Physica 106C (1981) 106-116 North-Holland Publishing Company

THEORY OF INTRAMOLECULAR ENERGY RELAXATION AND VIBRATIONAL OVERTONE RESONANCE ABSORPTION LINESHAPES OF ISOLATED POLYATOMIC MOLECULES

Mahendra PRASAD Department of Physics, McGill University, 3600 University St., Montrdal, Qudbec H3A 27"8, Canada

Received 2 May 1980

The vibrational dynamics of an isolated polyatomic molecule in interaction with a laser field is treated by using a graphical analysis of the molecular operators. The lineshape arises from any order of the perturbing intramolecular inter- action leading to an integral equation for the local mode scattering operator. Explicit calculation indicates that the identity of the local mode and the normal mode quanta can no longer be retained over hopping times which are comparable to mean free times. The renormalization method introduced here is quite capable of probing short time intra-molecular dynamics and improves on the strict Born approximation utilized by Helier and Mukamel earlier. Essential features of the lineshape remain the same except for the generalization to higher order effects and the inclusion of the interference processes.

1. Introduction

Discovery of an intense, monochromatic and coherent laser radiation has been playing important roles in different branches of physics and chemistry, too numerous in number indeed to spell out each explicitly [1 ]. Earlier investigations were made on the dynamics of a polyatomic molecule sitting in the ground vibrational state; however in recent years infrared laser photons have been injected to a molecule in the gas phase, which may lead to its vibrational excitation near the continuum. Intra-molecular dynamics of a polyatomic molecule in the excited region is hoped to provide additional information regarding the stability and the binding energy of the molecules. The excited state spectroscopy provides an interesting tool to study laser controlled chemical reactions involving large biological units where the polyatomic molecule may be forced to split into smaller units. Energy spacing of the vibrational spectrum is generally non-uniform due to the important role played by the anharmonic terms in the potential energy. Vibrational overtone resonance absorption lineshape arises when photons are absorbed by the active normal modes of the molecule such that n a > 1 (n a is the vibrational quantum number) and the mea- surements have been performed by Bray and Berry recently [2]. Absorption of the photons undergoes random interruption by the other modes which do not directly take part in absorption but simply perturb active modes while the quantum jumps are being completed. Random perturbation of the normal modes absorbing laser photons have been recently treated at length by Heller and Mukamel [3] (to be referred to as HM).

Due to the confinement of the vibrational excitation within the molecule the intra-molecular processes taking place over small length d and time scale r reveal additional information unavailable by other means of investiga- tions. Reduction in the length scale (d is of the order ofdm, the molecular dimension) means the additional renormalization of the energy of the vibrational states taking part in the absorption, whereas the short time dynamics (¢p = COp 1; COp being the average circular frequency of the molecule) imply that the memory effects can no longer be ignored as is customary with the usual methods of calculation.

In the present paper an earlier theory [4, 5] developed for treating constrained systems is extended to treat the vibrational overtone resonance absorption lineshapes of the isolated polyatomic molecules. The present theory is based on the proper connected diagram expansion of the super-operator resolvents and parallels the theory

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M. Prasad/Vibratlonal overtone resonance lineshapes 107

treated by HM, which employs the projection algebraic method used by Zwanzig [6] and Fano [7]. In some cases the results obtained by I-IM are confirmed, whereas in other, cases generalizations to include higher order effects and the interference processes may be found.

In section 2 representation of the basis and the hamiltonian of the system is defined in the notation followed by I-IM. General theory of the intra-molecular relaxation and the lineshape using the proper connected diagram expansion of the super-operator resolvents is formulated in section 3. Two limiting behaviours of the general expression are examined in section 4 and the connection is made with HM's result. Section 5 contains the contact calculation of the line-width. Section 6 gives a summary and a discussion of this paper.

2. The hamiltonian and representation

In this section the hamiltonian of a polyatomic molecule with special reference to benzene in interaction with the external laser field is discussed. Vibrational modes are ultimately split into two parts, one part contains all the six C-H stretching local modes which take part in the absorption whereas the other part includes the rest of the twenty-four stretching and bending normal modes playing the role of the perturbers. The purpose of the proper connected diagram expansion is similar to that described before [4, 5], namely to split the effect of the interrup- tion of the photon absorption by the fluctuating field of the perturbers into a relevant part described by the con- nected diagrams and an irrelevant part whose contribution is eliminated by the disconnected diagrams. Since the proper connected diagram expansion of the super-operator resolvents does not depend upon the representation, it holds irrespective of any chosen basis set.

