Resonance Raman scattering in molecular dimersA. R. Gregory, W. H. Henneker, W. Siebrand, and M. Z. Zgierski Citation: J. Chem. Phys. 63, 5475 (1975); doi: 10.1063/1.431283 View online: http://dx.doi.org/10.1063/1.431283 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v63/i12 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Resonance Raman scattering in molecular dimers * A. R. Gregory,tl W. H. Henneker,t W. Siebrand, and M. Z. Zgierskit§

Division of Chemistry, National Research Council of Canada, Ottawa, Canada KIA OR6 (Received 29 July 1975)

A systematic study is initiated of resonance Raman scattering in molecular systems with overlapping electronic absorption bands. Resonance Raman spectra, excitation profiles, and depolarization ratios are calculated for a simple molecular dimer in which the two overlapping Franck-Condon progressions are related through the permutational symmetry of the dimer. The model used is highly idealized, but can be solved exactly for a complete range of intermolecular coupling strengths. In the weak and strong intermolecular coupling regions, the exact results are compared with the physically more transparent results obtained from a low-order perturbation treatment. The spectra and profiles show properties characteristic of both totally symmetric and nontotally symmetric modes. The profiles and polarization dispersion curves are subject to both electronic and vibronic interference effects. Anomalous extrema of the depolarization ratio are often found to coincide with minima rather than maxima in the excitation profile. A pairwise comparison of the excitation profiles of Rayleigh lines, Raman fundamentals, and Raman overtones shows characteristic resonance-antiresonance coincidences. It is shown that interference effects of this sort are a general feature of systems exhibiting strong vibronic coupling.

I. INTRODUCTION

Raman scatteringl.2 can be described as a coherent absorption-emission sequence. The intermediate state in this process is usually represented by a linear com­bination of eigenstates of the target (assumed to be a molecule), each eigenstate being weighted according to its proximity to resonance. In other words, each eigen­state contributes a term which is inversely proportional to the amount by which it fails to conserve energy. The number and type of eigenstates effectively contributing to the scattering can be used for a rough classification of Raman processes. Far from resonance, the number of contributing states is large and includes many elec­tronic manifolds. In this case all energy denominators are large, so that the weighting factors are small and not selective. This leads to certain simplifications in the scattering tensor, since, firstly, only symmetric tensor components contribute significantly to the scat­tering, and, secondly, Franck-Condon type progres­sions of totally symmetric modes are very short, i. e., overtones have very little intensity. These two simpli­fications arise since the statistical superposition of many eigenstates leads to an intermediate state without a clearly defined symmetry and geometry.

Under resonance conditions, on the other hand, some of the energy denominators become small, leading to large weighting factors for the corresponding molecular eigenstates. The intermediate state in the scattering process now assumes the symmetry and geometry of these dominant states, leading to both Franck-Condon progressions and contributions from antisymmetric ten­sor components. The simplest resonance situation is that in which the molecular absorption spectrum con­sists of sharp electronic-vibrational lines and the in­cident light coincides with one of these lines. This is the resonance fluorescence limit, in which only one vi­bronic state contributes significantly to the intermediate state. If the absorption spectrum consists of broad overlapping bands or is continuous, it will be necessary to sum over many states, even under resonance condi­tions. However, the situation remains relatively sim­ple if all these states belong to the same electronic

manifold. Then the intermediate state may be said to coincide with an electronic rather than a vibronic state. Most resonance Raman spectra reported in the litera­ture correspond or are assumed to correspond to this special case. The basic properties of these spectra, i. e., their vibrational structure and polarization as a function of the incident light frequency, have been es­tablished in a number of recent papers. 3-12 These stud­ies show that resonance Raman scattering (RRS) is a powerful spectroscopic technique for the investigation of both ground and excited state properties. Due to the coherence of the absorption-emission sequence, the ef­fective spectral resolution that can be obtained is much higher than that of absorption and emission spectra tak­en independently of each other. However there is a price to pay for this additional information, since part of it is hidden in phase relations and thus not easily re­trieved. Much has been learned recently by comparing observed RRS spectra with the spectra calculated for Simple model systems. 5-12 Hitherto these calculations have considered only RRS with a single electronic band system. In the present paper we begin a study of RRS involving two overlapping electronic band systems. Such systems are often encountered in practice. Due to the coherence inherent in RRS, strong interference ef­fects may be expected which would tend to complicate a spectrum to the point where an analysis by simple in­spection becomes impOSSible. Some of these overlap­ping band systems have a special interest as models for (strong) vibronic coupling, in particular systems of bands related by symmetry. Examples are the molecu­lar dimer and molecules showing the Jahn-Teller ef­fect. In several respects RRS is a better method to study vibronic coupling in these molecules than absorp­tion and emission spectroscopy, because of the higher effective resolution that can be obtained.

In the present paper we restrict ourselves to one of the simplest possible examples of a system with two overlapping electronic bands, namely the molecular di­mer. In the dimer these two bands are related by per­mutation symmetry. They arise because the monomer excitation is shared by the two identical molecules. Ex-

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5476 Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers

change of this excitation gives rise to a symmetric and an antisymmetric electronic state. The corresponding splitting of the electronic absorption band may be of electronic origin (strong coupling), of vibronic origin (weak coupling) or of mixed origin (intermediate cou­pling). The model thus allows a complete range of pa­rameter values between the limit of completely degen­erate states, on the one hand, and the limit of widely separated electronic states, on the other. As a prelude to the actual model calculations, we discuss the two "trivial" limits briefly in Sec. II.

For the actual calculations, we use a conventional, much studied13 - 18 dimer model. It consists of two di­atomic molecules in a fixed spatial arrangement. One of the molecules is excited and the excitation ("exciton") is exchanged due to an intermolecular coupling which is taken to be constant. Electronic-vibrational coupling is assumed to lead to a shift in the intramolecular equilib­rium position of the monomer and thereby to a Franck­Condon progression in the monomer absorption spec­trum. Each of the monomer levels is split into a sym­metric and antisymmetric component by the intermolec­ular coupling. As a result the vibrations in the dimer will acquire properties characteristic of both totally and nontotally symmetric modes. The model has proved to be a good compromise in that it shows many features of more realistic models, 19 albeit in a simpli­fied form, and yet allows an essentially exact solution by a convenient numerical method. 15.18 In addition, low-order perturbation treatments are available13

•14

which are helpful in analyzing the "exact" results in the weak- and strong-coupling regions.

II. BASIC FORMALISM

We consider the time-independent (steady-state) scat­tering of a monochromatic beam of photons hvo = Eo by a molecule. The molecular eigenstates are represented by the complete set I{j}) consisting of states I j'), I j"), ... , with eigenvalues (i', (j'" ••• , respective­ly, and linewidths rr, r i ", ••• , respectively. The latter are assumed to arise from radiative and nonradi­ative damping processes which result in exponential de­cay and a Lorentzian lineshape for each state. Other line-broadening mechanisms, e. g., those associated with level shifts caused by solvent interactions, can be taken into account by suitably averaging over the dis­tributions of (i', (r', etc. We consider a Raman tran­sition I j ") - I j') corresponding to a scattered photon hVl = El = Eo + (i" - (J" This Raman scattering (and the corresponding Rayleigh scattering characterized by El = Eo) is formally described by a tensor {O!i'. J"} with Cartesian components1•

2

i''}'' _'\"',(j'l Mpl j)(jl Mali") (i'IMalj)(jIMpljll») O!pa -L..J E E 'r + E E·r'

j E J - i" - 0 - t J i - E i • + 0 - l i (1 )

where M is the electric dipole operator and the primed summation includes the complete set except for j' and j" .

