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Resonance Raman SpectraInterpretation by MolecularCoherent State RepresentationN.Yu. Trifonov aa 42-6 Kommunisticheskaya St., Minsk, Byelorussia 220002,USSRPublished online: 03 Dec 2010.

To cite this article: N.Yu. Trifonov (1992) Resonance Raman Spectra Interpretation byMolecular Coherent State Representation, Journal of Modern Optics, 39:1, 21-34, DOI:10.1080/713823352

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JOURNAL OF MODERN OPTICS, 1992, VOL . 39, NO. 1, 2 1 -34

Resonance Raman spectra interpretationby molecular coherent state representation

N. YU . TRIFONOV42-6 Kommunisticheskaya St ., Minsk, Byelorussia 220002, USSR

(Received 19 December 1990; revision received 17 April 1991)

Abstract . In this paper the problem of exact summation over vibrational statesof intermediate electronic level excited by resonance Raman scattering in amolecular system with an arbitrary number of vibrational modes is solved withthe help of the coherent state representation . For matrix elements of theresonance Raman (RR) amplitude the expressions in the form of an integral andmultiple series as well as recurrence relations are obtained . The interpretation ofRR spectra of linear-chain mixed-valence platinum complexes is given as anillustration .

1 . IntroductionResonance Raman (RR) band intensities in molecular systems are described, as a

rule, by the second-order perturbation theory [1] that connects intensities with thepolarizability tensor . The vast field of RR spectroscopy on crystals is usuallydescribed by the third-order perturbation theory [2] using a susceptibility tensor .Though historically the descriptions of RR experiments in molecules and solids havedeveloped independently, the fundamental physical processes are identical . Hence,in [3], an attempt to apply the perturbation theory of the third order to molecular RRscattering was made. As shown in [4], the methods [1] and [3] are equivalent .

The RR intensity of the v-w transition between vibrational levels of the groundelectronic state g of the cold (T=0 K) molecule under the influence of incident lightof frequency co o is connected with the dynamic polarizability tensor a„ r ,, . Hence wecan put down

1. ti (w0 -(OV)4lavwl 2 .

( 1 )

The vectors in this expression have n components according to a number of excitedvibrational modes in the system . Thus,

v=(v1,v2, . . .,v„)

corresponds to excitation of v 1 quanta in normal mode 1, v 2 quanta of mode 2, etc ., asa result of RR scattering in the ground electronic state . The initial molecularvibrational state is described by the vector

w=(w1, w2 , . . . , wa) ;

the vector

to =(w 1 , cot , . . . , (o„)

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represents a set of excited normal vibration frequencies of the system . The scalarproduct is defined by the usual means :

cov=CO 1V 1 +m2v2 + . . .+CO„v,, .

In standard approximations of RR theory [1] (adiabatic Born-Oppenheimer forelectron-nuclear motion, Condon for dipole electric momentum, harmonic forvibrational motion) f as well as in the suggestion of single resonant electronic state e(the rest excited electronic states are supposed to be removed), the polarizabilitytensor may be written as follows :$

<vImXmIW>IXvw= const

(2),r, AE,, +CD(m-w)-Coo - lhem

Here AEeg is the electronic energy gap, Fem is the relaxation rate of the state em, i.e .the total damping constant, which is determined by all the homogeneous broadeningmechanisms,§ <.I .> are overlap integrals for vibrational wave functions written inFock space [5], and variables

m=(m1,m2, . . .,m„)

are numbers of vibrational quanta in the intermediate electronic state .The expression (2) under review shows that in solving the problems of RR

intensity calculation one should make a summation over the multidimensionalvibrational space of the excited electronic state . The difficulty of this problemusually compels to solve it by fitting to the best coincidence with experimentaldata with regard to only some lower vibrational levels. One of the first attempts totake into account the contribution of all the intermediate vibrational states in one-mode molecular system was realized, evidently, in [6] . In the review [7] difficulties ofsumming-up over intermediate states in the case of multidimensional vibrationalspace as well as variants of getting them over with the help of a transform method (forexample [8, 9]) were examined. This method was developed from the time-correlatorformalism [10] on the basis of the theorem relating absorption and RR intensities asincident photon frequency functions .

