THE INFLUENCE OF INITIAL CONDITIONS ON DYNAMICAL PROCESSES IN MOLECULAR SOLIDS

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THE INFLUENCE OF INITIAL CONDITIONS ON

DYNAMICAL PROCESSES IN MOLECULAR SOLIDS

R. Hochstrasser

To cite this version:

THE INFLUENCE OF INITIAL CONDITIONS ON DYNAMICAL PROCESSES IN MOLECULAR

SOL

1

DS

R.M. Hochstrasser

Department o f Chemistry, University of PennsyZvania, PhiZadeZphia, PE 29104, U.S.A.

Résumé

-

Préparant par voie optique des excitations vibrationnelles dans les solides, nous proposons quelques idées nouvelles pour étudier les effets dynamiques qui en résultent. Nous avons prédit un effet significatif [voir référence 41. La compréhension des differentes contributions R la

largeur de raie Raman permet dlobtenir la valeur des temps de relaxation vibrationnelle pour les modee actifs en Raman.

Abstract

-

Racent ideas on the effects of opticai preparation on the dynamics of vibrational excitations in ~olide are presented. A significant effect is predicted [see reference 41. An understanding of the various contributions to the Raman linewidth in molecular crystals yields values for the vibrational relaxation timee of the Raman active modes.

1

-

INTRODUCTION:

In this report 1 will summarize some of our recent work on vibrational energy relaxation and pure dephasing in molecular crystals. In particular 1 will confine my remarks to the case of the benzene crystal.

Our approach to studying vibrational relaxation in crystals hae been to use the method of coherent anti-Stokes Raman scattering. In this technique the vibrational states (excitons) of the crystal are excited by two light fields whose frequency difference can drive an oscillating polarizability. This material oscillation may beat with a third field to generate coherent anti-Stokes light. Either the frequency dependence of the generated intensity, or the variation of signal intensity a s a result of delaying the third pulse, can be employed to measure the vibrational coherence 108s time. There can be many contributions to the coherence loss time but the situation for molecular crystals ie greatly simplified a s a result of the vibrational '8nergy transfer, or band formation, causing an averaging out of the inhomogeneoue distribution of transition frequencies /1-3/. The transitions that are observed are therefore essentially homogeneous and the linewidths or decay times yield interesting molecular properties direotly.

Much of our work during the past few years was aimed a t defining the various contributions to the Raman linewidth. The 992cm-1, totally symmetric ring stretching mode, has received the most attention. For this mode the four factor group (k=O) levels span f3vilcm-l, with the Ag (factor group) component a t the bottom of the band. The inhomogeneous distribution has a width of D

=

0.1 cm-1 as gauged from studies of isotopically mixed crystals /4/. Thus the effective inhomogeneous width /2/ that arises from averaging by exciton motion is (D/@)'Dz 10-~cm-~. This corresponds to an effective decay haiftime of Ca. 50ns, but the decay is not erponential and these caneiderations are juat guidelines. In any event, this contribution can be neglected in relation to picosecond timescale relaxation proceases.

Another important contribution to the linewidth of vibrational transitions involves iaotopic impurities. In recent work we showed that the

cl3

containing molecules in natural benzene crystals have significant effects on the vibrationai relaxation rates

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4 Thus experiments must be carried out ueing isotopically pure material. So far, the monodeuterobenzene was not removed, but experiments which involved deliberate doping have suggested that the effect of CgB5D in natural cryetals would not be eignificant. Experiments of thie sort and those discussed in the previous paragraph have covered al1 known possible contributions to the linewidth and lead us to conclude that the effects of intrinsic vibrational energy relaxation can be deduced by thia elimination.

II

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DESCRIPTION OF THE DOORWAY STATE:

As wae shown recently by Velsko and Hochetraseer /5/ the optical excitation of a cryetal or a mixed cryetal a t eufficiently low temperatures yields a rather special initial atate. The case of a neat cryetal ie weil-known: The initial etate ie approximately a k=O etate for which by definition the excitation amplitudes are equal for all moleculee. The mixed cryetal is a somewhat more wmplex situation, but the result is th'e same: The initial etate for a given configuration of impurity moleculee hae equal excitation amplitudes on ail hoet molecules. Thie result is dependent on there being no dephasing of the initially created excitations and aleo on there being separated bands. Each of theee approximations is diecuesed briefly below.