We are particularly concerned with Raman transitions between states differing only in vibrational quantum numbers. We therefore express the molecular eigen-

states Ij) in terms of adiabatic Born-Oppenheimer (ABO) states >VI,u

In = L Klu >VI,u(q,Q) I,u

>VI,u = <PI(q; Q)Alu(Q) ,

(2)

where q and Q denote the sets of electronic and nuclear Goordinates, respectively. Thus i is an electronic and U a vibrational quantum number. The <I>'s are paramet­rically dependent on Q and the A's on the i's as follows from the "definition" of ABO states:

(AI,u' I (<1>1' I JCM - TN I <1>1") + TN I Ai"u") = (I'u'<\" I" 0u', u" , (3)

where JCM is the molecular Hamiltonian and TN the nu­clear kinetic-energy operator. We now substitute (2) into (1) and make use of the fact that M is essentially an electronic operator, and.hence does not operate on the A's. This implies that we can express O!~:,}" in terms of electronic transition moments

(4)

which depend on Q only through the Q dependence of the <I>'s.

As a first example we consider a resonance Raman transition from Ij")"" >V.,o. i. e .• a vibrational ground state, to I j ') "" >V., v, i. e., a vibrationally excited state in the same electronic manifold. The scattering tensor elements can be written in the form

nv,nO _ '\"' (Anvl Pnl l AIJ(Alul aln I Ano) O!pa - L..J E T I,u (lu-(nO- 0- 1 iu

(A"v I a.11 Alu )(Alu I PI. I Ano) ) + (lu-(nv+EO-iriu

(5)

where the last step is valid if (Iu - (.0 - Eo"" ° for only one electronic manifold i and Pnl "* 0, aln"* 0, i. e., under typical resonance conditions. Note that we have ne­glected the "antiresonance" term (i. e., the second term between brackets) in (5). Although this approximation is justified close to resonance, it is undesirable when comparing resonance and off-resonance behavior, and will thus be avoided in later sections, notably in Secs. V and VI. To evaluate (5), we introduce the following standard assumptions. 2 We focus on a single normal mode of vibration, taken to be a one-dimensional har­monic oscillator, and expand the electronic transition moments in powers of this normal coordinate

Pnl =P~I +P~IQ+···

al. = aPn + alnQ+ .•• , (6)

where the primed quantities are Herzberg-Teller type vibronic coupling parameters. Limiting ourselves to terms that are either constant or linear in Q, we get from (5)

O!~o "" L (A(I. + Un-WI - Eo - irlutl(p~la?n(Vn I UI >(ull 0.) u

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Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers 5477

+ p~jO"fn(vn I Q I Uj )(Uj I On) + p~jO"ln(vn I Uj )(Uj I QI On»,

(7) where C>E: jn = E:jO - E:nO, nWj is the vibrational energy quantum in manifold i and a convenient shorthand nota­tion has been used for the vibrational states.

We have thus separated a~~ into electronic factors of the form p~jO"rn' P~iO"?n' and p~jO":n' and vibrational fac­tors consisting of products of generalized overlap inte­grals weighted by an energy denominator. The elec­tronic factors determine the symmetry properties of the transitions, i. e., the integrated intensity, the po­larization and the active modes. The vibrational fac­tors give rise to vibrational structure in the scattered intensity 11 plotted as a function of either Eo for con­stant Eo - E10 or Eo - E1 for constant Eo. The relation between this intensity and the scattering tensor is given by the standard formula 1

•2

(8)

where 10 denotes the incident light intensity and the hor­izontal bar an average over all molecular orientations. Most of our results will be expressed in terms of the quantities 8, defined by (8) and loosely referred to as scattering cross sections. By plotting 8v• o as a func­tion of Eo - E1 for constant Eo, we get a RRS spectrum. It shows progressions of vibrations associated with manifold n and is thus analogous to an emission spec­trum. By plotting 8v• o as a function of Eo for constant Eo - E10 we get a RRS excitation profile. It shows pro­gressions in vibrations associated with manifold i and is thus analogous to an absorption spectrum. The mir­ror image relationship frequently observed between ab­sorption and emission spectra, may thus be expected to hold also between RRS spectra and excitation profiles. In Secs. III and VI we shall investigate this relationship for the monomer and the dimer, respectively.

The analogy between absorption-emission and RRS spectroscopy can be extended to include polarization. Let us define a space-fixed Cartesian coordinate sys­tem {x, Y, Z}, such that the scattered photon propagates in the Z direction. The scattering cross section can then be written as a sum of two components

(9)

We assume the incident light to be linearly polarized and choose its polarization to lie in the XZ plane. This automatically defines right-angle scattering. The po­larization of the scattered light can be described in terms of the ratio

(10)

P, is known as the depolarization ratio for linearly po­larized incident light. (The corresponding depolariza­tion ratio for natural light is usually denoted by Pn') To express P, in terms of a tHI , we first write the asymmet­ric tensor {a} as the sum of a symmetric tensor {s} and an antisymmetric tensor {<l}:

{a} ={S}+{<l} ; (11)

Denoting the angle between Z and the polarization vector

of the incident light by cp, we can express P/( CP) by the standard formula1

_ 3y~+5y2 P ,( CP) - 10(3c sin2cp + y~(3 + sinicp) + 5Ya cos2cp

where

Y;=Tr{S}{St}-i3c,

Y; = Tr{<l}{<lt} ,

(:lc=tITr{s}12 ,

(12)

(13)

and t indicates the adjoint tensor. In practice most ob­servations are made for cp =i1T, so that

(14)

In general, P, depends on Eo as well as Eo - Eb so that the depolarization ratio will show dispersion.

The relation between the depolarization dispersion PI(Eo, E1) and the cross section 8v•o(Eo, E1) is clarified by regarding them as three-dimensional surfaces. Both can be drawn in Cartesian coordinate systems with two common axes, namely the axis Eo - AE:ln = unwj and the axis Eo - E1 = vnwn. Every point in the plane determined by these axes gives rise to two cross-section compo­nents 8x and 8Y , or, equivalently to a cross section 8 and a depolarization ratio P" Thus by introducing a third axis labeled 8 or p" respectively, we produce two three-dimensional surfaces. The "hills" of the 8(Eo, E1) surface correspond to vibrational maxima and can be identified by a pair of vibrational quantum numbers (u" vn ). According to (14), the hills of the p/(Eo, E 1)

surface indicate large asymmetric contributions to the scattering tensor and thus need not coincide with a pair of quantum number (u" vn).

III. MONOMER CALCULATIONS

We will first use our formalism to calculate 8(Eo, E1) and P/(Eo, E1) for a monomer. This will provide a basis for the interpretation of the dimer spectra reported in Sec. VI. It will be useful to consider two monomer models, namely one in which the vibrational mode under consideration is totally symmetric, and one in which it is nontotally symmetric. For convenience, we restrict the calculations in this section to the resonance fluo­rescence limit. This means that we take r lu to be so small, that the sum over U in Eq. (7) can be approxi­mated by the term for a single vibronic resonance Eo

= C>E:ln + unWI' The absorption spectra and Raman pro­files then consist of a progression of narrow lines.

A. Totally symmetric modeS

The vibrational structure associated with totally sym­metric modes is usually dominated by the Franck-Con­don principle, vibronic contributions being small in most cases. The opposite conclusion holds for nontotal­ly symmetric modes. For simpliCity, we assume that vibronic contributions are negligible for the totally symmetric mode and that Franck-Condon contributions are negligible for the nontotally symmetric mode to be treated in Sec. IIIB.