All these approaches are rather good for calculating RR spectra on powerfulcomputers, but are quite inconvenient for simple analysis . In this paper one can findsolutions of the problem formulated above using the coherent state (CS) represent-ation described in detail in [11]-[14] . CSs are used to describe a multidimensionalharmonic oscillator modelling vibrations in a large molecular system-so-calledmolecular CSs . In contrast to the approaches noted above, which are good forcalculating RR spectra with the help of powerful computers only, we develop anapproach that is convenient both for computer calculations and for a simpler andmore intuitive analysis .

t The object of interest is a large molecule, and so the effect of anharmonicity isinsignificant .

$ Herein and below, units in which c = h = 1 are used .§ Vibrational levels of the ground electronic state are operating initial and final states in an

act of RR scattering, while vibrational levels em of the excited electronic state are virtual .Thus, relaxation processes in the excited electronic state as well as inhomogeneous effects donot influence the RR band width . Therefore, for RR spectra, effectively, I'..=F.

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Resonance Raman spectroscopy

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In section 2, which is mainly methodical, a detailed description of the summationmethod for a case of excitation of a one-vibrational mode in molecula is given . Insection 3, on the basis of the developed method, the overall intermediate-state sum inthe common case of excitation of a multidimensional vibrational space is performed .Section 4 illustrates the use of the method for RR spectra interpretation .

2 . System with one vibrational degree of freedom2.1 . Notation

In this section we examine the case of excitation of one vibrational degree offreedom (n =1) only . Hence, we rewrite (2) in the following form :

06„Wconst<vIA(a')Iw>,

(3)

"0 IMXMI

A(Q) YoQ'+m-w'

(4)

o'=(AEeg -wo -iF)/w . (5)

Our first goal is the exact calculation (in the harmonic-oscillator approximation)of a sum in a matrix element of an operator (4).t For this purpose, let us use a Fock-Schwinger eigentime representation (see, for example, [15]) . In the framework ofthis representation a matrix element of RR amplitude may be written as

00

<vIA(a')Iw>= EJ: df exp[- fl(a'+m-w)]<vlm><mlw> .

(6)M=0 o

The matrix element (6) will be the subject of our investigation . We shall study thematrix element with the help of CSs for a harmonic oscillator, modelling vibrationsin a molecular system .

2.2. Expressions for amplitudes in coherent state representationThe matrix element

<ylA(6)Ia>=eXp[-z(IaI2+Iyl2)]o,w=o (v•w .

) ,/,aWy

<vIA(6)Iw>,

(7)

is a generating function for <vIA(a')Iw> . Replacing the expression (6) of theamplitude matrix element by the eigentime fi into equation (7) we obtain theexpansion

<yIA(a')Ia>=J

dfl exp [ -ilal2 (1 - e2P )] Y e- e( Q+')<ylm><mle 1a> .

(8)o

m=o

One can calculate the integral in equation (8) in several ways . It is moreconvenient to do this using a coordinate representation . We will model a vibrationalmode excited in a molecula by a harmonic oscillator . We should remember that thewave functions of ly> and lefla> concern the initial electronic state, g, while the Im>functions concern the excited one, e, the potential minimum position of which wewill consider to be displaced on the value of R . Into the expansion of (8) we cansubstitute coherent states of harmonic oscillator, representing Gauss exponents [11]

j' Note that the kernel of summation (4) is, in essence, the Green function of a one-dimensional quadratic system .

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and standard normalized wave functions manifested (for example [16]) via Hermitepolynomials, Hm , as well. In the obtained expression, we can accomplish thesummation over quantum number m using Mehler's formula for a product ofHermite polynomials [17] . Then, having integrated over coordinate variables weobtain, for an amplitude generating function,

<ylA(6)Ia>=exp [_1(IaI 2 +Iyl2) - S2 ]

Jdfe_

0"exp [(aefl-8)(y*-b)e -d+6(aes+y*)], (9)0~

where S is a dimensionless displacement of the oscillator potential minimum under atransition into the excited state

b=R(µ(o/2)1 12 ;

(10)

µ is a reduced mass of the modelling oscillator . The formula (9) is an integralexpression for the matrix element (6) of an RR amplitude in the representation ofcoherent states of the molecular oscillator .

The described method of obtaining the integral expression (9) is not suitable forgeneralization in the case of excitation of an arbitrary number of vibrations in asystem. So let us consider other ways .

