Calculations of absorption and emission spectra: A study of cisstilbene

Transcription

1 Calculatins f absrptin and emissin spectra: A study f cisstilbene David C. Tdd, Graham R. Fleming, and Jhn M. Jean Citatin: The Jurnal f Chemical Physics 97, 8915 (1992); di: / View nline: View Table f Cntents: Published by the AIP Publishing Articles yu may be interested in Femtsecnd Raman spectra f cis-stilbene and trans-stilbene with istpmers in slutin J. Chem. Phys. 137, (2012); / Viscsity dependence and slvent effects in the phtismerizatin f cis-stilbene: Insight frm a mlecular dynamics study with an ab initi ptential-energy functin J. Chem. Phys. 111, 8987 (1999); / Femtsecnd laser studies f the cisstilbene phtismerizatin reactins J. Chem. Phys. 98, 6291 (1993); / Vibratinal energy redistributin and relaxatin in the phtismerizatin f cisstilbene J. Chem. Phys. 97, 5239 (1992); / Flurescence upcnversin study f cisstilbene ismerizatin J. Chem. Phys. 93, 8658 (1990); /

2 Calculatins f absrptin and emissin spectra: A study f cis-stilbene David C. Tdd and Graham R. Fleming Department f Chemistry, Department f Physics, and The James Franck Institute, The University f Chicag, 5735 S. Ellis Ave., Chicag, Illinis Jhn M. Jean Department /Chemistry, Washingtn University, One Brkings Dr., St. Luis, Missuri (Received 13 May 1992; accepted 31 August 1992) Absrptin and emissin spectra are calculated by mdeling cis-stilbene as a system f 12 displaced harmnic scillatrs. We are able t btain gd agreement with the peaks f bth the rm temperature absrptin and lw temperature emissin spectra using parameters frm resnance Raman experiments by adjusting the psitin f the zer-zer transitin energy (E.) and slightly adjusting the displacements alng the nrmal mdes. The best fit value fr E. is cm- I. Using the displacements alng the 12 degrees f freedm, and a nrmal mde descriptin based n semiempirical quantum frce field calculatins (QCFF/PI), we determine a "relaxed" excited state gemetry which is twisted a maximum f apprximately 37 deg alng the ethylenic trsinal crdinate relative t the relaxed grund state gemetry. An extensin f the spectral calculatins is described which allws fr ne r mre f the mdes t be anharmnic and vibratinally unrelaxed. We apply this extensin t shw that cis-stilbene emissin can nt be riginating frm a gemetry with a 90 deg twist alng the ethylenic crdinate. Cmparisn f ur results with a recently btained rm temperature slutin phase emissin spectrum suggest that this emissin riginates frm vibratinally unrelaxed mlecules. INTRODUCTION The excited state ismerizatin f stilbene has lng been studied as a mdel system fr understanding the effects f slvent-slute interactins n cndensed phase chemical reactins. I The ismerizatin f trans-stilbene t cis-stilbene has been used extensively t study the rle f frictin in activated prcesses and as a testing grund fr unimlecular reactin rate theries. 2 Althugh the ismerizatin f cis-stilbene t trans-stilbene has histrically received less attentin, recent advances in ultrafast spectrscpy have given new life t the investigatin f this rapid prcess. The lifetime f ptically excited cis-stilbene at rm temperature is apprximately I ps in lw viscsity liquids 3 4 and nble gas clusters S and even faster in the gas phase 6 reflecting the (near) barrierless nature f the ismerizatin reactin. Althugh these shrt lifetimes make cis-stilbene an interesting system t study, they als make it difficult t btain structural infrmatin n this transient species. Detailed knwledge f the excited state ptential energy surface and the nuclear rearrangements that lead t ismerizatin, hwever, are required if attempts t quantitatively mdel the reactin dynamics are t be successful. Sme f the mst useful structural infrmatin btained n this system has cme frm grund state resnance Raman studies. 7 The relative intensities f the Raman active mdes yield infrmatin n the shape f the excited state ptential energy surface in the regin directly abve the relaxed grund state. Within certain assumptins this infrmatin can be used alng with the absrptin spectrum t calculate the displacements between grund and excited state minima alng the nrmal mde vibratinal crdinates. Cmbining the resnance Raman studies with the emissin spectrum can help us learn abut the nature f the flurescing species. Recently, the first rm temperature emissin spectrum fr cis-stilbene in a lw viscsity envirnment has been btained. 8 The lw quantum yield fr cis-stilbene emissin under these cnditins 8 9 (~1O- 4 ) and the cmplicatins f verlapping emissin frm trans-stilbene have limited mst previus wrkers t studies in highly viscus media The tw striking features f the absrptin and emissin spectra are the lack f structure and the large Stkes shift, even in lw temperature and nnplar envirnments. One issue we will investigate is the surce f the large displacement between the absrptin and emissin spectra. Des this displacement result exclusively frm relaxatin in displaced vibratinal mdes, r are there additinal relaxatin prcesses which cntribute? We knw frm the grund state resnance Raman intensities 7 that there are large displacements in at least 12 nrmal mdes. As a result, when exciting near the peak f the absrptin spectrum the initially excited cis-stilbene mlecules will cntain a large amunt f excess vibratinal energy. Hw des this affect the emissin spectrum and the reactin dynamics? It is well knwn that vibratinal cling f medium size plyatmics in slutin may ccur n time scales as lng as tens f ps.12 It is therefre quite reasnable t expect that during a lifetime f apprximately 1 ps, cis-stilbene will remain vibratinally unrelaxed. We explre this pssibility with a simple mdel in which ne vibratinal mde, with a large displacement, has a nn thermal ppulatin distributin determined simply by the grund state Franck Cndn factrs. Cis-stilbene can underg tw reactins after ptical ex- J. Chern. Phys. 97 (12).15 December American Institute f Physics 8915