Setting P~I = aln '" 0 in (7), we have

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5478 Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers

(\Iv,o =po aO (vn 1 Uj) (Uj IOn) (15) PC! nj jn A "'" •

.... Ejn + UnWI - Eo - tr lu

If Eo is chosen so that we are in resonance with the lev­el Elu, we get from (8) and (15)

ev,o(u) = (p~larn)2rj; 1 (vn 1 UI) (u; IOn) 12 , (16)

where

U= (Eo - !::.Ejn}/tiWI ;

v=(Eo -E1)/tiwn • (17)

Thus plotting ev,o(u) as a function of v gives a RRS spec­trum and plotting ev,o(u) as a function of U gives an ex­citation profile. For resonance with the spectral origin u=O, we get

-,,---Bv. 0(0) = (p~jar")2 r~ 1 (OJ IOn) 121 (Vn I OJ) 12 . (18)

In this case the Franck-Condon progression of the RRS spectrum has therefore the same structure 1 (vn 1 OJ) 12 as the Franck-Condon progression of a low-temperature emission spectrum. For nonzero values of u, the pro­gressions in the RRS spectra correspond to hot-band F.ranck-Condon progressions in the emission spectrum, gIVen by 1 (vn 1 Uj) 12. The Rayleigh scattering cross section for arbitrary U is given by

Bo,o(u) = (p~jar")2 ri! 1 (Uj 1 On) 14 . (19)

For constant r ju( = r j), the Franck-Condon progression of the Rayleigh scattering excitation profile thus has the same structure as the squared Franck-Condon pro­gression of the low-temperature absorption spectrum. Similar ly, the progressions for the fundamental and overtone RRS excitation profiles correspond to squared hot-band absorption spectra.

Simple expressions for the quantitative relation be­tween these cross sections can be obtained by setting Wj = wn( = w). The two vibrational potentials correspond­ing to the states i and n then have the same form and differ only in their relative position along the Q axis. We set Qn = 0 and Qj = Q, and assume the coordinate Q

to be mass-weighted. The vibrational overlap integrals in (16) can then be expressed in terms of the displace­ment parameter

(20)

The vibrational structure associated with the cross section ev,o(u) is given by the generalized Franck-Con­don factor

For the present model we have

l(vn luj)12= 1(u.lvj)12, l(vn IOj)12=exp(-b2)b2v/v! ,

I (On I Uj ) 12 = exp( - b2)b2u / u! ,

(21)

(22a)

(22b)

so that there will be a close relationship between the vibrational structure of RRS spectra, given by F as a function of v for constant U and that of RRS excitation profiles, given by F as a function of U for constant v. This relationship is made explicit by the following equivalent factorizations of Fv(u):

F(U)=I (vnluj) \21(U 10 >14

v (On 1 Uj > I n (23a)

Fv(u) = I ~~:: ~:~ \21 (Vn 1 OJ > (Uj 1 on)!2 (23b)

Here we have factored Fv(u) in a product of Poisson dis­tributions (22b) showing single maxima, and functions of the form

I (xI1)/(xl 0)1 2 =(x- b2Nb2

1 (xI2)/(xl 0) 12 = [(x- b2)2 - xJ2/2b 4

,

(24)

etc. which exhibit sharp minima. It follows that the Poisson distributions are different for RRS spectra and excitation profiles, but that the factors of the form (24) in Eq. (23) transform into each other under a u- v in­terchange. Hence Fv(u) shows the same minima as Fu(v). For instance, the RRS spectra corresponding to resonance with U = 1 show a minimum for

(25)

Similarly the RRS excitation profile for fundamental (v = 1) shows a minimum for U"'" b2

• The corresponding minima in the spectrum for resonance with U = 2 and in the profile for the first overtone are

u(or v) = 2 . (26)

In Table I we present values of e calculated for differ­ent values of U and v for a displaced oscillator with a displacement parameter 1 b 1 = 1. In this table the rows (u=constant) correspond to RRS spectra and the col­umns (v = constant) to excitation profiles. The maximum values of e which are underlined, lie roughly on a Con­don parabola.

The depolarization ratio is determined by the matrix elements (p~ja~n)2 and is independent of Eo and E1 for the present model. Polarization dispersion only occurs as a result of vibronic coupling which is assumed negligi­ble here. A typical result20 for totally symmetric modes in mOlecules with some symmetry is that only diagonal components of (\IpC! are different from zero. Un­der resonance conditions, i. e., upon neglecting all but one intermediate electronic state i, (\Ipp is fully deter­mined by the direction of (niMI i> so that usually only one of the three components (\Ipp, say (\I"", is different from zero. Then we have Tr{(\I} = (\I"" which gives P, =t. Alternatively n or i may be doubly degenerate, so that, say, (\I",,=(\Iyyand (\1 .. =0; in that case Tr{Cl!}=2Cl!"" and

1 P, =8.

B. Nontotally symmetric model 0

As mentioned in the preceding subsection, we assume that the vibrational structure associated with the non­totally symmetric mode is entirely due to vibronic cou­pling. To make the Franck-Condon contributions van­ish, it is sufficient to take wn = Wj( = w), as in Sec. IlIA, since b = ° by symmetry for asymmetric modes. We then have in the harmonic-oscillator approximation

(Ujl On) = 1i",0 , (27)

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Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers 5479

TABLE I. Vibrational structure of the RRS cross section calculated from Eq. (21) for a displaced harmonic oscillator with a displacement parameter I b I = 1. The maxima in each row and column are underlined to illustrate that they form a (dis-torted) Condon parabola. The notation 2(-4) means 2x 10-4•

~ 0 1 2 3 4

0 1 1 0.25 0.17 0.42(-1)

1 1 0 0.25 0.67 0.38

2 0.25 0.25 0.13 0.42(-1) 0.26

3 0.28(-1) 0.11 0.14(-1) 0.74(-1) 0.12(-2)

4 0.17(-2) 0.15(-1) 0.22(-1) 0.12(-2) 0.16(-1)

5 0.69(-4) 0.11(-2) 0.42(-2) 0.23(-2) 0.10(-2)

6 0.48(-4) 0.35(-3) 0.71(-3) 0.11(-3)

7 0.71(-4) 0.83(-4)

8 0.11(-4)

so that the Franck-Condon terms in (7) lead only to Rayleigh scattering. Since

(vn / Q/ Uj) (Uj I On) = (1f/2w)1/2 c\, 1 liu, 0 ,

(Vn / Uj) (Uj / Q/ On) = (1f/2w)1/2liv, lliu,1 ,

the RRS tensor (7) reduces to

(28)

Q!l,O = (!!... )112 ( p~j (]fn + Pgj (]!n ) Pa 2w AEjn-EO-irjo AEjn+lfw-Eo-irn

(29) The RRS spectrum shows thus only the fundamental, and the excitation profile shows just two resonances, cor­responding to U = 0 and 1.

To calculate the depolarization ratiO, we note that {3c, given by (13) vanishes for asymmetric modes, so that (14) reduces to

(30)

Limiting ourselves to nondegenerate states, so that all diagonal tensor components vanish for an asymmetric mode, we have

Y;,a=~L:(Q!pa±Q!ap)2 • P,C1

(31)

Far from resonance, we have Q!pC1 "" Q!C1P' so that Ya "" O. We can assume, without significant loss of generality, that in the resonance region all but one pair of tensor elements (Q!pC1' Q!C1P) vanishes or becomes negligible. 20

The relation between Q!PC1 and Q!C1P mayor may not be de­termined by symmetry, depending on the molecular point group and the species to which the mode belongs. There are three possibilities:

(1) Q!pC1 = Q!C1P by symmetry; in that case P, = t and shows no dispersion.

(2) Q!pC1 = - Q!C1P by symmetry; in that case P, =00 and shows no dispersion. Since Y. = 0, this mode has a very small scattering cross section outside the resonance region.