First, we make some calculations of an auxiliary character . Proceeding from thenotation of generating functions, we have

<yIS><SIa>=exp(-I CIZ)

I~I~m<yImXmIa> .

(11)m=o m.

Now we use the results of [18] ; hence the scalar product of CSs for harmonicoscillator can be transferred to

2x<vIa>=x ' 2 eXp[-i(I«I2 +IYI2 +a2)]exp[ (a-S)2+-(a-S)y*- y*2 +a5

x

2x

11

where the variables x and 1 are connected with the oscillator frequencies of w and (0 'in the states of Ia> and Iy>, respectively,

x=2(w~,)1~2 ,

= 2(ww,)1 1 2 .

(13)

In our case the oscillator frequency in both states is constant,

w=w'.

(14)

Taking into account the equalities (12)-(14), the expression (11) can be put down inthe form :

1

2"exp [ - iI«I 2 +IYI 2 ) -S2 +(a+y*)b]2~

dB exp {[(a- 5)e-`B+(y*-b)e`B]I~I}0

m0I~m! <ylm><mla)

.

(15)

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Resonance Raman spectroscopy

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An integral in this equality is a modified Bessel function of the first kind andzeroth order [17] with the following expansion in series :

Io(z) = E(Z2/4)'

(16)m=o

(M I)2

Using this expansion in equation (15) we obtain another representation for agenerating function of the amplitude matrix elements :

<YIA(a')lad=exp [-i{Iai2+Iy12 -b2 +(a-Y*)b]

x [(Y*-b)(b-a)]-(a -`Y[6'-w, (Y*-b)(b-a)] . (17)

In this formula y[ ., .] is the incomplete gamma function [19] .We may show that the expression (17) is equivalent to (9) . Generalization of (9)

and (17) is given in Appendix Al .

2 .3 . Expressions for amplitudes in the Fock representationNow let us pass from the integral expression for matrix elements of <yJA(a')la>

amplitude in the representation of a harmonic oscillator CSs to the expression for<vIA(o')Iw> matrix elements in the usual (Fock) representation . For this purpose, weshall use an expansion of the quadratic exponent into a Laguerre polynomial series[20] :

exp (y*x+ ay + y*az) = E Cvwy*vaw

,v,w=0

-1Cvw-

v+w 2Iv-wI(

x(v-w+lv-wl)/2y(w-v+lw+vl)/2Z(v+w-Iv-wb/2LIv+w-Iv-wl)12 -ZSubstituting this into (9) for the <yIA(a')Ic > matrix element, we expand CSs intodiscrete (Fock) states . We obtain the integral representation for the <vIA(a')Iw>matrix element of RR amplitude :

<vIA(6 ,)Iw>=Cmin (v, w)! \ 1/2 f'0

dfl exp [62(e# -1)-#Q ' ]max (v, w)!

o

x (26e -P/2shflIv-wIL

Imin (v, w) -462sh22 ,

(18)

where min (v, w) and max (v, w) are the minimum and maximum values of v and w,respectively .

If one considers separately only Stokes or anti-Stokes processes, one can simplifyequation (18) . For example, if v>,w, having replaced an integral variable, wecalculate that

w! 1/2

1

_6 2 t

a' 1 u-w

62 t 2<vIA(Q)Iw> =6`1

z0) J0dte (1-t) Lw

- 1-t

Integrals of the (18) and (19) type can be calculated and they represent a sum ofproducts of the gamma function F( .) and confluent hypergeometric function(D() [19] . For example,

<vIA(a')IO>= F(Q)bv(v!)1/24)(v+1,o'+v+(20)

F(o'+v+1)

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This formula coincides with the expression given in [6] . From the representation(17) one can obtain an expression equivalent to equation (20) :

<vlA(a')I0>=F((T') bi~°°T'(n+v+1) . (- bz)" -

(v!) /z n=O F(a'+n+v+1

)

n!

2.4. Recurrence relationsKnowing the generating function <yJA(a')Ia) gives us the opportunity to obtain

recurrence relations for the <vIA(a')Iw) matrix elements . Taking the derivative ofthis value with respect to the CS parameter a we find

a<YIA(Q)Ia)=(y*-b)<YIA(o-')Ia>+b<YIA(a'-1)Ioc) .