The hamiltonian of the molecule in interaction with the laser radiation is given by

='g~'M +'glaR +'g#'MR, (2.1)

where ~"M is the part of the total harnfltortian which describes the quantized vibrational spectrum and the inter- action among the modes of the molecule.~f' R is the quantized laser field part of the Hamiltonian and is given by

"~R = Y~. hwq(b~bq + ½). (2.2) q

)~'MR represents the coupling between the active mode of the molecule assumed to be only C-H stretches and the radiation field

"~blR = d " ¢~_, (b; + bq). (2.3) q

+ and bq create and destroy a photon of frequency Wq and wave vector q. d is the dipole operator and depends bq on the co-ordinates of the six C-H oscillators (R 1 . . . R6) only.

In order to apply the perturbation expansion, eq. (2.1) should be rearranged in the form

M'= ~0 + ~,3e" (2.4)

where the eigenfunctions and the eigenvalues for ~0 are supposed to be known and 9/. induces the transitions. ~, is an expansion parameter. ~t'~O is further written as

"~0 ="~l~ n) -l-'~r~'C~1) + ~R, (2.5)

108 M. Prasad/Vibrational overtone resonance lineshapes

where.*~61) describes the Hamiltonian of the six C-H local mode oscillators given by

6 2 .~e~l) = ~ __Pet + ,~(l)(R~)

a = l 2m~, (2.6)

and. ;~ n) describes the vibrational Hamiltonian of the rest of the 24 normal modes

30

ct=7 +½m o (2.7)

The operator ~e" describes the intramolecular coupling

(2.8)

~e'l~ll) stands for the interaction between the local modes, ~ri~nn) is responsible for the coupling between the normal modes and ~r'/~ln) represents the interaction between the local and normal modes. ~(I)(R~) in eq. (2.6) will be taken to describe an anharmonic term and the eigenvalues of the C-H osciUators are then given by [8]

6

4! )-- ~ = 1

{#R.oa(nc~ + ½) _ Xe#l~a(n, , + ~)2}. (2.9)

X e measures the anharmonicity while the eigenvalues for the bath oscillators (chosen to be the normal mode of the molecule) read as

30 e~n)= ~ ~t~a(na+½). (2.10)

a = 7

The eigenvalues are labelled in the occupation number representation as Inl, n 2 . . . . . n 7 . . . . . n30). It is further remarked that due to the molecular symmetry (6-fold degenerate local modes) only one mode may be assumed to couple radiatively to the radiation field. In that case the ground state is labelled as Ig) = 1000000) and the doorway state as Ira) = In00000).

3. Lineshape operator

Derivation of the lineshape is a fundamental problem in time dependent statistical mechanics. At least three approaches have evolved in the past few years. The stochastic theory of lineshape derived by Kubo [9], the pro- jection operator approach advanced by Zwanzig [6] and Mori [10], and the graphical perturbation method des- cribed earlier [4, 5, 1 I ] , to name a few, are relevant to the problem in hand. In the present section this last approach, modified and applied to a different system [4, 5], is followed to derive the renormalized expressions for the lineshape operator in the one-resolvent approach. The lineshape function is given by

K(~) = Im [Tr (d+R(w)d)], (3.1)

where d is the dipole operator and depends on the relative displacement of the six C-H oscillators from their equilibrium position. Angular brackets denote the average over the 24 normal modes and will be referred to as

M. Prasad/Fibrational overtone resonance lineshapes 109

bath variables further in the text. R(6o) is a super-propagator which describes the dynamics of the vibrational excitation within the molecule. This operator can be expanded in powers of the interaction perturbing the local modes involved in the photon absorption. The successive terms of this series are represented by the modified diagrams. The local modes are represented by black circles e, their interactions with the bath modes are denoted by vertical (.I.) lines and a horizontal line segment ( - - ) stands for the free streaming of the quanta arising from the C - H oscillators. A relaxation mode competing with the bath mode denoted by a vertical straight lines should be distinguished from it by drawing a zig-zag line ending at the blackened circle.[. This diagrammatic series is then resummed over the proper connected graphs (see fig. 1) to generate the lineshape operator - , defined by