3 8916 Tdd, Fleming, and Jean: Calculatins f spectra citatin, the cis t trans ismerizatin, and ring clsing t frm 4a,4b-dihydrphenanthrene (DHP). The imprtance f the secnd reactin (with a rughly 30% branching rati l3 ) fr understanding the excited state dynamics, particularly at early times, has recently been discussed. 14 We will be cncerned, hwever, mainly with the first reactin since we d nt expect the DHP prduct t cntribute t emissin in the regin f cis-stilbene flurescence. Because the energy difference between the cis-stilbene and DHP grund states is apprximately cm - I, 15 the DHP emissin will be substantially redshifted frm that f cisstilbene. DHP is als believed t have a very shrt excited state lifetime l6 and therefre a lw flurescence quantum yield. Our understanding f the excited state ptential energy surface f cis-stilbene is still quite limited. Examples f unanswered questins are What is the gemetry and nature f the species bserved in transient absrptin and flurescence experiments? Is there a barrier alng the cis t trans ismerizatin crdinate? If nt, at what pint d the excited mlecules mve ut f the bservatin windw? Studies f cis-stilbene in argn clusters frmed in a supersnic expansin reveal a lng lived state which fluresces with a 17.2 ns lifetime. 5,17 This result suggests a small barrier alng the directin f cis t trans ismerizatin. A recent investigatin f the temperature dependence f the flurescence quantum yield in slutin, hwever, indicates a negative activatin energy alng this reactin crdinate f -1.1 kcal/m1. 9 Studies f the excited state lifetime as a functin f pressure l8 and temperature l9 in alkane and alchl slvents suggest that a barrier, if it exists, is small (,kt). Earlier studies 20 f cis-stilbene shwed that viscsity, rather than temperature, is the mst imprtant factr in determining the rate f cis t trans ismerizatin and the cis-stilbene flurescence quantum yield. These studies als indicate the absence f a substantial barrier. The cis-stilbene excited state ptential energy surface has als been the subject f a recent theretical study.14 This wrk fcused n the ptential as a functin f tw internal crdinates-the symmetric in-plane bend and the symmetric phenyl ring twist in the cntext f the cisstilbene t DHP ismerizatin. This investigatin predicts excited state barriers t ring clsure t frm DHP-type cmpunds in cis-stilbene and a variety f hmlgus mlecules, which are in gd agreement with earlier experimental results. 21 Frm wave packet dynamics calculatins, this study als suggests that DHP frmatin plays an imprtant rle in the early time dynamics f excited cisstilbene. Hwever, the rle f relaxatin in ther crdinates, especially the ethylenic trsin has nt been thrughly investigated. In this study we calculate absrptin and emissin spectra fr cis-stilbene using varius mdels t learn abut the nature f the flurescing species. We use as a starting pint the infrmatin btained frm resnance Raman intensities. We base ur calculatinal technique n the matrix methd f Friesner et aj. 22 which allws the efficient calculatin f spectra fr large multimde systems. We incrprate a basis set calculatin t allw the inclusin f a vibratinal mde which is anharmnic r vibratinally unrelaxed. Specific details necessary fr the calculatin f emissin spectra are presented. Cmparing ur calculatins with experimental spectra we are able t put cnstraints n and btain sme specific infrmatin abut the nuclear rearrangements ccurring in the excited state f cis-stilbene. THEORY AND CALCULATIONS Absrptin Absrptin and emissin spectra are calculated frm the Furier transfrm f the transitin diple autcrrelatin functin. The absrptin line shape, as discussed by Grdn,23 is given by i(w) =~ J+ 00 dt e-ialt</l(o) '/L(t» 21r - 00 (1) where the brackets dente an ensemble average. This is related t the absrptin cefficient a (w) defined by Beer's law, l(z) =10 exp[ -a(w)cz] by the fllwing equatin: The kernel in Eq. (1), </L (0). /L (t) ), is calculated using the matrix methd f Friesner et al. 22 which generates the exact thermal average f a prduct f expnentiated peratrs f quadratic frm. The frequency spectrum can then be generated via numerical Furier transfrm. Equatin ( 1) des nt include any cntributin frm inhmgeneus bradening r lifetime bradening which are generally incrprated int ur calculatins. 24 Fr cis-stilbene we typically add an expnential damping term crrespnding t rughly 50 cm -I width t the integrand in Eq. (1) t accunt fr hmgeneus bradening as well as fr spectral cngestin arising frm lw frequency mdes which are nt explicitly included in the Hamiltnian. This cntributin is small in cmparisn t the ttal spectral width f apprximately 6000 cm - I. The use f these techniques t calculate absrptin spectra was discussed previusly.24 We will fcus n the mdificatins necessary fr calculating emissin spectra and fr the incrpratin f a vibratinal degree f freedm which is anharmnic r vibratinally unrelaxed r bth. The diple autcrrelatin functin is given by (2) </L (0). /L (t) ) = L Pi(i I /L (0). /L (t) I i) (3) i fr vibrnic states I i) with ppulatin distributins, Pi' A subscript f zer is added t the diple autcrrelatin functin (Le., < )0) when the Pi crrespnd t a Bltzmann distributin. We write the Hamiltnian as H=~lg)<gl +(~+E) le)(el, (4) where I e) and Ig) are the excited and grund electrnic states and Hf, and ~ are the crrespnding vibratinal Hamiltnians. E is the zerth rder energy difference between the grund and excited state surfaces. In the Cndn apprximatin, Eqs. (1), (3), and (4) give, J. Chern. Phys., Vl. 97, N. 12, 15 December 1992

4 Tdd, Fleming, and Jean: Calculatins f spectra 8917 x LPi(Xilexp(i.lf'Jlli)exp( -imtlli) IXi), (5) i where I Xi) labels vibratinal states. Up t this pint n assumptins have been made abut the values f the Pi and the nature f the nuclear Hamiltnians. Fllwing Friesner et al., 22 we adpt a harmnic mdel fr the ptentials. This allws the prduct f expnential peratrs appearing in Eq. (5) t be written in terms f a si!!sle expnential peratr with an effective Hamiltnian, H, which is als harmnic. 22,25 The thermal average f the effective peratr is evaluated using standard many-bdy phnn techniques. 22 Using the grund state nrmal crdinates as ur representatin, the vibratinal Hamiltnians are and 14= L (alai+ 1I2)Wi i Irv=m+ L [gi(ai+ a;) + Vi(ai+ a;)2], i (6) (7) where a; and ai are the bsn creatin and annihilatin peratrs. Fr absrptin, and when there is n mde mixing, the linear cupling parameter, g, and the quadratic cupling parameter, V, are given by and a} g=a.