(21)

Expanding into a and y* power series and equating coefficients of equal powers in theright and in the left sides of the expression, we obtain the recurrence relation

1/z

<vIA(Q)Iw+l)=+l

<v- 1IA(6)Iw)

+(wS

+1)1J2 [<vIA(a' -1)Iw>-<vIA(a')Iw)] .

(22)

Taking the derivative of the generating function with respect to y*° we obtain byanalogy :

1/z<v+ 1IA(~)Iw)= v+1

<vIA(a')Iw+1)

+ (v +1 )1/z[<vlA(a')Iw)-<vIA(~ +1)Iw)] .

(23)

The recurrence relations (22) and (23) connect matrix elements between themwhen arguments a' differ by 1 . For matrix elements <vJA(a')I0), with a vibrationlessinitial state, one can obtain recurrences connecting RR amplitudes with the samevalue a'. For this purpose, it is necessary to use their evident expression via specialfunctions .

For example, using the Gauss formula for contiguous relations betweenconfluent hypergeometric functions [19] we obtain

b<YIA(a +1)I0>= yea b<YIA(a)I0)-exp

_1Substituting this expression into (21), making differentiation procedures andequating again obtained coefficients of equal powers of CS parameters, we have

1/2

z<v+1IA(6)IO)=- v+1)

<v-1IA(a')I0)+ ~( +1+2<vIA(Q)IO) -bac ,

(24)

where 6,, 0 is the Kronecker delta . This recurrence relation (24) coincides with oneobtained by another method in [21] .

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3. System with a finite number of vibrational degrees of freedom3.1 . Formulation of a problem

In this section we consider a case of excitation of several normal vibrations in amolecular system . A vibrational structure of the ground electronic state is modelledby a multidimensional harmonic oscillator with the Hamiltonian

1 nAf

E wj(a,.a +aj a;),

(25)2j=1

where cod is the frequency corresponding to the j-th normal mass-weightedcoordinate xj , through which the generation operators at and annihilation operatorsaj of vibrational quanta are expressed in the usual way :

a,=T2 (x;(

(O

;)112+ip;l (

(O;)"2),

1aj = 1 (x;(w;)1i2-ip;l(w;)u2) .

It is suggested that in excited electronic state the molecular Hamiltonian takesthe form

n

.e'=- E wj(bb] +bj b;),

(26)2j=1

where w is the frequency corresponding to a normal mass-weighted coordinate

x' = xj- Rj .

(27)

Ri is the value of the molecular potential surface minimum shift in the excited state(relative to the ground state) along the j-th coordinate, bj is annihilation operator ofthe j-th vibrational mode quantum in the excited electronic state, and b; is theadjoint of bi . As in the one-dimensional case in the final formulae we suppose

vi

w;=coj .

(28)

We consider, as usual, that the initial molecules are in a vibrationless state . Thissimplifies the obtained formulae . Then, we can rewrite the polarizability tensor (2) inthe following form :t

Quantities in these formulae were explained earlier .

f The RR amplitude operator G(a) is proportional to the operator A(v') and, in contrast tothe latter, it is not measured in units of any normal vibration frequency with symmetrypreservation aim with respect to all n excited vibrations in the system .

a,,,=const<vIG(a)IO>, (29)

G(a)=~ImXmi , (30)

m Q+com

a=AEeg-coo -it . (31)

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For the RR amplitude matrix elements <vIG(a)IO), by analogy with the one-dimensional case (6), one may pass to an eigentime representation, where

<vIG(o)IO)=Y

d/3e -B(o- m )<vIm)<mIO> .

(32)m J o

~

And again, these matrix elements are the main consideration .

3.2. Expressions for amplitudes in the coherent state representationMatrix elements of RR amplitudes in the CS representation of a multi-

dimensional harmonic oscillator should be written in the following form :*v1

*v

<YIG(a)IO> =exp (-iIYI2)y Yl . . . y 1

/2 <vIG(o)10)(v l ! . . . V"!)

