellipIx Dagger Iydaggerhellip25I2

x iexcl 41Ix I y Dagger25I2ydagger

Dagger2I4zhellip107I2

x Dagger98I x Iy Dagger107I2ydagger iexcl 24I5

zhellipI x DaggerI ydagger

Dagger30xI zpermil16Ix I yhellipI x Dagger IydaggerhellipIx iexcl Iydagger2

iexcl 4I zhellip3I4x Dagger19I3

x Iy iexcl 36I2x I2

y Dagger19I x I3y Dagger3I4

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip69I2

x iexcl 106I x Iy Dagger69I2ydagger

iexcl 4I3zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggerŠ

Dagger90I2zx

2permil2I x Iyhellip3I2x Dagger2Ix I y Dagger3I2

ydagger

Spectral moments of CFs of asymmetric top molecules 1917

iexcl I zhellipI x Dagger I ydaggerhellip5I 2x iexcl 2I x I y Dagger5I2

ydaggerŠgm2

Daggerx I z f4hellipI x iexcl I ydagger2permil6hellipI x Dagger I ydagger3

iexcl I zhellip31I2x Dagger106I x I y Dagger 31I2

ydaggerŠ

Dagger64I2zhellipI x Dagger I ydaggerhellip5I2

x iexcl 7I x I y Dagger5I2ydagger

iexcl 4I3zhellip31I2

x Dagger26I x I y Dagger31I2ydagger Dagger24I4

zhellipI x Dagger I ydagger

Dagger15xI zpermil2hellipI x iexcl I ydagger2hellip3I x Dagger I ydaggerhellip I x Dagger3I ydagger

iexcl 2I zhellipI x Dagger I ydaggerhellip11I2x iexcl 14I x I y Dagger11I2

ydagger

Dagger2I2zhellip3I 2

x Dagger2I x I y Dagger 3I 2ydagger

Daggerx I zhellipI x Dagger I ydaggerhellip5I2x iexcl 2I x I y Dagger5I2

ydaggerŠg

In this derivation we have assumed that

hK ii0 ˆ 0 hellip i ˆ x y zdaggerand

hK iK ji0 ˆ 0 hL iK ji0 ˆ 0 for i 6ˆ j hellip i j ˆ x y zdagger

References[1] BURHSTEIN A I and TEMKIN S I 1994 Spectroscopy

of M olecular Rotation in Gases and Liquids (CambridgeUniversity Press)

[2] ST PIERRE A G and STEELE W A 1981 M olecPhys 43 123

[3] GORDON R G 1963 J chem Phys 39 2788

[4] LEICKMAN JCL and GUISSANIY 1981 M olec Phys42 1105

[5] AGUADO-GOMEZM and LEICKMAN JCL 1986 PhysRev A 34 4195

[6] AGUADO-GOMEZM and LEICKMAN JCL 1987 PhysRev A 35 286

[7] BORISEVICH N A BLOKHINA PZALESSKAYAG ALASTOCHKINA V A and SHUKUROV T 1984 IzvAkad Nauk SSSR Ser Fiz 48 709

[8] KALMYKOV YUP 1985 Opt Spektrosk 58 804 [1985Opt Spectrosc 58 493]

[9] KALMYKOV YU P and TITOV S V 1999 J molecS truct 479 123

[10] GORDON R G 1963 J chem Phys 38 1724[11] GORDON R G 1964 J chem Phys 41 1819[12] GORDON R G 1964 J chem Phys 40 1973[13] BERNE B J and HARP G D 1970 Adv chem Phys

17 63[14] LEICKMAN JCL and GUISSANIY 1984 M olec Phys

53 761[15] VARSHALOVICH D A MOSKALEV A N and

KHERSONSKII V K 1988 Quantum Theory of AngularM omentum (Singapore World Scienti c)

[16] LANDAULD and LIFSHITZEM 1976 M echanics ACourse of theoretical Physics 3rd Edn Vol 1 (OxfordPergamon Press)

[17] BOiumlTTCHERC JF and BORDEWIJK P 1979 Theory ofElectric Polarization Vol 2 (Amsterdam Elsevier)

[18] RODRIGUEZ R and MCHALE J L 1988 J chemPhys 88 2264

[19] BURSHTEINA I and MCCONNELL J R 1989 PhysicaA 157 933

[20] KALMYKOV YU P and MCCONNELL J R 1993Physica A 193 394

1918 Spectral moments of CFs of asymmetric top molecules

R lm0 ˆ 8f2m2I 2

x I2yhI x I zhellipL yK zdagger2 DaggerI y I zhellipL x K zdagger2

DaggerI x I yhellipL zK zdagger2i0 Daggerhellipx iexclm2daggerI2z permilI2

x hI x I zhellipL yK ydagger2

DaggerI yI zhellipL x K ydagger2 DaggerI x I yhellipL zK ydagger2i0 DaggerI2yhI x I zhellipL yK xdagger2