~ g (8) (9) where a is the dimensinless displacemene 6 between grund and excited state minima alng a particular nrmal mde. The grund and excited state nrmal mde frequencies, Wg and we> are given in cm -I. Fr emissin, the rles f m and ~ are reversed. If we als assume that the vibratinal Hamiltnian is cmpletely separable (i.e., n Duschinsky rtatin in the excited state) then the crrelatin functin in Eq. (5) can be factred int a prduct f ne mde crrelatin functins. The peratr exp(ihtlli) can then be written as N II expuii;t/li) (10) i=1 fr an N mde system where exp(ih;tlli) =expuif{"lili)exp( -im}iii). (11) Representing the vibratinal wave vectr <xii in secnd quantized ntatin as (nl,n2,n3,..,nni we can write LP;(x;!expUiitlli) Ix;) ; = L L'" LPn 1 n 2 '"nn(n"n2,''' 'nni nl n2 nn N X II exp(iii;t/ii) In"n2,"'nN)' ;=1 This can be rearranged t give, II L Pn(nil expuii;t/li) In)., I I ni The absrptin line shape can nw be written as (12) (13) ,,[ ( N-m _)( m i(w) =-J.L;e J dt'e-,wte'et n LPn/nil expuh;tlli) I ni) n L Pn/n;1 expuif{"lili ) 21T' = 1 ni 1=' ni Xexp( -lm}lli) I ni) ) ]. (14) Althugh we riginally assumed that the Hamiltnian is cmpletely separable, it is nly necessary that the N-m mdes in Eq. (14) are separable frm the ther m mdes. We will present calculatins in which m = 1 and the term invlving the N - I mdes is calculated using the techniques f Friesner et al. 22 The secnd prduct f sums in the integrand f Eq. (14) is evaluated using a harmnic basis set, and the tw terms are multiplied at each time pint prir t the Furier transfrm. By writing the absrptin line shape in this way we can intrduce anharmnic terms int ne, r a small subset f vibratinal mdes while retaining the advantages f the matrix methd in calculating the kernel fr the remaining mdes. We can als calculate spectra with nnthermal ppulatin distributins in the m mdes in Eq. (14), whereas the matrix methd is nly valid fr an equilibrium distributin. The secnd term in Eq. (14) is simply evaluated fr the case where m = 1. By inserting a cmplete set f states between the tw expnential peratrs we btain, L Pn(n I exp UIf{,tlli)exp ( -imtlli) In) n =L LPnl(nlj)1 2 exp[i (E j -En)tlli]. (15) n j The states (n I are grund vibratinal states and the states J. Chern. Phys., Vl. 97, N. 12, 15 December 1992

5 8918 Tdd, Fleming, and Jean: Calculatins f spectra (jl are vibratinal wave functins in the excited electrnic state and En and E j are the respective vibratinal energies f these states. In ur calculatins we represent the ptential energy surfaces fr this mde, in bth the grund and excited states, as sums f plynmials. T calculate eigenvalues and eigenstates fr a surface with anharmnic terms we write, and diagnalize, the Hamiltnian in terms f a cmplete set f harmnic scillatr states, with frequency CUI' In this representatin the apprpriate dimensinless crdinate, Q, is defined as Q= (CUI/CUO) 112q, where CU is the frequency f the nrmal mde which is being replaced. 26 The ptential energy surface in this representatin has the frm (16) Using plynmials simplifies the calculatin f the Hamiltnian matrix elements. Eigenvalues and eigenvectrs are determined using standard matrix diagnalizatin rutines. Flurescence The calculatin f emissin spectra requires nly a few mdificatins f the previus treatment f absrptin spectra. The mst imprtant change is that the rles f the grund and excited state vibratinal Hamiltnians are reversed. This requires a change in sign f the displacements and f cu in the integrand f the Furier transfrm. The ther significant change required fr the flurescence calculatin invlves the prper scaling f the intensities at different frequencies. The relative flurescence intensity F(cu) is related t i(cu) by (17) where n(cu) is the frequency dependent refractive index. In ur calculatins we include the cu 3 term, and ignre the frequency dependence f the index f refractin which will have nly a small effect fr cis-stilbene in alkane slvents. In cyclhexane, e.g., the variatin f the index f refractin results nly in a 4% change in n 3 ver the spectral regin frm 365 t 546 nmp Care is necessary when cmparing flurescence data with calculated spectra. The usual flurescence measurement, perfrmed with a grating mnchrmatr determines a spectrum with the units phtns s - I area - I nm -I. Such a spectrum must be cnverted frm a wavelength t an energy scale t cmpare it with ur calculated spectra. This cnversin invlves bth an abscissa and an rdinate change (i.e., the units f intensity are nt the same fr the tw scales). As a result the value f the intensity must be multiplied by 11.2/cnstant befre the abscissa is changed. 28 This cnversin insures that the integrated area is cnserved (i.e., the number f phtns) and can alter bth the width f the spectrum and the psitin f the peak. TABLE I. Experimental frequencies and displacements frm resnance Raman experiments (Ref. 7) and frequency f crrespnding mde frm QCFF/PI calculatins (Ref. 32). lu~xpt J A (t)qcffipi 165 cm cm " "Assuming this mde is harmnic in the excited state with a frequency f 560 cm- 1 (see the text). RESULTS Harmnic mdes In all ur spectral calculatins, cis-stilbene is mdeled either as a system f 12 displaced harmnic scillatrs, r 11 displaced harmnic scillatrs with the remaining mde being anharmnic. The 12 mdes chsen fr ur calculatins are thse indicated by the resnance Raman studies f Myers and Mathies 7 as having substantial displacements and are listed in Table I. It is these mdes that make an imprtant cntributin t the shape and the psitin f the absrptin and emissin spectra. Dimensinless displacements were given in that wrk fr 11 f the 12 mdes under the assumptin that there are n frequency shifts in these vibratins upn ptical excitatin. Since there is little infrmatin available abut the excited state frequencies, we als assume the grund and excited state frequencies are the same fr all mdes treated as harmnic in ur calculatins. When the linear displacements are large, as in cis-stilbene, quadratic interactins such as frequency changes and Duschinsky effect have a minr rle in determining the shape and psitin f the absrptin and emissin spectra. We nte that in cis-hexatriene, Duschinsky mixing f the C-C mtins with the lw frequency skeletal bend was invked t explain the difference in the absrptin spectra f the cis and trans ismers, hwever, the degree f mixing was small (crrespnding t a nrmal mde rtatin f apprximately 4.6 ) and the effects subtle. 29 The dimensinless displacement f the 560 cm - I mde was nt given in the resnance Raman study, but the slpe f this mde in the Franck-Cndn regin f the excited state was determined t be 785 cm- I 7 (The slpe has units f energy, since it is the derivative f a ptential with respect t a dimensinless displacement.) If we assume this mde is harmnic in the excited state, with a frequency f 560 cm -I, we btain a dimensinless displacement f a = Fllwing Myers and Mathies 7 we use QCFFIPI calculatins,30,31 prvided t us by Myers 32, t make rugh assignments f the nrmal mdes in terms f internal c- J. Chern. Phys., Vl. 97, N. 12, 15 December 1992