To simplify similar expression by analogy with Bargmann [22], we mustintroduce the designations

v! =v1 ! . . . vn !, y*v=y*vi. . . Yn v,

in accordance with which we can rewrite the previous expression in the form of

<YIG(o)IO>=exp(-iIYI 2)y_

Y 1/2 <vIG(o)IO>,

(33)(v!)

and the generating function for (32) will be

<YIG(c)IO)=Yf

dpe-P(Q +`°m)<YIm)<mjO> .m o

In accordance with section 2, where the method of calculation for the one-vibrational mode was set forth, let us consider the product of the multidimensionaloverlap integrals

2m

<YIC)<CIO)=exp(-Id1 2)Y ICI <YImXmIO)-m M!

The scalar product of two multidimensional CSs in the case of oscillatorcoordinate transformation under excitation according to the rule (27) can bepresented in the form of a product of one-dimensional ones :

n

<YIo = j=1II <Y;Idj) .

Having used for each of the one-dimensional scalar products the presentationsimilar to (13), we can rewrite the previous equality in the following way :

<YIC)=exp [- i( ICI' +IYI 2 +a 2)+CS]

x n -1/2 ex

S 21

O .

I~Ux;

p 2x' (~;- j ) +,(Cj-6j)Y*`-2~Y;` 2J

,

where

wj+wjwj-wjxi=

2(wwj) 1/2 ,

~ j2(w , ) 1/2

(34)

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where S is a vector compounded of dimensionless displacements S ; along eachnormal coordinate :

S=0" . . .' Sn)

Hence

<YIC><C10>=exp [-ICI Z -zIYI Z -SZ +y*8] II x; 1 exp

' [(Yi - g.l) 2 +Sj]

I=1

2xi

+-- [(y7 -S;)C;-C*S;] - 2z, (C; +~7 2 ) } .

Now let us fulfill the condition (28), i .e . let Vj x;=1, O;=0. Taking into account(34), we obtain

ISI2m

Y mi <YIm><mIO>=exp[ - zIYI Z-SZ +(y*- S)C- *8+y*S] .

Further, using the representation of the modified Bessel function Io in the formof a series, we obtain

<YIm><mIO>=exP( - iIYI2- S2 +y*S) fl[(S;-Y;')S;]'i .

=1

m;!

This enables us to write down the overall multidimensional sum (33) in CSsrepresentation : t

<YIG(~)IO> =exp [-iIYI2-(S-Y#)S]Q[(al-Y*)S1,(35)

In the expression

,ZMQ[z1, . . . , z1 ; w1' . . . , COn ; c]

(36)m m!(wm-c) .

Some properties of the Q-function are given in Appendix A2 .

3 .3 . Expressions for amplitudes in the Fock representationThe integral expression for generating functions of multidimensional RR

amplitude matrix elements follows from that for the special Q-function (seeAppendix A2) :

<YIG(a)IO> =exp (-iIYI2)f"0

dfe-P6 fl exp [-Sj(1-e-P°'j)+y,y,6;(1-e-Pwi)] .o

i=1(37)

Expanding the equality obtained into a series in powers of parameter y#, wededuce the integral expression for the RR amplitude matrix elements in Fock space :

<vIG(c')IO>= J o d&-O°[S(1(y'e1/2-)]v fl exp

(1-e-~° )] .

Resonance Raman spectroscopy

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(38)

]' The equality (35) can be deduced also by another method [23], by analogy with themethod described in Appendix Al .

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Calculating the integral (37) one can derive the amplitude matrix elementrepresentation in terms of a multiple series :

<vlG(a)I0>=8

112 [A e-'5' : b2k;(kji) -1

-1) i ;()I[v'

6+Jt

1wi(k,+l;)1-1

(V •)

1=1

kj=0

i' = o

lj

-

J

(39)

where\v'/

are binomial coefficients .l;

3 .4 . Recurrence relationsRecurrence relations for generating functions of RR amplitude matrix elements

with different arguments result from the relation (A5) for a special Q-function withcontiguous values of energy variable :

n(b;-y11)5 .w <yIG(a+w;)I0>+a<yIG(Q)I0> -exp( - i1v1 2 ) =0 .

i=1

Expanding matrix elements into a series in powers of the CSs parameter andequating terms of the same power, we obtain

n 6.0Y [bjw;<vIG(a+w;)IO>-b;w;(v;)1~2<v;-1IG(a+w;)I0>]+a<vjG(a)10>- (vi)1/2=0)j=1

(40)