(3) Q!pa * Q!C1P; in that case Q!PC1» Q!C1P (or Q!PC1« Q!C1P) for exact resonance yielding Y! = Y! and P, = 2. Since P,"" t far from resonance where Y!» y!, this case leads to

5 6 7 8 9

0.83(-2) 0.14(-2) 0.20(-4)

0.13 0.35(-1) 0.71(-2) 0.12(-2)

0.25 0.13 0.42(-1) 0.10(-1)

0.45(-1) 0.85(-1) 0.60(-1) 0.25(-1)

0.52(-2) 0.33(-2) 0.17(-1) 0.18(-1)

0.18(-2) 0.28(-2) 0.16(-4) 0.23(-2) 0.36(-2)

0.36(-3)

0.69(-5)

dispersion of the depolarization ratio.

The form of the dispersion curve has been obtained by Mortensen. 10 Using the definition

we get from (31)

2 If ( I 0 0 I )21 1 1 12 y s,a = 4w Pnj(]jn ± Pjn(]jn Zo - Eo ± Zl - Eo

with a depolarization ratio which according to (30) equals

_ ~ ~ I (Zo - EO)"l - (Zl - Eo)"l \ P, - 4 + 4 ( E )-1 ( E )-1 • Zo - 0 + Zl - 0

(32)

(33)

(34)

Thus for narrow bands (i. e., small r ju), the excitation profile consists of two Lorentzians but the depolariza­tion ratio shows a single maximum between the two Lo­rentzians. It follows from (34) that PI approaches a val­ue of t far from resonance, increases to a value of 2 at each of the resonances and increases further to a sharp maximum between the two resonances [if rju-o, (PI)max _ 00 J.

IV. DIMER MODEL

The dimer is taken to consist of two molecules, la­beled by indices A and B, in an arbitrary orientation relative to each other. The molecular eigenstates are approximated by ABO states. For each molecule we restrict ourselves to two electronic manifolds, corre­sponding to the ground state iJ>n and the excited state iJ>h and one normal mode. This mode is taken to be a dis­placed (i. e., b * 0) but undistorted (i. e., wn = Wj = w) harmonic oscillator. The intermolecular coupling, giv­en by matrix elements

VAB = (iJ>jA iJ>nB / V I iJ>jBiJ>nA) , (35)

is taken to be independent of QA and QB.

The total Hamiltonian of the dimer is

JeD =JeMA +JeMB + V= (Te+ TN + U)A + (Te+ TN + U)B + V. (36)

It is convenient to introduce an "exciton" -type notation,

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5480 Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers

viz. governed by electronic transition moments of the form

I IJiA) = !<P'A<PnB );

11Ji0) = ! <PnA<PnB ) , (37)

where 11Ji0) denotes the ground state. We now carry out the integrations over the electronic coordinates, viz.

(IJiA I (T. + U)A I If!A) =E,(O) +tw2(QA _ Q)Z ,

(IJiA , (T. + U)B' IJiA) = En(O) +tw2Q~ ,

(IJiAI(Te+U)BIIf!B)=(IJiAI VllPA)=O.

(38)

The last equation (VAA = 0) represents an additional ap­proximation which will break down if V is large. Sub­stitution of (38) into the Schrodinger equation of JeD leads to the vibrational equation13- 16

{TNA + TNB +~w2[(QA - Q)2 + Q~]+ VABG - E}A(QA, QB)=O, (39)

where G is defined as the operator that interchanges A and B.

Equation (39) can be separated by introducing "normal coordinates"

Q*=2-1/2(QA±QB) , (40)

yielding

[TN +tw2(Q+ - 2-1I2Q)2 - E+ ]X(Q+) = 0 (41)

[TN+tw2(Q--2-1/2Q)2± VAsG-E;]11*(Q-)=O, (42)

where

(43)

Equation (41) is a standard harmonic-oscillator equa­tion, so that

(44)

and Xv(Q+) is the vth harmonic-oscillator wavefunction. To evaluate (42) for which no analytic solution is avail­able, we use the numerical procedure of Merrifield, 15

and Fulton and Gouterman. 16 First we re-express it in units such that Ii = w = 1:

[ - ta2/(aQ-)2 + t(Q-)2 + bQ- ± cG - E!]1J! (Q-) = 0, (45)

where b is given by (20) and c = vABI liw. Then we ex­pand the 11!(Q-) in terms of harmonic-oscillator wave­functions Xk( Q-):

N

1J!( Q-) = ~ c:,. Xk( Q-) , 1<=0

where Ci .. is a (Hermitian) overlap integral and

- Uaz/a(Q-)2 _ (Q-)2 + 2k + 1]Xk(Q-) = 0 .

(46)

(47)

Using the well-known recursion formula for harmonic­oscillator wavefunctions, we derive the coefficients C: .. from the set of equations

(tk)1/2bC:_1 ... + [k + t ± ( - 1)kC - E! ]q ..

+[t(k+1)]1/2bCi+1 ... =0 . (48)

By choosing a large enough basis set Xk(Q-) in (46), one can in this way obtain results to any desired degree of accuracy.

The absorption and emission spectrum of the dimer is

(49)

where the moments are written as vectors to account for the relative orientation of the molecules in the di­mer. To calculate the vibrational matrix elements of O'ni> we use the Condon approximation, i. e., we assume O'nt to be independent of QA and QB'

The main virtue of this dimer model is that it allows an essentially exact solution with a minimum of param­eters. Most of these parameters can be taken directly from observed monomer spectra. Thus we equate w with the observed (adiabatic) vibrational frequency and b, through (20), with the displacement of the nuclear equilibrium position upon electronic excitation from <Pn

to <Pt. The intermolecular coupling parameter c and the angle between 0':, and 0':. are the principal structural parameters of the dimer. They can be obtained from the halfwidth and center of gravity of the observed di­mer absorption spectrum.

Despite its Simplicity and concentration on bare es­sentials, the model has previously been applied suc­cessfully to a number of experimentally observed spec­tra. 19• 21 -24 It is obviously subject to limitations. The use of the Condon approximation implies that the model in this form is not applicable when a substantial part of the vibrational progression of a totally symmetric mode is due to vibronic coupling with neighboring electronic states. Similarly, the assumption that c is independent of QA and QB is likely to break down' for very strong coupling (I c I »bZ), where the dependence of c on Q+ may contribute substantially to the vibrational structure of the spectrum. These and other limitations can in prin­ciple be overcome by suitable generalization of the mod­el. 19 We have omitted these refinements because they lead to additional parameters and thus reduce the trans­parency of the overall picture. They may be necessary and appropriate, however, when one tries to fit a spe­cific observed dimer spectrum.

V. THE SCATTERING TENSOR

We can now formulate the matrix elements of the scattering tensor for the dimer. We assume, as in Sec. III, that the initial state is not vibrationally excited, and is thus of the form Ij") = I 1f!0Xo(Q+>Xo(Q-». The corre­sponding final state is then I j ') = IlPoX.( q)Xw( Q-». In terms of the representation used in Sec. IV, the inter­mediate states can be written '

Ii) oo2-lIZ Xv(Q+)[If!A1J!(Q-)± If!BG1J!(Q-)] ;

G1J! (Q-) = 1J! ( - Q-) • (50)

Substitution of Ij), Ij'), and Ij") into (1) leads to tran­sition moments of the form

(j' I Mp Ii) = 2-112 S.vC ~ .. [~, ± ( -l)wp:d , where

(51)

(52)

is a vibrational overlap integral and C~ .. is given by

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Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers 5481

(46). Similarly, we have

( . I MI' ") 2-1/2 C' C ± (A B ) J 0 J = "1>0 I-' 0 0' In ± 0' In •

The energy differences between these states are (in wlits liw = 1)

(53)

€t,Vl-' -€n,oo=€c+vu:! -t(b2+1) ,

€t.VI-' -€n,sw=€c+ v +€! -s-w-t(b2 +1), (54)

where €! follows from (45) and €c is the energy of the Franck-Condon maximum in the monomer absorption band. Substitution of (52)-(54) into (50) yields

== ~ L {a: [P:I ± ( -l)Wp:I](O'tn ± urn) + a;[O':1 ± ( -l)W O':I]( ptn ± pfn)} , ±

(55)

where a: represents the resonance terms and a; the anti resonance terms (the subscript sign corresponds to the sign of the Eo term in the denominators). The reso­nance terms and the antiresonance terms sum to

(56a)

and

(56b)

respectively.