6 Tdd, Fleming, and Jean: Calculatins f spectra 6919 rdinates. The characterizatin f the fur lwest frequency mdes in the resnance Raman spectrum was presented in Table I f Ref. 7. These fur mdes receive substantial cntributins frm many internal crdinates. The 165 cm -I mde, fr instance, cnsists f phenyl twist, ethylenic trsin, ring distrtins, and hydrgen wag mtin in rughly that rder f imprtance. The 403 cm- I mde cntains primarily ethylenic trsin, with cntributins frm phenyl ring twist, hydrgen wag and ring distrtins. It is the 560 cm - 1 mde that receives the largest cntributin frm ethylenic trsin. Hwever, this mde is als cmpsed f hydrgen wag, phenyl twist, and ut f plane ring distrtins. The 963 cm -I mde is cmprised mstly f hydrgen ut f plane mtin, particularly ethylenic hydrgen wag. Hwever, this mde als cnsists f sme ethylenic trsin and sme phenyl twist mtin. Since we als single ut the 1629 cm -I mde in later discussins, we nte here that this mde is calculated t be a nearly pure ethylenic duble bnd stretch in the grund state. Calculated absrptin and emissin spectra, using the resnance Raman parameters, are displayed in Fig. 1. (The calculated absrptin spectrum f Fig. I uses nearly identical parameters t that in Fig. 2 f Ref. 7.) In these and all ther calculatins we assume the electrnic rigin, E,, t be the same fr absrptin and emissin, set kt=204 cm - 1 and add a 50 cm - 1 hmgeneus linewidth t smth ut the spectrum. 7 Fr the results presented in Fig. t E, was varied fr the best fit t the absrptin spectrum 7,33 giving a value f cm - I. Cmparisn f the experimental and calculated absrptin spectra was limited t energies t the red f the first absrptin peak because f the increasing cntributin f a secnd electrnic transitin t the experimental spectrum at higher energies. The calculated emissin is cmpared with the lw temperature spectrum f Stegemeyer et al. 10 btained in a mixture f ispentane and methylcyclhexane at 77 K. A mre recently btained experimental emissin spectrum f cis stilbene in 3-methylpentane,34 als at liquid nitrgen temperature, has the same peak emissin energy as the Stegemeyer spectrum (a value f cm- I ) and apprximately the same width. We have fund that fr ur mdel, with all harmnic mdes, there is n significant difference in the shape and psitin f the emissin spectra calculated fr rm temperature r fr liquid nitrgen temperatures. The agreement with the psitin f the flurescence is very gd cnsidering that n adjustments were made t the parameters determined frm the resnance Raman study and since this spectrum was btained in a glassy medium where a slight blueshift might be expected. Fr example, limited relaxatin in large amplitude mdes culd result in the 200 cm -1 discrepancy in the peak psitins. The surce f the rughly 1000 cm -I difference in the widths f the calculated and experimental spectra will be discussed in the next sectin. Since there is bviusly sme uncertainty in the dimensinless displacements, it is useful t determine hw large a change in thse parameters is required t exactly reprduce the Stkes shift between the absrptin and emissin spectra. Uncertainties in the ~'s arise bth frm experimental limitatins as well as the assumptins f n frequency changes, n anharmnicities, and n mde mixing. We have fund that with nly slight adjusts in the displacements in ne r mre f the nrmal mdes and small changes in E, we can simultaneusly fit the absrptin and the peak f the emissin spectra. Fr example, changing the value f the displacement in the 165 cm - 1 mde frm 4.4 t (a 4.6% change) and setting E. =29,000 cm- I we btain the spectra shwn in Fig. 2. Equally gd agreement between the calculated and bserved spectra can be btained by adjusting the displacements in ther mdes. Reducing the displacement in the 560 cm -I mde frm 1.40 t and setting E. t cm - 1 gives results indistinguishable frm thse in Fig. 2. A displacement f implies a slpe f 687 cm - 1 > l- 0.8 Ui 0.6 z w l- ~ > l- O. 8 Ui 0.6 z w l- ~ 0.4 O FIG. 1. Cmparisn f calculated and experimental absrptin (Ref. 7) and emissin {Ref. IO} spectra. Frequencies and displacements used fr the calculatins are thse determined frm resnance Raman intensities (Ref. 7) and listed in Table I and E.= cm- I. All spectra are nrmalized t 1.0. Experiment (-). Calculatin (-). FIG. 2. Calculated best fit t the absrptin spectrum (Ref. 7) and the peak f the emissin spectrum (Ref. IO). Displacement in 165 cm- I mde changed t and E.= cm- I. Experiment (-). Calculatin (-). J. Chem. Phys. Vl. 97. N December 1992

7 8920 Tdd, Fleming, and Jean: Calculatins f spectra > I- 0.8 ;n 0.6 z l1j I- ~ em- l FIG. 3. Cmparisn f experimental rm temperature spectrum f cisstilbene in hexane (Refs. 8 and 35) and calculated emissin spectrum frm Fig. 2. Experiment (-). Calculatin (-). fr the 560 cm -I mde in the Franck-Cndn regin f the excited state. The magnitude f the changes in the a's required t reprduce the experimental Stkes shift is quite small, and well within the estimated uncertainties fr these parameters. 32 Changes f this magnitude d nt strngly affect the width f the spectrum r the calculated excited state gemetry (vide infra). A cmparisn f ur calculated spectrum, r the lw temperature emissin spectrum, with the experimental spectrum f Saltiel et al. 8,35 taken in rm temperature hexane reveals a large discrepancy as seen in Fig. 3. The cis-stilbene in hexane spectrum [excited at apprximately 270 nm (Ref. 8)] has a peak at r 460 em-i t the blue f the experimental spectrum f Stegemeyer et al. The rm temperature spectrum is als 1510 cm - 1 wider than ur calculatins (r 2740 em-i wider than the lw temperature spectra). These discrepancies can nt be reslved by slightly adjusting the parameters in the mdel, and althugh there is certainly sme inhmgeneus bradening in this system, it prbably cannt explain the 1510 cm - 1 difference between ur calculatin and the rm temperature spectrum. In rder t understand the width and psitin f this experimental spectrum, we prpse that the shrt excited state lifetime f cis-stilbene in slutin strngly enhances the cntributin f unre1axed emissin t this spectrum. Vibratinally unrelaxed emissin In rder t gain further insight int the rm temperature emissin spectrum f cis-stilbene in hexane, we have calculated spectra with a crude mdel f vibratinally unrelaxed cis-stilbene. Even if intramlecular vibratinal relaxatin (IVR) were slw in cis-stilbene, excess vibratinal energy wuld be distributed in a number f nrmal mdes-in particular, thse mdes with large displacements. The emissin spectrum under these circumstances is, unfrtunately, t difficult t calculate with ur methd, s we shall assume all excess vibratinal energy is in ne nrmal mde. This was accmplished by replacing the Bltzmann distributin, fr that mde, with a ppulatin distributin determined simply by the Franck Cndn verlaps with the thermally ppulated grund states. This distributin crudely apprximates the t=o ppulatin distributin and is given, fr mde j, by the fllwing: (18) where E j are the energies f the grund vibratinal states and I Uul 2 are the Franck-Cndn factrs. [(Wj) is the spectrum f the excitatin surce which will be assumed fr simplicity t be a cnstant, crrespnding t a bradband excitatin surce. This assumptin tends t verestimate the degree f vibratinal excitatin in the unrelaxed mde. It is als assumed that this ppulatin distributin des nt change during the apprximately 1 ps lifetime f cis-stilbene in rm temperature