In this formula we use the designation

<v;-lIG(a+w;)IO>=<v1, . . .,v;-1, vi - 1, v;+1, . . .,vnlG(a+w;)IO>,

where b„0 is an n-dimensional Kronecker delta .The recurrence relation (40) connects the amplitudes with arguments differed on

a quantity of normal frequencies . It is interesting to get a connection of matrixelements of RR amplitudes with exciting only any one normal coordinate . Havingdifferentiated the integral expression for the matrix element <yIG(a)I0> (37) withrespect to the chosen y;*, and having done the same with expansion of this matrixelement with respect to the Fock states (33), let us equate the series obtained . We find[24] that

vi

(v;+1)1"2<v;+1IG(a)IO>=b;(<vIG(a)IO>-<vlG(a+ (o;)IO>),

(41)

where the following designation is accepted :

<v;+lIG(a)l0>=<v1, . . . . v;-1, v;+ l, vi+1, . . .,vnlG(a)I0> .

Using (41) in common relations (40) we can get rid of expressions with displacedarguments of the a+w; type. As a result, we obtain the following symmetricalrelations connecting the RR operator G(a) matrix elements :

nY_ [boi((v;)"2<v;-IIG(a)IO>-(v;+1)"2<v;+1IG(a)IO>)+ (b;+v;)w;<vlG(a)IO>]i=1

+a<vfG(a)IO>-b,,0=0 .

(42)

In these relations all the matrix elements depend upon one and the same argumentand differ by an excitation number of different normal vibrations . In the case of onlyone normal coordinate, the relations (42) are coming to recurrence relations (24)obtained earlier .

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Resonance Raman spectroscopy

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4. IllustrationLinear-chain mixed-valence platinum complexes are solid crystal salts under

standard conditions. The platinum exists in the ground state of these molecules intwo different environments, square planar and octahedral with formal oxidationstates, + 2 and + 4 respectively. Platinum ions are bridged by halogens . A moleculeformed in this way is a linear chain with alternate valences of chain-making platinumions. Physical properties of such compounds are highly anisotropic, and they may betreated as quasi-one-dimensional semiconductors [25] .

RR spectra of complexes of interest [26] are observed if molecules are irradiatedby light with frequencies in the absorption band . Electron transition is electricallyallowed, and RR spectra are characterized by long intense progression w 1v1i wherew 1 is the totally symmetric stretching vibration of the Pt +4-X chain; X is thehalogen. The anharmonicity constants for such modes are on the average -1 .0 cm -1for chlorides, and about -0 .3 cm -1 for bromides. This is in agreement with thegeneral observation that the vibration-forming long progression of RR overtones isharmonic. The intense progression of the main vibration w 1 is accompanied by otherweak progressions, the enabling mode of which is another Raman-active mode, whilethe progression-forming mode is, as for the main progression, w 1 .

With respect to equation (1), from the ratio of intensities of the RR bandscorresponding to different sets v 1 and v2 of vibrational quanta, we obtain

I„Zo -_(COO-wvz 4

Iv , o wo -wv1<v2IG(a)IO>

<v1IG(a)IO>

2

en = 1,2-diaminoethane H 2N-CHZCHZNH2 .

(43)

where matrix elements <vlG(a)10> are given by (38) and (39) and are dependent ondimensionless displacements S of the molecule potential surface minimum .

From experimentally measured values of the RR intensities of the first overtoneand the fundamental of the main progression-forming mode w 1i with the help ofequation (43), we found the magnitudes of the dimensionless displacement . Thedisplacements and the corresponding axial Pt ' 4-X bond length changes R (10) onexcitation from the ground to the resonant excited state for the compoundsinvestigated are listed in table 1 . Displacements along other excited normalcoordinates were obtained in the same way . So we have extracted the excited-stategeometry of such molecules from the RR spectra [27] .

With the help of displacement magnitudes by recurrence relations of section 3 .4we computed relative overtone and combination band intensities, i .e. intensities withrespect to the intensity of the main progression-forming mode w 1 . Calculated RRspectra of these complexes are in good agreement with experimental ones . Thistestifies to the reliability of the obtained excited-state geometry parameters .

Table 1 . Computed excited-state geometry changes for linear-chain mixed-valenceplatinum complexes .