To evaluate (55) we choose the following molecule­fixed axis convention: P:I is parallel to Z and P:I lies in the xz plane. 1 It follows that

X:I=Y:I=Y:I=O; z:I=IMI (57)

Z:I = IMI cosO; x:I = IMI sinO,

where 0 is the angle between P:I and P:I. Taking wlits such that I M I = 1, we get from (55)-(57)

a;;"OO = ( -1)W(N + R+) sin2 0,

a;:'oo ={( - l)W[(A+ + R+) cosO + R-] +A-} sinO, (58)

a:;"OO ={( - l)W[(A+ + ff) cosO +A-] + R-} sinO,

a::'oo = [1 + ( _1)W cos20](A+ + ff) + [1 + ( - l)W](A- + R-) cosO,

and all components containing the index y vanish. Thus, in the notation of Eq. (11), {a} = 0 if w is even, so that { a} then reduces to a symmetric tensor {s }. If w is odd, the tensor is asymmetric and has a zero trace, i. e., i3c, given by (13), vanishes. It appears therefore that Q- behaves as a nontotally symmetric normal co­ordinate, as distinct from Q+ which remains totally symmetric.

To probe these relationShips further, we now consider several limiting cases. The limit of weak intermolecu­lar coupling is characterized by the conditions13• 14 I c I «1 and I c I «b2

• It follows from (45) and (46) that in this limit the Hamiltonian of Q- reduces to that of a har­monic-oscillator, so that the C~k reduce to overlap in­tegrals SI-'W and the energies E:~ to a harmonic progres­sion /J. + j. Similarly, the quantum numbers sand w re­ducetot(vA+vB)andjlvA-vBI, respectively. The RRS spectrum and excitation profile will thus reduce to that for a Single, totally symmetric normal mode, rep­resented by Eqs. (18)-(26). Correspondingly, the de­polarization ratio will show no disperSion.

The opposite limit of strong intermolecular coupling is characterized by the conditions13,14 I c I »1 and I c I »b2

• In this limit, the Hamiltonian of Q-, displayed in Eq. (45), separates into two weakly coupled harmonic­oscillator Hamiltonians. Thus again the C!k reduce to SI-' w, but the E:! now turn into J.1. + j ± c. Thus while the RRS spectrum will resemble that of a single totally symmetric mode, the excitation profile separates into two components, shifted by c and - c relative to the monomer profile. If b"'" 0, the vibrational overlap ma­trices will be essentially diagonal, so that only Rayleigh scattering is allowed, i. e., v = J.1. = s = w = 0, and (55) re­duces to

aOO,OO"" !.[(P:# +P:.)(O'ta+ufn) Po 2 E:c + C - Eo - ir

+ (~I - P:. )(O'tn - O'ra) ] E:c - C - Eo - ir + • . . , (59)

where the dots represent the anti resonance terms la­beled A in (56) which are small in the resonance region. Equation (59) has the same form as Eq. (29). Hence the corresponding depolarization ratio will show the same dispersion as for a nontotally symmetric Raman fundamental, given by (34).

In the intermediate coupling region, the behavior will of course be more complex, but some regularities can be noted. Let us assume for simplicity that 0 = tn, so that the two components have the same RRS cross sec­tion, and that I A I « IR I which should be valid in the resonance region. For even w, Eq. (58) now reduces to

a;~ 2P,OO = a;~ Zp,oo = K

a;~2P. 00 = a~2P'OO = R- , (60)

so that {as,2P,OO} is a symmetric tensor to be denoted by is}. For odd w, we have on the other hand

a;~2P+l,OO = _ a:~/ilP+1,OO = _ K

a:~2P+1,OO = _ a:~2P+1,OO = _ R-, (61)

so that {as,ZP+l,OO} is an asymmetric tensor which we separate into its symmetriC and antisymmetric compo­nents:

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5482 Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers

=(-: : :)+(: : -:-). (62)

\ 0 0 R+ K 0 0

The corresponding depolarization ratios are according to (14)

PI{S} =t [(1 + 3r2)/(2 + r2)] (63)

PI{as} = i(3 + 5r2) ,

where r=K/R+. The following special cases can be recognized:

I R' I » I wi: P ,{ s } = t ; I R' I = I wi: P ,{ s } = t ;

PI{as} = t PI{as} = 2

IR+I« Iwl :PI{s}=L PI{as} =00

(64)

It follows that for the dimer, as opposed to the mono­mer, not only the {as} tensor but also the {s} tensor shows polarization dispersion, depending on the ratio

(65)

Assuming r to be small, we conclude that I r I = 1 if either I a: I » I a: I or I a: I « I a: I, i. e., at the two resonance positions

(66)

At some point in between these two resonances we will have the condition I a: I = I a: I, so that r=oo or 0, de­pending on whether or not C;" ... C~.o has the same sign as C;;' ... C;.o. If rgoes to zero between the two reso­nances, it will go to infinity .ar from both resonances, and vice versa. Thus in general PI will have either a maximum or a minimum between a pair of resonances but will not peak at a resonance position. At these po­sitions defined by (66), we have, according to (64) and (65): t::s PI::S 2, whereas between the resonances the lower limit is P, =t and the upper limit PI =00 (for r- 0).

VI. RESULTS AND DISCUSSION

To obtain a more detailed understanding of RRS in di­mers, it is desirable to carry out numerical calcula­tions. Accurate numerical solutions of the model can be obtained if the basis set of harmonic-oscillator wavefunctions used in (46) is large enough. Our calcu­lations are based on a set of 30 such functions which is ample for the range of parameter values considered. To facilitate the physical interpretation of these results, we also compare them with the results obtained from perturbation theory which are less accurate but physi­cally more transparent. Such a comparison becomes tenuous in the intermediate coupling region where the perturbation approaches are known to converge poor­ly.13.14

As in Sec. III, we calculate the RRS spectrum, exci­tation profile and depolarization ratio. While varying Eo so as to scan the entire absorption band, we calcu­late excitation profiles and polarization dispersion curves only for Rayleigh scattering and the Stokes Ra­man fundamental and first overtone. Most calculations

are carried out for fJ = hand r = 0.2, as these values lead to particularly instructive results. An angle fJ = h gives equal scattering cross-section to both components into which the intermolecular coupling splits the spec­trum. Angles fJ = 0 or 7T would give all intensity to one of the components. Similarly a linewidth r = 0.2 leads to well-resolved but relatively uncluttered spectra and profiles. In practice the observed linewidth is of course not only due to r, but also to solvent effects, etc. which should be taken into account by averaging the tensor over the energy level shifts caused by these effects. The basic coupling parameters band c, representing elec­tronic-vibrational and intermolecular coupling, respec­tively, are each independently varied between 0 and 2, so as to cover the complete range from weak to strong intermolecular coupling. Throughout the calculations we take the electronic energy gap Ec equal to 15. The results are insensitive to modest changes in this param­eter. It governs the AIR ratio which remains much smaller than unity in the resonance region for Ec» 1.