liquids. Tw such calculatins are cmpared with the experimental result in Figs. 4(a) and 4(b). These have unrelaxed emissin frm the 560 and 165 em - 1 mdes, respectively. Fr Fig. 4(a) we use 60=1.225 fr the 560 cm- I mde, and use the experimental frequencies and displacements listed in Table I fr all the remaining mdes and E, = cm -I. The parameters fr the calculatin in Fig. 4(b) are the same as thse used fr the emissin spectrum in Fig. 2. We chse t lk at these mdes because they bth cntain a large degree f ethylenic trsin accrding t QCFF /PI calculatins 7,32 and the 165 cm - 1 mde in particular because it has a very large displacement. Leaving the 560 cm -I mde unrelaxed has very little effect n the emissin spectrum. The calculatin where the 165 cm -I mde is unrelaxed, hwever, is significantly altered. The excellent agreement with experiment in Fig. 4(b) is prbably smewhat frtuitus, but this result is indicative f the degree and the manner in which the spectrum can be altered by vibratinally unrelaxed emissin. We als examined the case in which the 1629 cm -I mde was unrelaxed. Since this mde is cmpsed f mstly ethylenic stretch it is relatively islated frm the slvent and may remain unrelaxed n the time scale f ismerizatin. This yielded a spectrum slightly wider and with a lnger tail n the red edge, than the spectrum illustrated in Fig. 4(b) (fr 165 em - 1 mde). These results strngly supprt the suggestin that vibratinal excitatin abve the thermal distributin culd explain the blueshift and the larger width f the rm temperature spectrum in hexane slutin. It shuld be nted that fr calculatins f emissin spectra f cisstilbene btained under cnditins f shrt pulse excitatin, it may be necessary t include vibratinal phase cherences. Inclusin f an anharmnic mde Finally, we have als perfrmed calculatins t test the feasibility that flurescence is riginating frm the scalled twisted intermediate. Althugh this wuld be unlikely in a lw temperature glass because f sterie effects, it J. Chem. Phys., Vl. 97, N. 12, 15 December 1992

8 Tdd, Fleming, and Jean: Calculatins f spectra > l- 0.8 v; 0.6 z lj.j I I' ~ (a) em-i (a) DIMENSIONLESS DISPLACEMENT > I- 0.8 Vi 0.6 z w I- ~ 0.4 > l- 0.8 v; 0.6 z lj.j I O. 2 O. 2 (b) em-i (b) em-i FIG. 4. (a) Experimental rm temperature emissin (Refs. 8 and 35) and calculated emissin with 560 cm -I mde vibratinally unrelaxed (see the text). Using 11 = in the 560 cm -I mde, E.= cm -I, all ther parameters the same as thse f Fig.!. (Using these parameters t calculate vibratinally relaxed emissin yields spectra indistinguishable frm the calculatins in Fig. 2.) Experiment (-'-). Calculatin (-). (b) Experimental rm temperature emissin (Refs. 8 and 35) and calculated emissin with 165 cm- 1 mde vibratinally unrelaxed. All ther parameters same as in Fig. 2. Experiment (-). Calculatin (-). FIG. 5. (a) Ptential energy surfaces used t calculate emissin in Fig. 5(b). Excited state surface given by 21.58(!-!.53{f+59.15Q'. Grund state surface given by (!+ 1480Q'-118.9{!+3.74<j". Q is the dimensinless crdinate using a 50 cm -I basis set fr the calculatin. A displacement f Q= 1.0 implies a change in ethylenic trsin f 25.3 accrding t QCFFIPI calculatins n the grund state f cis-stilbene (Ref. 32). (b) Cmparisn f experimental spectrum f Saltiel et al. (Refs. 8 and 35) and calculated emissin spectrum using surfaces in Fig. 5(a). Experiment (-). Calculatin (-). is certainly plausible in the case f rm temperature emissin. Fr these calculatins we replaced the harmnic 560 cm - 1 mde with an anharmnic mde which crudely represents the ismerizatin crdinate. While adding large anharmnicities t nly ne mde will be an apprximatin, we believe in the cntext f the present wrk that it still allws us t capture the essence f the prblem. The grund state ptential energy surface alng the anharmnic mde is given minima at the cis-stilbene and trans-stilbene gemetries and a maximum at a trsinal angle f 90. The barrier height crrespnds rughly t the grund state thermal barrier frm the cis-stilbene side f 42.8 kcal/ml. 36,37 The frequency f bth wells appraches a 560 cm -1 harmnic scillatr at lw energies. The excited state ptential surface has a minimum directly abve the grund state maximum and a slpe f 785 em -1 in the regin directly abve the grund state well fr cis-stilbene. A surface which fulfills these criteria is shwn in Fig. 5(a) alng with the calculated emissin spectrum in Fig. 5(b). The surfaces are described with Eq. (16) and the parameters are given in the captin f Fig. 5 (a). The hrizntal axis in Fig. 5(a) crrespnds t the dimensinless crdinate f the basis set used in this calculatin. A displacement f 1.0 alng this crdinate crrespnds t a 25.3 rtatin arund the ethylenic trsin as determined by QCFF /PI calculatins. 32 The rest f the 90 rtatin arund this bnd is assumed t result frm rearrangements in ther mdes. The parameters fr the ther mdes are thse determined by the resnance Raman studies 7 ( see Table 0. It is evident that there is n resemblance between the calculated and the experimental spectra in Fig. 5(b). We have tried a variety f ther similarly shaped surfaces, but in all cases the agreement is very pr. Obviusly these J. Chem. Phys., Vl. 97, N. 12, 15 December 1992

9 8922 Tdd, Fleming, and Jean: Calculatins f spectra calculatins are versimplificatins given the limited knwledge available n the ptential energy surface in the regin f the twisted state. Hwever, the features f a very large red shift and a width much larger than the experimental spectra seem t be a general result f the large increase in the grund state surface at the 90 gemetry. It is therefre likely that emissin is nt emanating frm an excited state gemetry even clse t that crrespnding t the grund state maximum. DISCUSSION The psitin f the flurescence spectrum can be reprduced withut changing the zer-zer transitin energy frm absrptin t emissin, and with nly a slight mdificatin f the displacements given by the resnance Raman intensities. This implies that the spectral shift f the emissin can be simply explained as a result f the substantial displacements in a large number f vibratinal mdes. It is therefre unnecessary t invke ther relaxatin surces. This result is als cnsistent with the lack f a rise time in any f the transient decays btained fr excited cis-stilbene. 3,4 The parameters in ur calculatins can be used t determine the apprximate gemetry assciated with the flurescing species. Relaxed gemetry The dimensinless displacements alng nrmal mdes which served t calculate the varius spectra can be used t determine displacements alng internal crdinates. The relatinship between the tw is given by the fllwing: 38 8;= ~.t4;fllT1/21:!./=~pij' (19) where the 8 j are internal displacements, the I:!.j are the nrmal mde dimensinless displacements and OJj are the nrmal mde frequencies given in cm -I. The nrmal mde cefficients, Ai» are just the slutins t the Wilsn FG matrix equatin,39 A(FG) -AA=O. (20) We btain Au's fr cis-stilbene frm the results f grund state QCFF /PI calculatins. 