DaggerI yI zhellipL x K xdagger2 DaggerI x I yhellipL zK xdagger2i0Š

Dagger16I x I yI z fhK zL x K yi0permilhellipx iexclm2daggerI x I zhellipI y DaggerI z iexcl I x dagger

iexcl3m2I y I z Š iexclhK zL yK x i0permilhellipx iexclm2daggerI yI zhellipI x DaggerI z iexcl I ydagger

iexcl3m2I x I z ŠDaggerm2I x I ypermilhK x L yK zi0hellip2I x iexcl2I y iexcl I zdagger

iexclhK yL x K zi0hellip2I y iexcl2I x iexcl I zdaggerŠ

DaggerhK x L zK yi0permilhellipx iexclm2daggerI x I zhellip2I y DaggerI z iexcl I xdagger iexcl3m2I x I yŠ

iexclhK yL zK x i0permilhellipx iexclm2daggerI y I zhellip2I x DaggerI z iexcl I ydagger iexcl3m2I x I y Šg

Dagger2hK 2x i0I yI z fx I zpermil16I x I yhellipI y iexcl I x dagger

iexcl2I yI zhellipI x Dagger9I ydagger Dagger4I xhellipI x iexcl I zdagger2

Dagger9xI yI zhellipI x Dagger3I ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I x I zhellipI z iexcl I xdagger2 DaggerI yI zhellip32I2x

iexcl7I xhellip8I y iexcl I zdagger Dagger45I yI zdagger

Dagger18hellipx iexclm2daggerI y I zhellip2I x I y iexcl I x I z iexcl3I yI zdagger

iexcl9m2I y I2zhellipI x Dagger3I ydaggerŠg

Dagger2hK 2yi0I x I z fx I zpermil16I x I yhellipI x iexcl I ydagger

iexcl2I x I zhellip9I x DaggerI ydagger Dagger4I yhellipI y iexcl I zdagger2

Dagger9xI x I zhellip3I x DaggerI ydaggerŠ Daggerm2permil8I x I yhellipI x iexcl I ydagger2

iexcl4I yI zhellipI z iexcl I ydagger2 DaggerI x I zhellip32I2y iexcl7I yhellip8I x iexcl I zdagger

Dagger45I x I zdagger Dagger18hellipx iexclm2daggerI x I zhellip2I x I y iexcl3I x I z iexcl I y I zdagger

iexcl9m2I x I 2zhellip3I x DaggerI ydaggerŠg

Dagger8hK 2zi0I x I y fhellipx iexclm2daggerI zpermilI xhellipI x iexcl I zdagger2

DaggerI yhellip I y iexcl I zdagger2 DaggerI x I yhellip4I z iexcl I x iexcl I ydaggerŠ

Daggerm2I x I ypermil9hellipx iexclm2daggerI zhellipI x DaggerI ydagger Dagger18m2I x I y iexcl8I2z Šg

R lm1 ˆ 16hL zK zi0I2

x I2ym

2 f2I2z iexcl 2hellipIx iexcl I ydagger2 iexcl 3xI zhellipIx Dagger I ydagger

Dagger3m2hellipIx I z Dagger I yI z iexcl 2I x I ydaggerg

Dagger4hLx K x i0I2yI

2z fhellipx iexcl m2daggerpermil4I2

x iexcl 4hellipI y iexcl Izdagger2

iexcl 6I zhellipIx iexcl Iydagger iexcl 3x IzhellipI x Dagger3IydaggerŠ

Dagger3m2permil3I zhellipIx iexcl I ydagger DaggerxhellipI zhellipI x Dagger3I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl Ix I z iexcl 3I yI zdaggerŠg

Dagger4hLyK yi0I2x I2

z fhellipx iexcl m2daggerpermil4I2y iexcl 4hellipI x iexcl I zdagger2

iexcl 6I zhellipIy iexcl I xdagger iexcl 3x Izhellip3Ix Dagger IydaggerŠ

Dagger3m2permil3I zhellipIy iexcl Ixdagger DaggerxhellipI zhellip3I x Dagger I ydagger iexcl 4I x I ydagger