LR(w)) = (.~a 0 - w - -'- + ia)-l; a ~ 0+, (3.2)

",~co) = (--A,.~I + ;k2.i~'l (.i~' 0 -- 6o -- "-- + ia)-l.i#l + . . . ) , (3.3)

~0 is the unperturbed IAouville operator containing artharmonicities and ~c 1 contains the effects of bath perturba- tions. Most calculations including those of Fano and HM would obtain the following expression for "- in the Born approximation

",~(6~) = (--X.~ + X2-~'I(.L/' 0 - co + ia)-l.~¢l + . . . ) . (3.4)

However, in the present case an extra resummation over the proper connected graphs is made to obtain the integral equation for the lineshape operator given by eq. (3.3). This therefore is the point of departure of the present approach which goes beyond the Born approximation employed by HM. Some diagrams and definitions are given in fig. 1. Fig. la is a disconnected graph because it is broken into two by cutting a free line segment. Fig. lb is a third-order connected diagram. Fig. 1 c is a proper connected diagram and is obtained from fig. 1 d by suppressing the inner tetradic self-energy parts. This single diagram should be understood as representing both types of pro- cesses, namely outward and inward flow of local mode quanta. Fig. 1 d is the lowest order improper diagram. Lowest order interference processes occur in the fourth order which is displayed in fig. 1 e. This process is not considered any further and therefore only the relaxation due to a single bath is assumed throughout.

q q' q q'

(a) (b}

q

q // (cl (d)

q q'

(e)

Fig. 1. (a) Typical disconnected graph; (b) connected diagram; (c) proper connected graph; (d) improper diagram; (e) lowest order interference diagram.

110 M. Prasad/Vibratiorial overtone resonance lineshapes

Applying the procedure discussed the lineshape function K(w) [eq. (3.1)] is evaluated to yield a familiar formula [4]

1 Fram' m z - ' m - + ':,

(3.5)

where tram' and A m r a, are the width and shift respectively, associated with the (m, m')th vibrational transition of the molecule. When the dipole operator connects the ground state Ig) to a single doorway state Im) within the space of the local mode states as conjectured earlier, the above expression collapses to the form

r(~o) =-2 (oJ - eg - era _ ~,n)2 + ~ ~-2gra Idgm 12' (3.6)

where

1 lqnl~M n) Im'>l 2 Fgra, ~.-- r~. +- E 2 ~, (~o +e~,-e~- ~.,)~+~r~,

(3.7)

In the discussions to follow the attention is confined to the width; shifts are dropped altogether as they can be included by the lineshape translation. The expression for l"g m is calculated in terms of "gain" and "loss" factors (appendix A) and is given by

rv. =/'in- rout,

Tin = ~ E L°(a~qn '~l~ In)Im~[2 - qn'{3Dv'~In)Ira'~(mal~v'~ In)Ima>}]

(3.8)

r~, ( e m _ e ra ' + ec~ _ e#)2 +~F~,m , l 2

-Tout= Y E Lo(~)(tqn,~l:~In)Im'~I 2 - (meI~n)ImoO 6,n'N't~ln) lm'~) ]

(3.9)

Pm'~ (e m, - era + e~ - ca) 2 + :~ r2m,g"

(3.10)

p(a) describes the equilibrium distribution of the bath modes. Tin represents the term that accounts for the rate at which the vibrational quanta arrive in the state Imob by making a transition due to the coupling potential; similarly Tou t accounts for the rate at which the vibrational quanta leave the state Im@ again by making a transi- tion due to the coupling potential. The above result is in full conformity with the generalized master equation approach [12].

The Lorentzian lineshape expressions (3.5), (3.9) and (3. l 0) are derived in the spirit similar to that of previous papers [4, 5, 14]. However the expression for Fer n given in eqs. (3.8)-(3.10) differs significantly from those of HM. The processes which give rise to the relaxation of the vibrational transitions are classified into two main cate- gories: (a) Flow of the energy bundle in the forward direction, (b) quantal loss of energy in the reverse direction, yielding eqs. (3.8)-(3.10).