32 With Eq. (19) we have calculated the displacement alng the ethylenic trsin fr the parameters used in the absrptin and emissin spectra shwn in Fig. 2. We assume that the QCFF /PI descriptin f the nrmal mdes is reasnable and that there is n Duschinsky effect and that there are n frequency changes in the excited state. The QCFF /PI mdel is believed t prduce accurate frce fields fr plyenes, and t be applicable t relatively large systems. 40 The I:!. values fr the calculatin and the 8;/s, which are given in radians fr a trsin, are listed in Table II. It can clearly be seen that fur nrmal mdes make substantial cntributins t this displacement. Unfrtunately, the sign f the I:!.'s is nt determined frm the resnance Raman experiments. If we assume that the sign f the fur largest fju's is such as t increase the value f the ethylenic trsinal angle, we determine a displacement f 35. Increasing this trsinal angle is cnsistent with the changes in bnd rder which result frm excitatin, and TABLE II. Cntributins f nrmal mde displacements, Ai' t ethylenic trsin, (), calculated with Eq. (19). Displacements are thse used fr Fig. 2. CL)~xPt JA) Aij J.5ijJ () J 165 cm deg ttal cm deg ttal 2.56 allws fr a reductin f the steric interactin between the phenyl grups. If we assume the displacements in all the mdes cntribute cnstructively t the trsinal angle we btain a maximum displacement f apprximately 37. Hwever, we have less cnfidence that the higher frequency mdes all add psitively t the displacement f the ethylenic trsin. Given a predicted grund state gemetry with a 9 twist alng this crdinate,30 the gemetry f the relaxed excited state (r at least the emitting species) has an apprximately 44 twist in the ethylenic trsin. This is in reasnable agreement with the relaxed excited state gemetry predicted by Warshel 30 with an ethylenic trsin f 35. We nte that if the size f the barrier fr cis t trans ismerizatin is determined by the gemetry where the S 1 and S2 electrnic surfaces crss, this large trsinal angle fr the relaxed excited state culd accunt fr the lack f a substantial barrier fr this prcess. Electrnic rigin and spectral width Frm the calculatins shwn in Fig. 2 we btain ur best estimate fr E, f cm -I. The brad spectra f cis-stilbene in practically all envirnments have made an accurate experimental determinatin f this value difficult. In rder t see any structure in the absrptin band, Dyck and McClure 41 were required t investigate cis-stilbene in diphenylmethane plycrystals at 20 K. Cnsidering that little structure was present even under these extreme cnditins their estimate 41 f cm -I must be cnsidered a rugh apprximatin. The wavelength f the zer-zer transitin was als estimated t be greater than nm by Petek et al. 13 fr cis-stilbene in nble gas clusters frmed in a supersnic expansin. This implies an E, f less than cm -I, in gd agreement with the estimates frm ur calculatins. If we assume that mirrr image symmetry42 applies fr the 90 K absrptin 41 and the 77 K emissin 10 spectra f cis-stilbene in alkane envirnments, we determine a value f E 0 f em - I, als in gd agreement with the values used in Figs. 1 and 2. J. Chem. Phys., Vl. 97, N. 12, 15 December 1992

10 Tdd, Fleming, and Jean: Calculatins f spectra 8923 Since we assume that the electrnic rigin des nt shift between absrptin and emissin, we cnsider the pssible effects f slvent relaxatin and the cmparisn f ur calculated emissin spectrum with a lw temperature emissin spectrum. A significant shift f the zer-zer transitin energy between absrptin and emissin resulting frm slvatin f the excited state is unlikely because all the experimental spectra used fr cmparisn with ur calculatins were btained in shrt chain alkane slvents. There als seems t be n shift in E, frm absrptin t emissin fr trans-stilbene in n-alkane slvents. 43 We have investigated the effect f the temperature dependence f the index f refractin n ur estimate f E. We estimate (using the frmula f Bayliss44) fr cis-stilbene in alkanes a shift in E, between rm temperature and liquid nitrgen temperature in the range f 60 t 140 cm -I. The estimate f E,, frm ur spectral calculatins wuld change by nly half this value, r 30 t 70 cm -I, indicating this is nt a substantial effect. There is an bvius discrepancy between the widths f ur calculated emissin spectra and the steady state spectrum f Stegemeyer et af. 10 btained in a lw temperature glassy envirnment. Using a mdel with 12 harmnic mdes, as in Fig. 1, we fund that temperature alne culd nt accunt fr this discrepancy. Dyck and McClure 41 bserved that the experimental absrptin spectrum is much narrwer and redshifted in glassy material at liquid nitrgen temperatures than in rm temperature slutins. A qualitative explanatin fr this effect was ffered by Brmberg and Muszkat 45 based n calculated anharmnicities in the phenyl twist mde arising frm steric interactins. The degree t which this affects the emissin spectrum has nt been extensively explred, but wuld be an bvius extensin f the present wrk. We d nt expect, hwever, a shift with temperature in the emissin peak as large as that seen in absrptin because this mde is prbably mre nearly harmnic in the excited state. This wuld likely result frm changes in the equilibrium gemetry and a reductin f steric interactins. There is als sme questin as t whether Brmberg and Muszkat's treatment f this crdinate as a nrmal mde is reasnable. The symmetric phenyl twist seems t be apprximately a nrmal mde in the cis-stilbene hmlg 1,2 diphenylcyclbutene, \3,14 hwever, QCFF /PI calculatins 1,28 suggest that this mde in cis-stilbene is highly mixed with ther internal crdinates in a number f nrmal mdes. Ismerizatin and vibratinal cling Our calculatin invlving a highly anharmnic mde indicates that emissin is nt riginating frm an excited state gemetry clse t that at the grund state maximum r what is usually referred t as the twisted intermediate. If emissin des nt arise frm this species then an interesting questin arises. Frm transient absrptin experiments we knw that the delay between the disappearance f excited cis-stilbene and the appearance f grund state transstilbene is less than ca. 150 fs. 4 It des nt seem likely that cis-stilbene culd twist frm a gemetry which has an ethylenic trsin angle f apprximately 44 t ne with a 90 twist and then internally cnvert t the grund state manifld in such a shrt time. One pssibility is that the scalled phantm state has a gemetry quite different frm the (rigid) 90 trsin which is usually