All calculations are based on the tensor components a,/:' 00 given by (55)-(58). The RRS cross sections and depolarization ratios are calculated from

fJ;' 0 = t i3c sin2cp + to y~(1 + t sin2cp) +t y~ cos 2cp (67)

together with (9) and (10), where v = s + w. This corre­sponds to averaging over all orientations of freely ro­tating dimers.l Plotting fJv.o(Eo) against Eo - El =V leads to RRS spectra. We shall not consider the shape of their vibrational peaks but represent them by lines for convenience. In general we calculate RRS spectra for each prominent vibronic band in the absorption spec­trum. Plotting fJv.o(Eo) against Eo leads to excitation profiles which we have calculated only for v = 0, 1, and 2. Here the effect of the finite linewidth r is nontrivial and has been fully included in the reported profiles.

The depolarization ratio PI(~7T) is calculated for lin­early polarized light and right angle detection by means of Eq. (14). This calculation is a straightforward gen­eralization of that represented by Eq. (63). In general we have combined the graphs showing the dispersion of the depolarization ratio (i. e., P, versus Eo) with those shOwing the corresponding excitation profiles.

We shall group the results in five subsections, de­pending on the parameter values used. Subsections A through C are based on fJ = hand r = 0.2; they differ only in the values of the coupling parameters band c. In subsections D and E, we show the effect of varying fJ and r, respectively, for some selected b, c combina­tions.

A. Weak intermolecular coupling

In Sec. V we showed that for extremely weak inter­molecular coupling the dimer spectrum reduces to the monomer spectrum. Under these conditiOns, P, shows no dispersion and the RRS spectra have a vibrational structure closely related to that of the excitation pro­file, as shown in Sec. III. In Fig. 1, we compare the RRS spectra corresponding to u( = J.l + II) = 0, 1 and 2 with

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Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers 5483

03

(C)

':102 0

@

01

0 03

-;- 02 0

@

01

0

06 (0)

904 0

@

02

012 4 E, 2 10 14 16 18 Eo 20

FIG. 1. Comparison of the vibrational structure of resonance Rayleigh or Raman scattering spectra with that of excitation profiles in the weak intermolecular coupling limit (b =2, c = O. 05). The spectra are represented by lines with cross sec­tions 8(Et), the profiles by curves with cross sections 8(Eo). The spectra in (a), (b), and (c) correspond to excitation in the 0-0 (Eo =11), 0-1 (E o=12), and 0-2 (EO =13) absorption band, respectively. The profiles in (a), (b), and (c) correspond to the Rayleigh line, the Raman fundamental and the first over­tone, respectively. The other parameter values used (r = 0.2, 8 =!7r and €c= 15) apply also to all subsequent figures except as indicated.

020 ft (b) 05 " I' I' I'

N 015 " (\, , , 6 ' \ I,

04 / I I, (, @ I , I ,

010 \1 ,I 1 J \ N \/ \/ ,

',I 6~ 03

(a) 0.6 025

020 05

6015 ft 6~ @ " /I 04 ,

010 I ,_

Eo

FIG. 2 Comparison of RRS excitation profiles (solid lines) with the dispersion of the depolarization ratio (broken lines) under weak intermolecular coupling (b=2, c=0.5) for the Ra­man fundamental (a) and the first overtone (b).

@~~6~9L:

@U~5b:8l: @ I "I~O" I Eol4ll, Eo'17 ~

O~~H0246810 V V V V

FIG. 3. RRS spectra corresponding to Fig. 2. The 8 scale is linear and the cross sections for each graph are normalized to unity for the Rayleigh line (v = 0).

the excitation profiles corresponding to v( = s + w) = 0, 1 and 2, respectively. These vibrational structures and their relationships follow directly from Eqs. (23)-(26). The envelope of these structures has the same basic shape as IXu(Q) 12 or IXv(Q) 12, respectively, i.e., a (distorted) Gaussian modulated by a squared Hermite polynomial. The zeros of the Hermite polynomials lead to minima in the scattering cross section for the corre­sponding values of u or v [cf. Eqs. (25) and (26)].

In Figs. 2 and 3 we show the effect of increasing c within the weak-coupling regime. The RRS spectra and profiles are slightly deformed by the increased coupling, but a more drastic effect is the change of P, which now shows a substantial dispersion. To explain this behav­ior, we recalculate the results by perturbation theory, using Vas a (small) perturbation. The zeroth-order weak-coupling wavefunctions are of the form

w!O)(p., II) = 2-112 [ IJiA X" (Q- + b) ± ( -1)" IJiBX,,( Q- - b)]X,,( Q+ + b) (68)

in units n=w=l, so that b=2-1I2 Q". Thus the first-order energies can be expressed as

E!1)(P., II)=P.+ 11+1- b2 ± (-1)"cS",,(2b), (69)

where we use the notation

(70)

Thus if r is small enough the maxima in the absorption spectra and excitation profiles will show a splitting equal to 2c times a vibrational overlap integral.

Carrying the calculation one step further, we get the first-order corrected wavefunctions

W!zl)(p., II) =w~(p., 11)+ ('f 1)"2-1/2CXv(Q+ +b)

X{IJiA[L (p.' - p.t1S",,,(2b)x,,.(Q- - b~ ""~jJ. 'J

± IJiB[L (p.' - p.t1S"".(2b)X".(Q- + b)]}. ",,~IJ.

In the special case p. = II = 0, this result reduces to

W!l )(0,0)

(71)

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5484 Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers

= w~o)(O, 0) ± cSo1(2b)w~O)(0, 1) ± tcS02W!O)(0, 2) + ..•• (72)

Hence weak intermolecular coupling mixes the (zeroth­order) harmonic-oscillator wavefunctions of the Q­mode, the mixing coefficients being equal to c times a vibrational overlap factor. Whether the mixing involves components of the same or of different sign depends on whether the two vibrational wavefunctions differ by an even or odd number of vibrational quanta. For small b (i. e., b2 ~1), the mixing is predominantly between states differing by one quantum and thus involves com­ponents of different sign. As a result the scattering tensor acquires an antisymmetric component leading to values of p, > to the off -resonance value. The maxima in 0, can again be understood as interferences between zeroth-order states of different symmetry caused by a nontotally symmetric mode. The interacting states dif­fer in vibrational as well as electronic quantum num­ber, so that the interference is of vibronic origin just as in the Mortensen model. 10 However, these states may be separated by an energy gap much smaller than unity, so that they are not resolved for the chosen value of r =0.2. The maximum in p, will then be very close to the maximum in the excitation profile. A high-reso­lution calculation shows however that the maxima in p, are still located between the maxima in 8 for the inter­acting states, in agreement with the general behavior pattern discussed in Sec. V. The similarity between the p,(Eo) and 8(Eo) vibrational structures in Fig. 2 is therefore partly due to the chosen low resolution. It is significant, however, that p,(Eo) does not show the same minima in the vibrational progression as 8(Eo). This can be understood on the basis of perturbation theory. However, a detailed perturbation calculation of the structure of o,(Eo) is too complicated to shed much light on the results of the "exact" model calculation. We limit ourselves therefore to a calculation of the relative p, values within the progression without investigating the form of the individual vibrational lines. The corre­sponding vibrational structure will be compared with that of the excitation profile given by a straightforward generalization of Eq. (21):

(73)

Thus, analogous to Eq. (25), the excitation profile of the fundamental shows a minimum in F1(1l, II) for Il + II = b2

• The vibrational structure of the polarization dis­persion curve is given by the anti symmetric component of the scattering tensor, the symmetric component leading to a constant value of t. The antisymmetric part of the scattering tensor involves coupling between fI.!*l) (Il, II ) and fI~1>(IJ.', II). Using (71), we can write the correspond­ingvibrational overlap factor F.A)(u, II) in the form

F~)(Il, II) = F~(Il) = tc2{L: (Il - 1l')-l [S"." (2b)Sv" (b)S,,'o(b) ,,'