The exietence of eeparated bands, ae illuetrated in Figure 1, implies that

Fig. 1

-

Separated band pulsed excitation.

the excitation bande of the two components, hoet A and guest B, are not simultaneouely excited by the frequency components of the light pulse of width wp.

In order that the bande are not mixed significantly we must have their widths e~

and t e coneiderably lese than their separation A. In thie situation a sufficiently long light pulse (rp

>>

1/A) centered a t the frequency will interact in a firet approximation only with A-band molecules.

In the separated band limit the full cryetal Harniltonian has eigenetates that a r e essentially of A and B type and the r e h t i o n s of A molecules ie governed by the potentiel

Vjk is the lattice coordinate dependent intersite potentiel. The vibrational relaxation occure through both VAA and VAB pathways, with the former corresponding to vibrational energy reorganization amongst A molecules and the latter in trapping of a host vibrational energy by B molecules.

The vibrational energy redistribution occurring in a neat crystal involves relaxation at a single eite as well a s relaxation which requires the creation of vibrational excitations a t different sites. Since the moleculee of the lattice are all related by symmetry, it is not poesible to dietinguieh theee pathways. However by introducing

email amounts of isoîopic impurities, it should become possible in distinguish the varioua relaxation mechanisms.

The state of a crystal at time t, IS(t)>, resulting from the ground etate,

I%>,

interacting with excited states through the additional potential V(t) is given by:

t

- i w j t

l ~ ( t ) > - 1 $ ( 0 ) > = zl+j>. d t , e i l o t , <+jlV(t.)l*o> (2)

J

--

where ej,

=

~ j - 0 ~ , and l+j> are the e~enfunctione of the crystal. The additional potentiel in optical experimenta ia -E(t)*p for linear excitations, -BI (t)Ea (t):a for quadratic excitations, and eo on. When the Fourier transforme of the fielde are introduced, the nature of the prepared &te can be written in the famiiiar form:

where on is the appropriate combination of field frequenciea used t o drive the syetem and d l

=

p, d a

=

a, d r

=

y , etc., where p ie the dipole moment, a polarizability, y the hyperpoiarizability operator and so on. The quantity

F ( ~ ~ ~ - U ( " ) ) is in general an nth order convolution of the spectra of the fielde. In

the linear came thie is just the magnitude of the driving field epectrum a t the reeonance, E(wjo-o). In many of our experiments aimed at vibrational relaxation we use a Raman interaction (dl), in which case:

with u(l)

=

ai-a, where wl and e, are the center frequencies of the two driving fields of a CARS experiment. Equation (3) expresses the fact that a change in the wavefunction occurs only for thoee transitions for which dn and a field frequency component exbts.

In experiments with light pulses that persiet for times short compared with the dynemical evolution of the eystem, we can coneider that the pulee prepares a etate which then evolvee under the influence of the system Hamiltonian. The prepared state, or doorway state, for the Raman interaction ie:

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III

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EFFECT OF INITIAL STATE ON RELAXATION:

It wae aeen above that the doorway state in the firet approximation, equation (6) corresponds to one in which the excitation is equal on al1 host molecules even in the presence of many B molecules within the excitation volume. The energy relaxation of host molecules can now occur through the terme VAA and VAB.

Obviouely the presence of B moleculee impliee the absence of A molecules in this relevant volume so that A+A tranefer ie changed by adding B. Simiiarly A+B tranefer now dependa on the epecific topology of the disordered A lattice. Thie way of loohing at energy trapping ie somewhat unueual eince we are not coneidering the flow of energy to traps via some wavepacket in the hoet lattice, but instead that the doorway state collapees into a particuiar configuration of B

moleculee. The predicted rate of relaxation is then the average of the rates for many different nearest neighbor configurations. This mode1 predicte a nonmonotonic variation of the rate of trapping ae a function of the concentration of B moleculee /5/ which ie coneietent with the preliminary results for relaxation in

mixtures /3,4/.

It ie of great interest to consider the effect on reiaxation of changes in the nature of the prepared etate IS>. One of the assumptions in defining IS> ie that the dephasing of the excitations ie slow cordpared with the preparation time. Certain general aepecta of the other limit where dephaaing ie rapid compared with the energy relaxation, can be eeen without calculation with reference to Figure 2.

+

_PHASED

T R A P P I N G

-

DEPHASED

T R A P P I N G

Fig.

:

-

Illustration of phaeed and dephaeed trapping processes.

The situation described so far involvee 'phaeed trapping' where the final states. (If>) corresponding to combinations of interna1 and external degreee of freedom, are reached directly from a well defined distribution of exitation amplitudes, whose oscillations a t each aite are exactly predictable.