imagined. A mdel fr the excited state nuclear rearrangements assciated with cis t trans ismerizatin which is cnsistent with ur results and previus studies will be presented. The nuclear mtins which fllw excitatin f cis-stilbene can be described rughly as a tw step prcess. In the first step, a new gemetry is established which results primarily frm ut f plane displacements f the ethylenic carbns and hydrgens. Althugh btaining a relaxed gemetry als invlves twisting f the phenyl grups tward a mre planar gemetry, this mtin may ccur n a much slwer time scale than that f the ethylenic hydrgen and carbn mtin r even that f the excited state lifetime. Since there is little r n barrier fr cis t trans ismerizatin, this new gemetry can best be cnsidered a quasiequilibrium state. The imprtant pint is that "paddle wheel" mtin f the phenyl grups arund the ethylene bnd, is nt required t attain a substantial increase in the ethylene trsinal angle. If we define a plane, in a space fixed crdinate system, by the grund state axis f the ethylene bnd and the C 2 symmetry axis, then in the paddle wheel mtin the ethylene bnd remains in this plane, and the phenyl grups sweep ut a large vlume as the ethylenic trsinal angle increases. In the picture we are prpsing there is very little phenyl grup mtin required fr the initial increase f the ethylenic trsin t its equilibrium excited state value (prbably arund 45 ). This picture is very similar t that discussed by Myers et al. 1 in the cntext f the early time dynamics predicted frm resnance Raman intensities and a mre recent prpsal by Sensin et af. 46 In the latter wrk they suggest that the initial gemetry changes result in part frm a tendency f the ethylenic carbns tward Sp3 hybridizatin in the first excited singlet state. 46 The secnd step in ur mdel invlves mtin tward the gemetry which allws rapid internal cnversin t the grund state. Since this prcess must be extremely fast,4 it als can nt invlve significant paddle wheel mtin. Instead, we prpse that this prcess invlves a significant increase in the ut f plane displacements f the ethylenic carbns and hydrgens-at which pint all duble bnd character f the "ethylenic" carbns will be destryed. This mtin will inevitably require sme displacement f the phenyl grups, sme f which is clearly required in rder t change frm cis- t trans-stilbene. Hwever, we suggest that the mtin is a slicing mtin thrugh the slvent alng the directin f increasing the in-plane bend angle. After the establishment f a gemetry similar t that at the grund state maximum, relaxatin tward cis- r trans-stilbene is rughly equally likely in accrdance with the expected branching rati fr this prcess. The mtins just described wuld be very rapid and weakly cupled t the slvent since large displacements f slvent are never required. The suggestin that vibratinal cling in cis-stilbene ccurs n a lnger time scale than its excited state lifetime in a lw viscsity slutin is quite reasnable. As discussed J. Chem. Phys., Vl. 97, N. 12, 15 December 1992

11 8924 Tdd, Fleming, and Jean: Calculatins f spectra by Elsaesser and Kaiser,12 vibratinal relaxatin n the picsecnd time scale is quite cmmn fr medium size plyatmics. Studies f anthracene and naphthalene in the grund electrnic state and in slutin revealed time scales fr vibratinal relaxatin tward a Bltzmann distributin f 7 t 10 ps.12 Recent studies f the grund state relaxatin f vibratinally ht trans-stilbene frm antistkes Raman intensities 47 indicate decay times n the rder f 10 t 20 ps. This is cnsistent with earlier experiments in which grund state trans-stilbene is prepared by ptically exciting cis-stilbene. 48 This system initially cntains a large excess f vibratinal energy and was bserved t cln a 14 ps time scale. 48 These studies strngly supprt the idea that the rm temperature emissin spectrum f cisstilbene in lw viscsity slvents arises frm a vibratinally unrelaxed state. Whether the excess vibratinal energy is in mdes that d, r d nt participate directly in ismerizatin is nt clear. In ur calculatins we made the simplistic assumptin that nly ne mde was vibratinally unrelaxed. The actual distributin f energy between varius mdes is strngly dependent n the time scale f IVR and the strength f the cupling f each mde t the slvent bath. We als treat the varius unrelaxed mdes quantum mechanically-in terms f the ppulatin distributin f their eigenstates. Unrelaxed emissin culd als result frm a mde (r set f mdes) which is s strngly cupled t the slvent that its mtin is verdamped and best treated in terms f classical diffusin. Unfrtunately, ur frmalism des nt allw us t treat such a situatin quantitatively. We expect that the inclusin f a slwly relaxing verdamped mde wuld result in changes in the emissin spectrum similar t thse calculated when we included a vibratinal mde with a ppulatin distributin determined by the grund state Franck-Cndn factrs, i.e., a brader spectrum, shifted t higher energies. A vibratinal mde which may be verdamped and which als has a large displacement between the grund and excited state is the symmetric phenyl twist. The 165 cm -I mde cntains a substantial amunt f phenyl twist character accrding t the QCFF/PI calculatins. 32 Using A=4.196 fr this mde, and Eq. (19), we btain a displacement f 14.5" fr the phenyl twist mde between the grund and the excited state. This can be cmpared with the displacement determined directly frm QCFF /PI calculatins (withut spectrscpic input) f Whether this mde shuld best be described as a nnthermal quantum scillatr r an verdamped classical scillatr in the cntext f the present wrk has nt been established. The timescale f a similar phenyl twisting mtin is believed t cntrl the excited state lifetime f triphenyl-methane (TPM) dyes. 49 The lifetime f the TPM dye crystal vilet varies frm 2.2 ps in methanl t 32 ps in decanl 49 suggesting that the phenyl twist mtin in cis-stilbene may be verdamped. We als nte that if vibratinal cling is ccurring n the same time scale as the ppulatin decays we wuld predict a slightly faster decay time f the flurescence at the edges f the spectrum, particularly the blue edge (shrter wavelengths). Yet there have been n cnclusive bservatins f spectral evlutin in the transient flurescence r absrptin experiments. If, hwever, the vibratinal cling is ccurring n a time scale lnger than the lifetime f excited cis-stilbene, then there wuld be nly minr changes in the lifetime acrss the transient absrptin r emissin spectrum. The implicatin f these ideas and previus results 7 is that the validity f a ne-dimensinal stchastic mdel f ismerizatin must be carefully examined. Given the mixed nature f the nrmal mdes in terms f the ethylenic trsin, phenyl twist and hydrgen wag mtins it is reasnable t expect that multidimensinality may be imprtant in the descriptin f the cis t trans ismerizatin. The develpment f multidimensinal theries f unimlecular rearrangement SO is relatively recent and cnsiders mdes which can be described in the classical limit. One predictin f these theries is that if there is an anistrpy in the effective frictin experienced by the varius reactive degrees f freedm, the reactin crdinate may vary between media f different viscsity. 