+ (-1}" S"".(2b) S"v(b)So",(b)] r ' where Il - Il' is odd and [cf. Eq. (70)]

Sv" (b) = (Xv(Q-) I X,,(Q- + b»

S" 'o(b) = (X", (Q- - b) I Xo(Q-» ,

(74)

(75)

etc. For Simplicity we restrict ourselves to b2 $1, so that Il - /J. I = ± 1, and consider only the fundamental (v = 1). Making use of the fact that S"'" = - S"". for /J. - /J.' odd, we get from (74)

Ft(/J.) "" 2C2{ "~%l (/J. - /J./)-lS"." (2b)

X[Sl"·(b)S,,o(b) + Sl,,(b)S"'o(b)]} 2

(76)

From the recurrence formula for harmonic-oscillator states, we have

and

Sl."+lS".O = (/J. + 1t1l2(/J. + 1 - b2)S~0 ,

Sl." S"+l. 0 = (/J. + 1 )-1I2(/J. - b2)S~ 0 ,

St."-l S".O = /J.1/2b-2(/J. -1 - b2)S~O , (77)

S"+l." (2b) = - (/J. + 1)-1I2[ /J.1/2 S" -1." (2b) + 2 bS"." (2b)]. (78)

Substitution into (76) yields

Ft(/J.) = 2c2S!.o(b)[ 4bS"." (2b)

1/2(2/J.- 1 _ 2b2+1)S (2b)]2

+/J. ~ /l+1 ,,-1." • (79)

The first term within the square brackets is generally a slowly varying function of /l. In the region of /l val­ues where it dominates, Ft(/l) will thus behave simi­larly to a Rayleigh excitation profile. This is the case in Fig. 2, based on b = 2 which means that the second term within the square brackets vanishes near /J. "" b2

•

Hence the first term dominates in the region where the excitation profile has a minimum, and gives rise to a smooth maximum in the polarization dispersion in that region. Note, however, that this argument is not valid for either very large or very small values of b2

•

B. Strong intermolecular coupling

In Sec. V we showed that for extremely strong cou­pling, the Rayleigh scattering excitation profile re­duces to a doublet separated by 2c and centered at E: e •

For 8=h, the two lines have the same scattering cross section. The depolarization ratio peaks at E: e,

i. e., halfway between the two lines. This behavior is analogous to that observed for RRS by a nontotally sym­metric mode in a monomer. If b", 0, Raman scattering occurs along with Rayleigh scattering. In Fig. 4 we show the structure of the Rayleigh and RRS excitation profiles and depolarization dispersion in the extremely strong coupling limit. It is seen that the Rayleigh dou­blet is basically maintained for RRS, but that the lines of the doublet now exhibit vibrational structure. For the fundamental, this structure is completely analogous to that of a Mortensen-type doublet: The two resonances occur for the 0-0 and 0-1 transitions for both compo­nents. If I b I «1 and 8 =t7T, the resulting four compo­nents all have the same cross section which is very small in absolute value. Under these conditions the scattering is similar to that of a nontotally symmetric mode. If I b I »1, however, within the strong coupling regime characterized by b2 « I c I, the vibrational

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Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers 5485

N

° @

003

-;- 002 o

@

o 6

,., " \ -' '

14 16

Eo

(c')l 110 j i N

:Os 0<>:, . -..... J

. ·-10 1

(0) : 110 I

........ --- J

ios 6<>:,

18-~~~0

FIG. 4. Same plot as Fig. 2 for very strong intermolecular coupling (b = O. 1, c = 2); (a) represents the Rayleigh line, (b) the Raman fundamental, and (c) the first overtone.

structure of each of the two components takes the form of a typical Franck-Condon progression. The RRS, which is much stronger now, thus becomes similar to

that of a totally symmetric mode. A modest move in this direction is seen upon comparing Figs. 3 and 5.

This behavior is readily explained in terms of per­turbation theory. The zeroth-order strong-coupling func tions are

with the corresponding energies

(81)

The perturbation is of the form bQ- and mixes states with different values of J.L through

Q-X" (Q-) = 2-1IZ[J.L1IZX,,_1 (Q-) + (J.L + l)lIZX"+l (Q-)]. (82)

Thus the first-order wavefunctions are of the form

\11 ~1l ( J.L, II) = \11;0) (J.L, II) + 2-1/Z b[J.L 1Iz\I1!0) (J.L - 1, II)

so that, in particular

>11;°(0,0) = [2-1IZ(I/JA ± IPB>XO(Q-)

+ib(lJiA ~ lJiB>X1(Q-)]xo(Q+ + b) • (84)

It follows that the + component corresponding to J.L mixes to first order with the - component of J.L ± 1 and, by the same token, to second order with the + compo­nent of J.L ± 2, etc. To first order in b, we have thus for the RRS fundamental

(11 po-I 0) == 2-1/Z L (lJiO[X1 (Q->Xo( q) + Xo( Q-)x1 (Q+) I Mpl >1I!l) (J.L, II» (>1I!l)(J.L, II) I Ma II/Jo Xo(Q->Xo(Q+» ",V

In this approximation which is valid for small I b I, the RRS excitation profile of the fundamental shows four maxima of equal cross section, corresponding to J.L + II = 0 and 1 for the + and the - component. The interfer­ence between the components J.L = 0 (+) and J.L = 1 (-) and that between Jl = 0 (-) and J.L = 1 (+) leads to a maximum in the depolarization ratio P, for a photon energy Eo given by

EG=€c -1 +-H€}0)(OO)H~0)(10)]

= €c - 1 + H€!G)(10) +€~O)(OO)]

=€c+i(l- bZ)<=><€c+i , (86)

which is just the "center of gravity" of the correspond­ing excitation profile.

The treatment is readily extended to higher orders, as required for overtones and larger values of I b I. In first order, the first overtone has an excitation profile with only two maxima, corresponding to J.L = 1 (±). How­ever, since the corresponding cross section is of order bZ

, a second-order treatment is required and yields four additional maxima, corresponding to Jl = 0 (±) and J.L = 2 (±). A straightforward second-order evaluation of the matrix element (21 po- I 0) yields the following rel-

(85)

ative cross sections 9zo(Jl, ±), all of order b2, for these six maxima:

~o(O, +) = 920(0, -) = 9ZG(2, +) = 9zo(2, -)

8zo(1, +) = 9:!O(1, -) = 23/29zG(O, +) ,

as shown in Fig. 4(c).

(87)

If I b I is not small, higher-order contributions be­come important and the perturbation approach tends to become unwieldy. Figures 5 and 6 show in which direc­tion the behavior changes. If I b I is so small that the vibrational structures of the + and - components do not overlap significantly, these two components have quite similar RRS spectra and excitation profiles. The dis­persion of P, shows much less symmetry, however. This is due to the fact that the interference between vi­bronic levels in the same electronic component (+ or -) shows a different pattern than the interference between the two electronic components which, as we have seen [cf. Eq. (86)] is fairly symmetric. The vibronic inter­ference can be interpreted as an indirect coupling of the states J.L(±) and Jl'(±) via J.L"(~). This coupling is thus asymmetric since the vibronic levels of a component form a Franck-Condon-Uke progreSSion if I b I is not

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5486 Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers

04 (b) \ 06

/' I ~ I,

N " 05 N

" C I' o~, @ \ ':

02 \) \j \ ~ " 1\1 n4 " I ,j

03

u

(0) 09

10 ~ ~ " " " 07 · , -, ,

0 ' , 0Q:' @

, I

· , · / 05 \ /\ ,

05 ,/

'J

FIG. 5. Same plot as Fig. 2 for strong intermolecular cou­pling (b = O. 75, c = 1. 5).

small. Since PI tends to peak between bands and may reach nontrivial deviations from the off-resonance val­ue PI =t even for bands with very small cross sections, PI(Eo) tends to have a more extended structure at the high-energy side of Ea. The absolute maximum in PI(Eo), corresponding to the electronic interference, tends to shift to the low-energy side of Ea, however, due to the term b2 in (86). The combination of these two effects accounts for the asymmetric form of the PI(Eo) curves in Fig. 5.