Daggerm2hellip4Ix I y iexcl 3I x I z iexcl I yI zdaggerŠg

R lm2 ˆ 15permil2Ix Iy iexcl IzhellipI x DaggerI ydaggerŠpermil8I2

x I2y iexcl 8I x I yI zhellipIx Dagger I ydagger

Dagger I2zhellip5I2

x iexcl 2Ix I y Dagger5I2ydaggerŠm6

Dagger15f16Ix I yhellipI x iexcl Iydagger2permilI2x I2

y iexcl I zhellipIx Dagger I ydaggerŠ

Dagger2I2zhellip3I4

x Dagger43I3x Iy iexcl 84I2

x I2y Dagger43I x I3

y Dagger3I4ydagger

Dagger3xI zpermil8I2x I2

yhellipI x Dagger I ydagger iexcl 4I x I yI zhellip3I2x Dagger2I x Iy Dagger3I2

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip5I2

x iexcl 2I x I y Dagger5I2ydaggerŠ

iexcl I3zhellipI x Dagger Iydaggerhellip47I2

x iexcl 78I x Iy Dagger47I2ydagger

Dagger2I4zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggergm4

Dagger f16I x IyhellipI x iexcl I ydagger2hellip3I2x iexcl 2I x I y Dagger3I2

ydagger

iexcl 8I zhellipIx Dagger IydaggerhellipIx iexcl I ydagger2hellip3I2x Dagger34I x I y Dagger3I2

ydagger

Dagger2I2zhellipI x iexcl Iydagger2hellip107I2

x Dagger602I x I y Dagger107I2ydagger

iexcl 32I3zhellipIx Dagger Iydaggerhellip25I2

x iexcl 41Ix I y Dagger25I2ydagger

Dagger2I4zhellip107I2

x Dagger98I x Iy Dagger107I2ydagger iexcl 24I5

zhellipI x DaggerI ydagger

Dagger30xI zpermil16Ix I yhellipI x Dagger IydaggerhellipIx iexcl Iydagger2

iexcl 4I zhellip3I4x Dagger19I3

x Iy iexcl 36I2x I2

y Dagger19I x I3y Dagger3I4

ydagger

Dagger I2zhellipI x Dagger Iydaggerhellip69I2

x iexcl 106I x Iy Dagger69I2ydagger

iexcl 4I3zhellip3I2

x Dagger2Ix I y Dagger3I2ydaggerŠ

Dagger90I2zx

2permil2I x Iyhellip3I2x Dagger2Ix I y Dagger3I2

ydagger

Spectral moments of CFs of asymmetric top molecules 1917

iexcl I zhellipI x Dagger I ydaggerhellip5I 2x iexcl 2I x I y Dagger5I2

ydaggerŠgm2

Daggerx I z f4hellipI x iexcl I ydagger2permil6hellipI x Dagger I ydagger3

iexcl I zhellip31I2x Dagger106I x I y Dagger 31I2

ydaggerŠ

Dagger64I2zhellipI x Dagger I ydaggerhellip5I2

x iexcl 7I x I y Dagger5I2ydagger

iexcl 4I3zhellip31I2

x Dagger26I x I y Dagger31I2ydagger Dagger24I4

zhellipI x Dagger I ydagger

Dagger15xI zpermil2hellipI x iexcl I ydagger2hellip3I x Dagger I ydaggerhellip I x Dagger3I ydagger

iexcl 2I zhellipI x Dagger I ydaggerhellip11I2x iexcl 14I x I y Dagger11I2

ydagger

Dagger2I2zhellip3I 2

x Dagger2I x I y Dagger 3I 2ydagger

Daggerx I zhellipI x Dagger I ydaggerhellip5I2x iexcl 2I x I y Dagger5I2

ydaggerŠg

In this derivation we have assumed that

hK ii0 ˆ 0 hellip i ˆ x y zdaggerand

hK iK ji0 ˆ 0 hL iK ji0 ˆ 0 for i 6ˆ j hellip i j ˆ x y zdagger

References[1] BURHSTEIN A I and TEMKIN S I 1994 Spectroscopy

of M olecular Rotation in Gases and Liquids (CambridgeUniversity Press)

[2] ST PIERRE A G and STEELE W A 1981 M olecPhys 43 123

[3] GORDON R G 1963 J chem Phys 39 2788

[4] LEICKMAN JCL and GUISSANIY 1981 M olec Phys42 1105

[5] AGUADO-GOMEZM and LEICKMAN JCL 1986 PhysRev A 34 4195

[6] AGUADO-GOMEZM and LEICKMAN JCL 1987 PhysRev A 35 286

[7] BORISEVICH N A BLOKHINA PZALESSKAYAG ALASTOCHKINA V A and SHUKUROV T 1984 IzvAkad Nauk SSSR Ser Fiz 48 709

[8] KALMYKOV YUP 1985 Opt Spektrosk 58 804 [1985Opt Spectrosc 58 493]