50 The presence f nnthermal ppulatin distributins in reactive mdes, r mdes which d nt participate in the ismerizatin(s), may als cmplicate the descriptin. Finally, we shuld cmment n the limitatins f ur spectral calculatins. There are a few bvius apprximatins in the ptential surfaces used in ur mdelling. We assumed there were n frequency changes and n Duschinsky rtatin in the excited state fr the calculated spectra nt invlving an anharmnic mde. These assumptins were nt impsed by the theretical methd, but were necessitated by the limited experimental infrmatin available n the excited state vibratins. Althugh inclusin f these refinements shuld imprve the agreement f ur calculatins with experiment, we d nt expect their missin t affect the general nature f ur cnclusins. We als assume that QCFF /PI calculatins f the grund state f cis-stilbene give a reasnable representatin f the nrmal mdes f the excited state in terms f the internal degrees f freedm. This apprximatin shuld nt qualitatively alter ur cnclusins abut the excited state gemetry. We als assume that the transitin diple mment des nt depend n the vibratinal crdinate. At sme pint alng the mtin frm the relaxed gemetry t a mre twisted state, hwever, there is very likely a change in the electrnic nature f the excited species such that the radiative rate decreases markedly. In the cntext f ur mdel fr the ismerizatin dynamics, this wuld be the pint at which the initially excited mlecules leave the windw f bservatin f transient flurescence measurements. CONCLUDING REMARKS We have described calculatins which shw that, using experimental values fr the displacements f 12 Franck Cndn active vibratinal mdes, it is pssible t reprduce bth the absrptin spectrum and the Stkes shift f the flurescence spectrum f Cis-stilbene. These results suggest the existence f an excited state having a displacement This article is cpyrighted as indicated in the article. Reuse J. Chem. f AIP Phys., cntent Vl. is subject 97, N. t 12, the 15 terms December at: Dwnladed t IP:

12 Tdd, Fleming, and Jean: Calculatins f spectra 8925 in the ethylenic trsin substantially less than 90. Calculated spectra with an unrelaxed mde strngly suggest that the rm temperature flurescence spectrum f Saltie1 and c-wrkers s cntains cntributins frm ne r mre vibratinally unrelaxed mdes, and therefre that vibratinal relaxatin f at least certain degrees f freedm is slw cmpared t the ismerizatin time scale. The spectrscpic parameters enable us t suggest an apprximate gemetry fr the relaxed excited state and prpse a mdel fr the cis t trans ismerizatin prcess. This mdel implies a relatively weak dependence f the ismerizatin rate n the slvent viscsity, when viscsity is changed by changing slvent, since it invlves mtin f smaller mlecular grups. This is cnsistent with the weak viscsity dependence bserved in a series f alkane slvents. 3,4 A strng crrelatin f the ismerizatin rate with viscsity is bserved when viscsity is changed in a single slvent by changing pressure: s The cnnectin between these tw pints is discussed in detail in Ref. 19. If we fit the bserved decay times 3,4,19 fr cis-stilbene in n-alkanes t a frm k=alllq+khp, and assume the reactin rate fr DHP frmatin is equal t 2.7X loll S-I (Refs. 18, 51) and independent f alkane slvent, we determine a value f a =0.25. The value f a determined fr trans-stilbene in n-alkanes is 0.32, I and values as large as 0.39 have been suggested. 52 This suggests that the initial mtin fr trans t cis ismerizatin may displace mre vlume and interact mre strngly with the slvent envirnment than the reactive mtin fr cis t trans ismerizatin. Althugh ne might naively think these mtins shuld be similar, the initial relaxatin and the reactive mtin frm the cisstilbene side is prbably strngly influenced by the large steric interactins which are nt present in trans-stilbene. ACKNOWLEDGMENTS We thank Anne Myers (Rchester) fr supplying us with a cpy f the QCFF IPI calculatin fr cis-stilbene and t Jack Saltiel (Flrida State University) fr the cisstilbene emissin spectrum and t bth fr preprints fr their wrk and many valuable cnversatins. This wrk was supprted by a grant frm the Natinal Science Fundatin. ID. H. Waldeck, Chern. Rev. 91, 415 (1991). 2G. R. Fleming, S. H. Curtney, and M. W. Balk, J. Stat. Phys. 42, 83 (1986); M. Lee, G. R. Hltrn, and R. M. Hchstrasser, Chern. Phys. Lett. 118, 359 (1985); J. Schreder, D. Schwarzer, J. Tre, and F. VP, J. Chern. Phys. 93, 2393 (1990). 3D. c. Tdd, J. M. Jean, S. J. Rsenthal, A. J. Ruggier, D. Yang, and G. R. Fleming, J. Chern. Phys. 93, 8658 (1990). S. Abrash, S. Repinec, and R. M. 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Brezin, Nuv Cirnent 64,37 (1969). 26The dimensinless crdinate, q, alng a nrmal mde with frequency /lj is related t the Cartesian crdinate, x, by q= (j1./ljifi) 112X. j1. is either the reduced mass r reduced mment f inertia, depending n the units f x. 27 Handbk f Bichemistry and Mlecular Bilgy, 3rd ed., edited by G. D. Fasman (CRC, Cleveland, 1976). 28C. A. Parker, Phtluminescence f Slutins (Elsevier, Amsterdam, 1968). 29p. Petelenz and B. Petelenz, J. Chern. Phys. 62, 3482 (1975). 30 A. Warshel, J. Chern. Phys. 62, 214 (1975). 31 A. Warshel and M. Karplus, J. Am. Chern. Sc. 96, 5677 (1974). 32 A. B. Myers (private cmmunicatin). 33 Absrptin Spectra in the Ultravilet and Visible Regin, edited by L. Lang (Academic, New Yrk, 1967); Vl. IX. 34G. Hhlneicher, M. Muller, M. Demmer, J. Lex, J. H. Penn, L.-X. Gan, and P. D. Lese!, J. Am. Chern. Sc. 110, 4483 (1988). 35 J. Saltiel (private cmmunicatin). 36J. Saltiel, S. Ganapathy, and C. Werking, J. Phys. Chern. 91, 2755 (1987). 37 G. B. 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Phys. 93,9185 (1990). 49D. Ben-Amtz and C. B. Harris, J. Chern. Phys. 86, 4856 (1987). SON. Agmn and S. Rabinvich, J. Chern. Phys. (submitted fr publicatin); A. M. Berezhkvskii and V. Y. Zitserman, J. Chern. Phys. 95, 1424 (1991); M. M. Klsek, B. M. Hffman, B. J. Matkwsky, A. Nitzan, M. A. Ratner, and Z. Schuss, J. Chern. Phys. 95, 1425 (1991), and reference therein. si2.7xio" is the average f the values determined (Ref. 18) in n-pentane, n-hexane, n-ctane, and n-nnane at rm temperature and atmspheric pressure. 52J. Saltiel and Y.-P. Sun, J. Phys. Chern. 93, 6246 (1989). J. Chern. Phys., Vl. 97, N. 12, 15 December 1992