C. Intermediate coupling

In the intermediate coupling region, where b2"" I c I ,

the vibrational and electronic splittings become compa­rable in magnitude, so that the weak and strong cou­pling expansions break down. If b2

"", I c I < 1, these splittings occur within the individual vibrational bands, so that the excitation spectra remain reminiscent of a

[' Eo~l3lw: [ Eo'6 024602460246

V V V

FIG. 6. RRS spectra corresponding to Fig. 5.

04

03 n<)

0J

c£l? ['7

C:I O"J

n (H ,~

I ~, (0)

II

- 10 09 0

,-) '" @ Q.

C7

05

05

0 - 03 V 18 2CJ

FIG. 7. Same plot as Fig. 2 for weak-couplinglike intermedi­ate coupling (b = c = O. 5).

Franck-Condon progreSSion, as they would in the weak­coupling limit. This behavior is illustrated in Figs. 7 and 8. If, on the other hand, b2

"" I c I > 1, the excitation spectra show the splitting in two electronic components characteristic of the strong-coupling limit, illustrated in Figs. 9 and 10. On closer inspection, however, these spectra turn out to be irregular and very rich in detail. The complexity of the underlying structure which is partly masked by the fairly large value of r chosen, shows up indirectly in the large number of res­onance-antiresonance coincidences between Rayleigh and Raman fundamental or between Raman fundamental and first overtone. As shown in Sec. III such coinci­dences are typical for the higher excited states of har­monic OSCillators, and thus are a direct consequence of

@ rrll E,O:B o 2 v4 6

@ lL: trl I ~017 lXI*, E

o'4 @

024 V

l~~11 ~016 024

V

FIG. 8. RRS spectra corresponding to Fig. 7.

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Gregory, Henneker, Siebrand, and Zglerski: Resonance Raman scattering in molecular dimers 5487

04r

03

1

1

2:: 02 @ ;

01'

o 05

, " " , ,

04 ; II "

! I :\

i \ f\ I'

08

06 , N

\ 6~ \! 04 Q.

02

~08

06 6 03: 1\ ! \ ,/ \ i\ @' I \ 1 \1 \ \! II f\

02 1 /," \: '"' \,' \ /" [ _____ ' v \/

01

~----- ----~--

12 14 16 18 20

FIG. 9. Same plot as Fig. 2 for strong-couplinglike inter­meditate coupling (b = c = 1. 5).

the large number of terms required in the expansion (46). The dispersion of the depolarization ratio shows this underlying complexity even more convincingly. The strong interferences lead to both strong minima and strong maxima in p,(Eo), especially near Eo = (c' Al­though the existence of these maxima and minima is readily understood on the basis of the general consider­ations in Sec. V, it would be futile to look for a simple perturbation-type explanation of a particular extremum in the intermediate coupling region. For spectra of this complexity no other explanation can be given than the one contained in the "exact" solution of the model.

D. Variation in orientation

If e, the angle between the two monomer transition moments, differs from t7T, the perfect balance between the two electronic components (+ and -) is lost. As a

@~~~l~ @~l:~ @Lblw;~

o 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 V V V

FIG. 10. RRS spectra corresponding to Fig. 9.

,1

" ,\ , , , ' , ' 02

1" //~ \, , , ," --- ,/"\} \: -" ..... -IJ

o - -----I

i I

15'

11

" "

" " , \ ) \ h I I fl I I II

--;- I 1\ ;\ 010 f \ 1\ @I·!\!\j'

I' ,

I ,'I

05 : 'd , V

i--- j

OL 10 12 14

106

I ~ . OQ:' 04

02

~08 I _

J 6~ 06

02 20

F1G. 11. Effect of varying the molecular orientation (8 = !7r). Same plot as Fig. 9.

result, one of the components will have a larger cross section than the other. If e=o or 7T, one of the compo­nents has a zero cross section. Again PI(Eo) shows a more complex behavior than e(Eo) in that the weaker component may show the larger vibrational structure in PI(Eo), as shown in Fig. 11. The RRS spectra are es­sentially independent of e (cf. Figs. 10 and 12), except for the reduced cross section in the region of the weak component.

E. Variation in linewidth

If the linewidth is increased from r = O. 2 to r = O. 5, the resolution in e(Eo) and p,(Eo) is greatly reduced with a corresponding loss of information, as shown in Fig. 13. The RRS spectra remain sharp, but the structure of the vibrational progressions shows much less detail (Fig. 14). In the intermediate coupling region, the

@L:l&:L: @Lw:l:~

II. Eol2 I Eo=15 I Eo=18

@~~~ V V V

FIG. 12. RRS spectra corresponding to Fig. 11.

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5488 Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers

005'

0 1 --

015

I r---

DOS:

0-10 12 14

...... \ , ,

---~O

08

'_., 06

16

, , O'¥

'j'\ ,04 Q \._"'~' .... :

18 20

FIG. 13. Effect of varying the linewidth (r = o. 5). Same plot as Fig. 9.

spectra represented by Figs. 13 and 14 would probably be too broad to allow a meaningful assignment.

VII. CONCLUSION

This paper represents a first attempt to study RRS for molecular systems with overlapping electronic ab­sorption bands. The results show that such bands give rise to complex interference effects. This is a gener­alization of Mortensen's results10 for RRS leading to excitation of nontotally symmetric modes. According to the Herzberg-Teller picture, this scattering is due to vibronic coupling between the resonant state and at least one other electronic state. Although the two ab­sorption bands do not need to overlap in this picture, the second electronic state may be said to superpose its properties on those of the first by means of vibronic coupling. Accordingly the interference is of vibronic rather than electronic origin and the splitting of the Mortensen-type doublet in the excitation profile is equal to a vibrational frequency.

In the case of two overlapping absorption bands, the basic splitting is of electronic origin, and represents an electronic coupling parameter modulated by a vibra­tional overlap factor. The two components themselves are not simple Lorentzians, as in the Mortensen model, but show vibrational structure. These vibrational split­tings may range from much larger to much smaller than the electronic splitting, depending on the system at hand. The vibrational structure varies accordingly from a Mortensen-type doublet (in the fundamental band) to a Franck-Condon progression. In addition to the electronic interference between the two band systems, there will also be vibronic interference between the vi­brational components in each band system. This inter­ference gives rise not only to maxima in the depolariza­tion ratio between the two interfering bands, as in the Mortensen model, but also to minima.

I, Eo'2 I Eo=15 E =18

® L ~ ~+:c+IIIIIIIO. o 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

v v v FIG. 14. RRS spectra corresponding to Fig. 13.

It should be realized that the dimer, because of its permutation symmetry, represents a very simple ex­ample of a system with overlapping absorption bands. If the two band systems are not related by symmetry, the resulting spectra and profiles will generally vary in a less systematic manner, although the basic interfer­ence patterns outlined here should remain valid.

Finally the question may arise how these predictions compare with experiment. The answer is, unfortunate­ly, that, for the time being, our predictions cannot be directly tested since no dimer scattering data are avail­able. The results reported to date on other molecular systems with overlapping absorption bands will be con­sidered in a future publication.

*Issued as NRCC No. 14967. tNRCC Postdoctoral Fellow. tPresent address: Department of Chemistry, MIT, Cam­

bridge, MA. §Permanent address: Department of Theoretical Chemistry,

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Gregory, Henneker, Siebrand, and Zgierski: Resonance Raman scattering in molecular dimers 5489

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