It should be m~a2oned in passing that Carmeli and Nitzan [ 15 ] recently introduced a random coupling model in the two-resolvent formalism. In this model the expansion of Green's function is first attempted. Binary Green's

M. Prasad/Vibrational overtone resonance lineshapes I 1 1

functions are then constructed to obtain observable quantities. However, the determination of the latter quantities required a contour integration in the complex energy plane. By restricting oneself to the super-operator expansion the construction of the binary quantities and the contour integrations are avoided. The formalism developed by Carmeli and Nitzan may be looked upon as a diagrammatic interpretation of Mori's theory [10], whereas the present technique is a direct generalization. Nevertheless it should be interesting to translate the one in terms of the other to explore the suitability in the related problems.

4. Limiting solution

The above general solution (3.8) possesses two interesting limiting behaviours which are examined below.

4.1. I n c o h e r e n t sca t t e r ing l im i t

In this limit the identity of the individual vibrational quanta is retained and the scattering process is essentially incoherent. The importance of the tetradic self-energy diagram insertions and therefore the width of the inter- mediate levels is diminished. Then the following approximation for the Lorentzian function in eq. (3.8) is justified:

lim ~rm'g--,0 (em - e m, + e~ - e#) 2 +~2F2,g rrS"em - em' ea - e . ( + h) (4.1)

Introducing the above replacement in eq. (3.8) we obtain

Fg m = 7r E E L°(cO{((m'~l~ln)lma))2 - (rn '~l~l~ln) lm'[J)(mctD~r~ln) lm~}6(em - era' + ea - e#)

In' or)

+ p(fl){(qnotDe~ln)lm'/~)2 - (maD¢'l~ln)imob (m'/31"//'l~ln)lm'/~} 6(era , - e m + e# - %)1. (4.2)

Apart from an additional term the above expression is merely the result derived by HM. However, to demonstrate the physical meaning of the additional terms we may define an unit vector it pointing in the direction of the ket Ima)- in the Hilbert space. The term inside both curly brackets may be written in the following form:

{(0n'/31~e'/~ln)lmt~)2(1 - h'/~}.

The unit vector/] points in the direction of the ket Im'D (see fig. 2). Therefore the introduction of the additional term has amounted to adding a term of the type (1 - cos x) in the general expression of the scattering rates miss- ing in the paper of Jortner and Mukamel [ 1 3]. The importance of such terms in the transport type calculations is explained separately [14]. In the present problem we have merely made such inclusions to emphasize the asym- metry in the flow of vibrational quanta thereby giving rise to the irreversible nature of the involved processes.

4. 2. M a n y - q u a n t u m c o h e r e n c e l i m i t

This limit plays an important role near the resonance and the value ofhI 'g m, exceeds the other energy differ- ences occurring in the denominators of the lorentzian functions of eqs. (3.9) and (3.10). Physically this limit may be described as a process in which the local mode quanta interact with an average field generated by all the rest of the normal mode quanta. One may also like to consider it as the "statistical limit" due to the potential energy distribution being statistical in character. In the sum over m ' in eq. (3.8) we may retain the m ' = m term because

112 M. Prasad/Vibrational overtone resonance lineshapes

II,p'~

Fig. 2. The scattering angle x showing the projection of the final scattering ket Im'~ on the initial ket Ima>.

other terms give larger energy denominators and therefore contribute nothing in the asymptotic limit. Further- more the energy difference terms may be disregarded. To that end we obtain a simplified new solution for I'g m,

F 2 = ~ Lo(a){l(rn 3Pt"~ln)lmobl2 - (m31~t~ln)lm~(mo~l~n)lmob} ct,

+ p(fl) {l(motl~ In) Im~l 2 _ qnal~ln)Imob ¢,'n~l~t~ in) Im3)}l • (4.3)

The above result may also be derived using the moment technique [9] after making the above-mentioned correc- tions. The expression (4.3) obtained above simply describes the second moment of the molecular hamiltonian at higher energies. The problem of the intra-molecular energy relaxation is believed to fall in between the two limits discussed above. Under these circumstances eq. (3.8) is an interesting interpolation formula of wide utility.