[9] KALMYKOV YU P and TITOV S V 1999 J molecS truct 479 123

[10] GORDON R G 1963 J chem Phys 38 1724[11] GORDON R G 1964 J chem Phys 41 1819[12] GORDON R G 1964 J chem Phys 40 1973[13] BERNE B J and HARP G D 1970 Adv chem Phys

17 63[14] LEICKMAN JCL and GUISSANIY 1984 M olec Phys

53 761[15] VARSHALOVICH D A MOSKALEV A N and

KHERSONSKII V K 1988 Quantum Theory of AngularM omentum (Singapore World Scienti c)

[16] LANDAULD and LIFSHITZEM 1976 M echanics ACourse of theoretical Physics 3rd Edn Vol 1 (OxfordPergamon Press)

[17] BOiumlTTCHERC JF and BORDEWIJK P 1979 Theory ofElectric Polarization Vol 2 (Amsterdam Elsevier)

[18] RODRIGUEZ R and MCHALE J L 1988 J chemPhys 88 2264

[19] BURSHTEINA I and MCCONNELL J R 1989 PhysicaA 157 933

[20] KALMYKOV YU P and MCCONNELL J R 1993Physica A 193 394

1918 Spectral moments of CFs of asymmetric top molecules

iexcl I zhellipI x Dagger I ydaggerhellip5I 2x iexcl 2I x I y Dagger5I2

ydaggerŠgm2

Daggerx I z f4hellipI x iexcl I ydagger2permil6hellipI x Dagger I ydagger3

iexcl I zhellip31I2x Dagger106I x I y Dagger 31I2

ydaggerŠ

Dagger64I2zhellipI x Dagger I ydaggerhellip5I2

x iexcl 7I x I y Dagger5I2ydagger

iexcl 4I3zhellip31I2

x Dagger26I x I y Dagger31I2ydagger Dagger24I4

zhellipI x Dagger I ydagger

Dagger15xI zpermil2hellipI x iexcl I ydagger2hellip3I x Dagger I ydaggerhellip I x Dagger3I ydagger

iexcl 2I zhellipI x Dagger I ydaggerhellip11I2x iexcl 14I x I y Dagger11I2

ydagger

Dagger2I2zhellip3I 2

x Dagger2I x I y Dagger 3I 2ydagger

Daggerx I zhellipI x Dagger I ydaggerhellip5I2x iexcl 2I x I y Dagger5I2

ydaggerŠg

In this derivation we have assumed that

hK ii0 ˆ 0 hellip i ˆ x y zdaggerand

hK iK ji0 ˆ 0 hL iK ji0 ˆ 0 for i 6ˆ j hellip i j ˆ x y zdagger

References[1] BURHSTEIN A I and TEMKIN S I 1994 Spectroscopy

of M olecular Rotation in Gases and Liquids (CambridgeUniversity Press)

[2] ST PIERRE A G and STEELE W A 1981 M olecPhys 43 123

[3] GORDON R G 1963 J chem Phys 39 2788

[4] LEICKMAN JCL and GUISSANIY 1981 M olec Phys42 1105

[5] AGUADO-GOMEZM and LEICKMAN JCL 1986 PhysRev A 34 4195

[6] AGUADO-GOMEZM and LEICKMAN JCL 1987 PhysRev A 35 286

[7] BORISEVICH N A BLOKHINA PZALESSKAYAG ALASTOCHKINA V A and SHUKUROV T 1984 IzvAkad Nauk SSSR Ser Fiz 48 709

[8] KALMYKOV YUP 1985 Opt Spektrosk 58 804 [1985Opt Spectrosc 58 493]

[9] KALMYKOV YU P and TITOV S V 1999 J molecS truct 479 123

[10] GORDON R G 1963 J chem Phys 38 1724[11] GORDON R G 1964 J chem Phys 41 1819[12] GORDON R G 1964 J chem Phys 40 1973[13] BERNE B J and HARP G D 1970 Adv chem Phys

17 63[14] LEICKMAN JCL and GUISSANIY 1984 M olec Phys

53 761[15] VARSHALOVICH D A MOSKALEV A N and

KHERSONSKII V K 1988 Quantum Theory of AngularM omentum (Singapore World Scienti c)

[16] LANDAULD and LIFSHITZEM 1976 M echanics ACourse of theoretical Physics 3rd Edn Vol 1 (OxfordPergamon Press)

[17] BOiumlTTCHERC JF and BORDEWIJK P 1979 Theory ofElectric Polarization Vol 2 (Amsterdam Elsevier)

[18] RODRIGUEZ R and MCHALE J L 1988 J chemPhys 88 2264

[19] BURSHTEINA I and MCCONNELL J R 1989 PhysicaA 157 933

[20] KALMYKOV YU P and MCCONNELL J R 1993Physica A 193 394

1918 Spectral moments of CFs of asymmetric top molecules