In the other limit the doorway state decaye rapidly into its dephaeed replica where now the amplitudes on different hoet mites are random phaeed, signified by the parameter ai in Figure 2. Now the energy relaxation ia a 'dephaeed trapping' into the set of final etates (If>). It appears as if the rate of relaxation ehould be

populated a s a result of a apontaneoua relaxation procesa the subaequent relaxation could begin from quite different statea whose character would depend on the mechaniam of the relaxation.

IV

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VIBRATIONAL RELAXATION RESULTS:

A s indicated elsewhere /4/ the vibrational relaxation times of the fundamentala of benzene do not correlate well with the number of available relaxation channels. This has suggested that there are apecific mode couplinga involved. For example, the coupling of the 991cm-1 C-C stretch with the 854cm-1 C-H out-of-plane bend /6,7/. However the rates generally increase with the total vibrational energy content as sumrnarized in Figure 3.

h

-

i

V I B R A T I O N A L R E L A X A T I O N I N

' * C ~ H ~

: I '0

lol'

U W l-

-_---

A-

o.

a

y---

œ

1 1

a---G--T-

5

1

0l0

9-

FREE ACCESS REGION

i

1 RESTRlCTED I ACCESS REGION

-

EVIB

(cm-'

)

Fig. 3

-

Vibrational relaxation rates of aome Raman active modes in benaene. The dashed line is intended to draw the eye to the two regiona of behavior of rate versus total vibrational energy. (Data from Ref. C41)

The benzene cryetal i s sufficiently harmonic that we might expect Av

=

*l selection rules to determine the relaxation pathways, where Av indicates changes in both the internal and the external, relative motional, degrees of freedom. If thia ie correct, then the variations of rate with energy in Figure 3 have an obvious interpretation in terms of the internal anharmonicities of the benzene modes.

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molecular anharmonicities becoming significant. A t a certain state density still in the region of the fundamentals, the interna1 harmonie modes might be sufficiently mixed to allow relaxation into many channels while still maintaining the Av

=

1

selection rule for external modes. Interna1 mode mixing was previously invoked to describe the relaxation of CH modes in liquids /8/ specifically in that case in terms of their ooupling to CH bending motions.

I t wiii be interesting to discover by studies of the infrared active modes of benzene whether the trend of Figure 3 is upheld. The basic ideas proposed here might carry over into other types of molecules. Of course significant differences may be expected in the energy a t which the anharmonic effects become dominant. For example in larger molecules where there are many lower frequency modes than found for benzene the onset of the free access region may be a t quite low total energy. The lowest frequency modes may then be b u close to the lattice modes for there to be a significant restricted region. In such cases one may actually observe a minimum in the rates a t some intermediate energy where the modes are freed from significant lattice involvement. I t is interesting that the trends indicated here for benzene crystals are not remarkably different from those seen for benzene molecules undergoing wllisions with isopentane /9/. In that case the excitation of some of the combinations involving u, va' and u, in the electronically excited state yields relaxation times that are not very iensitive to mode type or to total energy content. In fact one could imagine a similar curve drawn through these gas phase data a s is drawn a s a dashed line in Figure 3. I t would be of great interest to have results for ground state relaxation resulting from benzene-benzene collisions, and for liquid benzene.

Acknowleduement

This work was supported by grants from NSF and NIH.

References

1. R. M. Hochstrasser and 1. 1. Abram in Liuht Scattering in Solide, J. Birman, H. Cummins and K. Rebane eds., Plenum NY (1979).

2. 1. 1. Abram and R. M. Hochstrasser, J. Chem. Phys.,

12

(1980) 3617.

3. S. Velsko and R. M. Hochatraaser, J. Phys. Chem. @ (1985) 2240.

4. J. Trout, S. Velsko, R. Bozio, P. L. DeCola and R. M. Hochstrasser, J. Chem. Phys.

a

(1985) 4746.

5. S. Velsko and R. M. Hochstraeser, J. Chem. Phys.

82

(1985) 2180.

6. F. Ho, W. S. Taay, J. Trout and R. M. Hochstraeser, Chem. Phys. Lett.

83 (1981) 5.

-

7. R. Righini, P. F. Fracassi and R. G. Della Valle, Chem. Phys. Lett.

97 (1983) 308.

-

8. A. Fendt, S. F. Fischer and W. Kaiser, Chem. Phys.

52

(1981) 55.