5. Explicit calculation

In order to carry out an explicit calculation it is obvious that a precise form of the potential energy surfaces below the threshold energy is needed. For the energies of the bands we may adopt the model derived by Albrecht and co-workers and Henry and co-workers [8] displayed in eq. (2.9). Furthermore when the interest lies in the vibrational quantum number dependence of the linewidth we may assume the following approximation for the matrix element of the potential

lOnl~n)Im'>l 2 = C.:(%), (5.1)

where C is a constant independent of na and f (n, , , ) is a function of % , chosen to be n a presently. Under the above assumptions, the solution for the linewidth may simply be obtained in the two differing limits discussed in section 4.

(i) In the coherent scattering limit the energy difference term in eq. (3.7) dominates over FSm at higher energies and it merely reconfirms the narrowing limit of HM, namely

_ Cnc, Fg m 1 (5.2) Pgm =~ co~Xe(n~ _ 1)2 o~ na(1 - l/nct) 2"

However we have retained the resonant contribution and the zero point term in the energy which may be scaled out. (ii) In the limit of incoherent scattering the energy denominators in eq. (3.7) vanish due to the strict conserva- tion of energy in the scattering and the linewidth simply becomes proportional to the square of the matrix element of the interaction. In this limit the hnewidth varies linearly with the vibrational quantum number n~ as

M. Prasad/Vibrational overtone resonance lineshapes 113

obtained by HM. However we have shown that the narrowing is related to the energy being approximately con- served instead of obeying an exact conservation law. It may be further noted that the linewidth goes through a minimum value when plotted as a function of n~. The minimum value of n a depend on the frequency of the local modes, the force constants and the width of the open scattering channels. Therefore further experiments may be performed to analyze the nature of the transition taking place near the width minima. Bray and Berry's [2] experiments seem to apply to the region below the transition.

6. Discussion

Eq. (3.8) describes the processes discussed above in an unified manner and presents the generalization of the calculation quoted by HM which is based on the Born rates. The proper connected diagram expansion was introduced [ 11 ] to calculate the transport coefficients in the extended systems and is a general artifice to uncover the microscopic dynamics in a graphical manner. This is one of the routes to link the macroscopic properties of a statistical system to its underlying scattering dynamics. In the present paper it has been explicitly shown that the operators representing the intra-molecular dynamical processes can be analyzed through a set of graphs which are easily amenable to physical interpretation, whereas in the projection operator formalism any two distinct sets of processes can be unlinked from one another by projections to obtain a meaningful result. A detailed comparison between the two methods applied to electron transport has just appeared [12] and should provide an extra con- trasting source where these methods need to be used in chemical systemL

The narrowing of the lines observed by Bray and Berry and illuminated by HM is also confirmed in the present discussion as the same arguments which were used by HM apply. Nevertheless, it should be stressed that narrowing should be suppressed by extra na dependent factors in the function f ( n ~ ) and the same should also apply to the linear broadening of the absorption line. One of the main points here has been to show how one can take account of all the higher order terms in the evolution operator defined by .~'. The answer is achieved by the resummation of a graphical series which generates the collision operator and therefore the lineshapes. One may then like to ask the question as to "what is the effect of all these higher order terms on the rate obtained by Fermi's golden rule formula", which otherwise shows an oscillatory diffraction type pattern. The answer is that these oscillations are damped by the higher order terms with the relaxation modes characteristic of the physical system and are fully treated here. It is however not guaranteed that this type of damping should provide the necessary and sufficient condition for the local mode quanta to relax by a diffusion or hopping-like process. Further investigation is needed for settling this issue, however, it is the general belief that the main features should remain unaltered. To summarize, a general theory of the intramolecular relaxation and the vibrational overtone resonance absorption lineshapes of the isolated polyatomic molecules has been formulated in the diagrammatic terms. The general expression is examined in two different limits and the explicit calculation is discussed in order to illustrate the power of the present approach. The numerical analysis as well as further applicatior, - to other deuterated benzene molecules should be reported in later publications.