Complex S R (A)

[Pt(en) 2][Pt(en) 2C1 2](C104)4 2.54 0. 21[Pt(en) 2][Pt(en) 2Br2](C104)4 1 .77 0. 15[Pt(en)Cl2][Pt(en)C1 4 ] 2.00 0. 17[Pt(en)Br2][Pt(en)Br 4] 1 . 99 0.17

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N. Yu. Trifonov

The author of [26], who did not succeed in extracting such information from RRspectra of these platinum complexes using other methods, suggested, in terms ofchemistry, that the Pt +4-X bond length changes on excitation, perhaps by sometenths of an angstrom; this agrees with the magnitudes we obtained .

5 . ConclusionWe now consider the principal results obtained in this paper . With the help of the

CS method, we carried out the exact summation of the contributions from all themultidimensional vibrational diversity of the intermediate electronic state into thepolarizability tensor of RR scattering in a polyatomic system . For the RR amplitudematrix elements, closed expressions, using molecular system parameters andexperimental conditions, were obtained . Amplitude matrix elements are expressedusing a Q special function . Recurrence relations for Q-functions and some types ofrecurrence relations for matrix elements of RR amplitudes are deduced . In the caseof the excitation of one normal mode, the expressions obtained are as previouslyachieved by other methods .

The developed formalism enables one to solve the problem of extracting excited-state geometry parameter according to RR spectroscopy data as well as RR bandintensity calculation .

Application of the method to the spectra of concrete species has been carried out .For illustration, we chose linear-chain mixed-valence platinum complexes, thecalculations for which cannot be provided by other methods .

AcknowledgmentThe author wishes to thank E. V. Doctorov for numerous helpful discussions .

AppendixAl . Generalization of equation (9)

In view of possible subsequeant generalizations we obtain an expression formatrix elements in the coherent state representation without the condition (14) .Remembering [17] that the exponent exp [2xz-z2 ] is a generating function for theHermite polynomials Hm(x) and taking into account that the scalar product <yIa) is agenerating function for the product <m1a) in a mixed basis we obtain from equation(12) the necessary expression for the product <mIa) included in equation (8) :

<mIa)=expC- Ja 12 + 2 (a-S)2 +ab-Z82I (Z~mx ) J2Hm ((2XO ) 1/2) (A1)Substituting the equality (A 1) into the initial expansion (8) and using, as it was

done above, the Mehler formula for the Hermite polynomials product, we obtain anintegral expression for the matrix elements of RR amplitude in the coherent staterepresentation in the following form :

<YIA(o')Ia)=exp [-z(IaI 2 +IYI 2)-S2]

x J o dfl (x2 x02e

x)1/2 exp fx2e~s~ _. [(ae p(x+w)2-g2)+(Y* -b)2 ]

[aeP(x +W)2 - 6](Y* -6) l+2eP-1P2e_s

+S[aeP(x+y)2+y*]} .

(A2)X

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33

This integral expression (A 2) has a more common nature than (9) becausededucing the first one it was suggested that frequencies of the normal mode in theinitial and in the excited states are different . If one takes a condition (14) offrequencies equality, i .e. X=1 and =0, one will come back from (A2) to theexpression (9) .

A 2. Special Q-functionThe Q-function like (36) appeared [12] in connection with investigations of

various systems in the CS representation of quadratic Hamiltonians . The functiondepends on 2n + 1 variables. It is convenient to name the first n variables, zj, as themain variables, the second group of variables, wj, as frequency variables, and the lastvariable as energy .

Somehow this function is one of the natural generalizations of a confluenthypergeometric function in a multidimensional case . Namely, with the identicalfrequency variables w 1= . . .wn =w the Q-function comes to a confluent hypergeo-metric one :

1

n

a

nQ[z1, . . .,zn; w, . . .,w; -a]=-exp > zj 4 1,1--, - Y .zj -a

j=1

w

j=1

In the case of a one vibrational mode the expression (35) comes to (20) . If onerewrites (36) in eigentime fi representation and sums over states m, one can obtain theintegral representation for the Q-function :

f 000

nQ[z1, . . ., zn ; wl, . . .,w,, ; -Q]=

dfl exp -/iQ+

zjexp(-fiwj) . (A 3)0

j=1

Hence, we immediately have a formula for a Q-function differentiation along anymain variable :

aQ[z1, . . .,zn ; wl, . . .,wn ; -a]=Q[zl, . . .,zn ; wl, . . . ,wn; -a-wj] . (A4)Ozj