Acknowledgement

Part of the present work was done while the author held a post-doctoral research assistantship in the Depart- ment of Theoretical Physics and the Clarendon Laboratory. of Oxford University during 1977-1979. The au.thor wishes to express his sincere gratitude to Professor R. J. Elliott, F.R.S., and Professor R.A. Stradling for providing him with the opportunity and the facilities for working during his stay.

114 M. tVasad/Vibrational overtone resonance lineshapes

Appendix A

In this appendix the (m - n) representation [6] for calculating the matrix elements of the generalized line- shape operator is described and the steps which lead to eq. (3.8) are indicated. The lineshape operator is given by

".~(Z)=k2(~ln)(.~e0-Z-Z)-13v'~ln)), Z = - w + i a , a-~0 +. (A.1)

a hat ( ' ) on a letter denotes the Liouvflle operator associated with it. Define the function operator • as

= (-% - Z - ~-Id (A.2)

or

(.% - Z - ~ = d. ( A . 3 )

Consider the matrix element of

qnl'-~4)[ n> = (#~,q~)M(N) = E (M[ZIM'~M'(N)' (A.4) M'

where

M = m - n, (A.5)

N = ½(m + n). (A.6)

Next it is assumed that the dominant contribution to -~ comes from its diagonal elements and therefore

QPI [ -" [M~) = ( M [ " ~ t ) ~ MM , = [ ~(JV) ] M~ MM, (A.7)

and (A.4) becomes

(m[#~[n) = (M[ Z(N) [M~M(N ). (A.8)

Also (M[~(N) [M) using (A.1) is given by

(MI#~(N)IM)=)2 ~ ~ C~t/13e-~ln)lM,)(M,[(.o%_ Z - X)-IlM ~) (M'I3e'~In)IM). (A.9) M' M"

Matrix elements of any quantum operator A in the (m - n) representation are given by

(mlAIn) = CM + ½NL4 bl4 - ½AO (A.10)

and the matrix elements of a I iouville operator .L~ are given by

~[..~°(~ ') = wM'J~°~_ M,17-M -- rI-M'o~I~N)_ M,W M , (A. 11 )

M. Prasad/Vibrational overtone resonance lineshapes 115

where the ¢/'s are the shift operators acting on a function f o f N such that

n ~ f(.,v) = f (N +- ½M) n ~M. (A.] 2)

It is then easily verified that

(N + ½MI.~AIN- ½M) = (.~A)M(N) =- ~ (MI£a(Ar)IM')AM,(N), (A.13) M'

(MI-~o(N)~M') = (e N + ~M -- eN- ~M) ~MM', ( A. ] 4)

(MI~In)~I'[')=(N + ~M]~~In)kV-b M ' - ½M)~M'-M-(IV-M' +½MITV'~In)~ r- ~M)rl -M' +M (A.I 5)

and

(M'[(..~) - Z - x)-l[/l~f ') =(eN+ ~M -- eN_~M -- Z -- [X(N)]M)-I~M,M,,"

Using the above series of the equations and the properties of the shift operators (eq. (A. 12)), eq. (A.4) is reduced to

[ ':(~0]zCz(~0 = Tin - Tout,

Tin = h 2 ~ [([0V+ ½Mlg/'~ln)kV+ M ' - ½M)12q~ N) M'

- ( N + ½Ml~e'r~ln)lN + M' - ½M)(N + M ' - ]Mlqr~ln)~/V- ½M) ~bM(N + M' - M))

X {eN+ M, _ -~M -- @N- ~-M - Z - [~,(N -1- ~M t - ~M)] M, )-1],

Tout = )2 Z [{[(N- M' -t- ~Ml~C/'l~ln)~tY- ~M)I2~f,(N) M'

- (iV- M' + ½M[le'~In)IN- ½M)(N + ½MIIV'I~In)IN- M' + ]M) ¢~/(N- M' +M)}

x (EN+ ~M - ~N+~M-M' -- Z - [ x ( N - ½M' + ½M)]M, }-I].

(A.16)

(A. 17)

(A.18)

(A.19)

Making a further averaging over the bath modes and replacing [-:(N)] M by AMN -k iFMN the equation for FM, N given by eqs. (3.8)-(3.10) are derived.

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116 M. Prasad/Vibrational overtone resonance lineshapes

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