Integrating the representation (A 3) by parts, one obtains the recurrence relationconnecting the Q-function with different energy variables ; that is,

n

Y, wjzjQ[z1i . . . , zn ; w1, . . . , wn ; -a-wi]j=1

+UQ[z1, . . .,z,, ; COI, . . .,wn; -v]=exp( n Z). (A 5)r=1

If we consider the one-dimensional function Q[z ; w ; -a], there remains onlythree terms in recurrence relation (A 5) and we come to Gauss relations forcontiguous confluent hypergeometric functions [19] .

References[1] ALBRECHT, A. C., 1961, J. chem. Phys., 34, 1476 .[2] LOUDON, R ., 1964, Adv. Phys ., 13, 423 .[3] PETICOLAS, W . L., NAME, L., STEIN, P., and FANCONI, B., 1970, J. chem . Phys ., 52,1576 .

[4] GAROZZO, M ., and GALLUZZI, F ., 1976, J. chem . Phys ., 64, 1720 .[5] FOCK, V ., 1932, Zs. f. Phys ., 75, 622 .

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[6] PREEM, R., 1963, Trudy Inst . Fiz. Astron . Acad. Nauk Est. SSR, 20,114 (in Russian) .[7] CHAMPION, P . M., and ALBRECHT, A . C., 1982, A. Rev. phys . Chem ., 33, 353 .[8] TONKS, D. L., and PAGE, J . B., 1979, Chem. Phys . Lett ., 66, 449 .[9] TANNOR, D . J., and HELLER, E . J ., 1982, J. chem. Phys., 77, 202 .[10] HIZHNYAKOV, V ., and TEHVER, I ., 1967, Phys. Stat . Sol ., 21, 755 .[11] GLOUBER, R . J ., 1963, Phys. Rev., 131, 2766 .[12] MALKIN, I . A., and MAN'KO, V . I ., 1979, Dynamical Symmetries and Coherent States of

Quantum Systems (Moscow : Nauka) (in Russian) .[13] PERINA, J., 1984, Quantum Statistics of Linear and Nonlinear Optical Phenomena

(Dordrecht : D. Reidel) .[14] PERELOMOV, A . V., 1986, Generalized Coherent States and Their Applications (Berlin :

Springer) .[15] FoCK, V., 1937, Phys. Zs . Sowiet ., 12, 404.[16] FLUGGE, S ., 1971, Practical Quantum Mechanics, Vol. 1 (Berlin : Springer) .[17] BATEMAN MANUSCRIPT PROJECT, 1953, Higher Transcendental Functions, Vol . 2 (New

York: McGraw-Hill) .[18] MALKIN, I. A., MAN'KO, V. A., and TRIFONOV, D . A., 1970, Phys. Rev ., D2,1371 .[19] ABRAMOVITZ, M ., and STEGUN, I . A. (eds), 1972, Handbook of Mathematical Functions

(New York: Dover) .[20] Husimi, K ., 1953, Prog. theor. Phys., Osaka, 9, 381 .[21] SONNICH MORTENSEN, 0 ., and HASSING, S ., 1980, Adv. Infrared Raman Spectrosc ., 6, 1 .[22] BARGMANN, V ., 1982, Rev. mod. Phys ., 34, 829 .[23] DOKTOROV, E. V., MALKIN, I . A., MAN'KO, V . I ., and TRIFONOV, N . Yu ., 1983, Soviet

Spectrosc.-Lebedev Inst . Repts., 58, 1034.[24] TRIFONOV, N. Yu., 1984, Doklady Akad. Nauk BSSR, 28, 804 (in Russian) .[25] TRIFONOV, N . Yu., 1988, Proc. of the Intern. Conf. and Intensive Tutorial Course on

Semiconductor Materials (New Delhi: University of Delhi), p . 152 .[26] CLARK, R . J . H., 1984, Adv. Infrared Raman Spectrosc ., 11, 95 .[27] TRIFONOV, N. Yu., 1988, Proc. of the 11th Intern . Conf. on Raman Spectroscopy

(Chichester : John Wiley), p . 601 .

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