Nonequilibrium phooinduced elecron ransfer Minhaeng Cho and Rober J. Silbey Ciaion: J. Chem. Phys. 13, 595 (1995); doi: 1.163/1.4794 View online: hp://dx.doi.org/1.163/1.4794 View able of Conens: hp://jcp.aip.org/resource/1/jcpsa6/v13/i2 Published by he American Insiue of Physics. Addiional informaion on J. Chem. Phys. Journal Homepage: hp://jcp.aip.org/ Journal Informaion: hp://jcp.aip.org/abou/abou_he_journal op downloads: hp://jcp.aip.org/feaures/mos_downloaded Informaion for Auhors: hp://jcp.aip.org/auhors Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions
Nonequilibrium phooinduced elecron ransfer Minhaeng Cho and Rober J. Silbey Deparmen of Chemisry, Massachuses Insiue of echnology, Cambridge, Massachuses 2139 Received 15 November 1994; acceped 3 April 1995 We consider phooinduced elecron ransfer, which is inrinsically a hree-sae sysem consising of elecronic ground, elecronic excied elecron donor, and elecron accepor saes. I is assumed ha he bah consiss of a collecion of harmonic oscillaors. Using an elemenary ime-dependen perurbaion heory, i is found ha he nonequilibrium Golden rule formula proposed by Coalson e al. J. Chem. Phys. 11, 436 1994 can be rigorously obained in a cerain limi of our resuls. Invoking a saionary phase approximaion, a simple resul analogous o he Marcus expression is obained, excep for he presence of ime-dependen reorganizaion energy. he mulidimensional naure of he solvaion coordinae sysem is discussed furher. Finally a few numerical calculaions are presened. 1995 American Insiue of Physics. I. INRODUCION Ulrafas laser echniques 1 3 developed in he las wo decades enable direc measuremen of populaion changes of eiher he elecron donor or accepor saes 4 9 in elecron ransfer reacions in condensed media. By using a subpicosecond laser pulse, one can creae a populaion sae on he elecron donor sae, which is in his case coupled o he elecronic ground sae radiaively. here are wo popular ways o deec he populaion changes in ime. One of he wo is o probe he ransien absorpion inensiy of he donor sae radiaively coupled o anoher elecronic sae. 1 his ransien absorpion experimen is useful when here is an elecronic sae accessible by he opical field. he oher mehod is o measure he simulaed or sponaneous emission inensiy from he donor sae. In conras o he sponaneous emission measuremen, he simulaed fluorescence measuremen uilizes an addiional pulse o simulae he emission of phoon from he accepor sae. 11 hese wo mehods, ransien absorpion and ligh emission measuremens, are basically relaed o he general pump probe-ype experimen. he former differs from he laer by he probing mehod. In some cases, inerpreaions of hese resuls are difficul because one has o have a full knowledge of he poenial surfaces and inramolecular dynamics of he arge molecular sysem. Mos of he heoreical sudies on elecron ransfer in condensed media are based on he assumpion ha he iniial sae is a hermal equilibrium sae, which is saionary, on he donor surface. In his convenional siuaion a single solvaion coordinae represening he flucuaing bah degrees of freedom by projecing heir flucuaions ono a onedimensional coordinae is chosen. his solvaion coordinae is collecive in naure since i represens he mulidimensional poenial energy surfaces consruced by he bah degrees of freedom. However one of he remarkable oucomes of his reducion procedure is ha one can use an approximae picure for he ime evoluion of he solvaion coordinae, such as generalized Langevin equaion or equivalenly generalized Fokker Planck equaion. 12 17 Furhermore, as Marcus showed a long ime ago, he elecron ransfer rae can be fully described by a single quaniy, he classical solven reorganizaion energy, in he classical high emperaure limi. 18 he solven reorganizaion energy represens he magniude of he overall coupling srengh of he bah degrees of freedom wih he elecron ransfer pair hus we shall use hese quaniies as measures of coupling srengh in his paper. In conras o his convenional siuaion, phooinduced elecron ransfer involves an addiional ground sae opically coupled o he donor sae. Unless we ignore he iniial relaxaion process of he opically creaed wave packe on he donor surface, we canno selec a single solvaion coordinae in his hree-sae sysem coupled o mulidimensional bah degrees of freedom. he opical exciaion sep is governed by anoher solvaion coordinae whose flucuaion induces broadening of specra as well as relaxaion of he nonequilibrium wave packe on he donor surface. One of he complexiies is ha he wo coordinaes, one associaed wih he opical ransiion and one wih he elecron ransfer, are no necessarily correlaed wih each oher. his acually induces a grea deal of difficulies since we have o deal wih a ruly mulidimensional solvaion coordinae sysem in his case. As one can expec, only for imes longer han he relaxaion ime on he donor surface, will he elecron ransfer rae reach is equilibrium value. In his paper we will explore his nonequilibrium naure of he phooinduced elecron ransfer in he nonadiabaic regime. Recenly Coalson e al. 19 considered a similar problem. Insead of direcly considering he opical process in heir formulaion, hey proposed a nonequilibrium Fermi Golden rule formula for he case when he iniial preparaion of he donor populaion is in he nonequilibrium sae on he donor surface. hey replaced he iniial saionary sae wih nonequilibrium ime-dependen sae in he usual Golden rule formula. In order o es he nonequilibrium Golden rule formula, hey carried ou compuer simulaion sudies wih a spin-boson Hamilonian used by Garg e al. 16 We presen in his paper a rigorous derivaion of he nonequilibrium rae kernel for he nonequilibrium elecron ransfer process when he preparaion of he donor populaion is performed by an ulrafas opical pulse. In some limiing case we show our resuls reduce o heir so-called nonequilibrium Golden rule formula. We furher show ha a simplified analyic expres- J. Chem. Phys. 13 (2), 8 July 1995 21-966/95/13(2)/595/12/$6. 1995 American Insiue of Physics Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions 595
596 M. Cho and R. J. Silbey: Phooinduced elecron ransfer sion for he E rae kernel can be obained by using he saionary phase approximaion. I urns ou ha he dimensionaliy of he solvaion coordinae sysem plays a crucial role in undersanding he nonequilibrium naure of he phooinduced E. We presen some simple numerical evaluaions of our formal resuls o help undersanding he resulan equaions. We organize his paper as follows. In Sec. II, we skech he enire picure of he nonequilibrium E process qualiaively. We nex formulae he ime-dependen nonequilibrium E rae kernel by using elemenary ime-dependen perurbaion heory. he nonequilibrium generalizaion of he Marcus expression for E rae consan is obained by invoking he saionary phase approximaion. In Sec. IV, we presen numerical resuls for a few cases. We finally summarize our resuls in Sec. V. II. QUALIAIVE PICURE A hree-sae sysem consising of an elecronic ground sae, an elecronic excied sae which acs as an elecron donor sae, and an elecron accepor sae is considered. g, D, and A are he ground, donor, and accepor saes, respecively. I is assumed ha he ground sae g is radiaively coupled o he donor sae D which is he elecronic excied sae. Furhermore, he donor sae D is coupled o he accepor sae A by a nonzero elecron exchange marix elemen,. Before we presen a heoreical descripion of he nonequilibrium elecron ransfer in condensed media, we will briefly skech he enire picure. I is assumed ha he sysem is iniially in he ground sae in hermal equilibrium wih he bah. herefore, he iniial sae is saionary and can be defined by he saisical disribuion in he phase space. he bah is modeled by a collecion of he harmonic oscillaors, which are coupled o each sae linearly. he poenial energy surface of he elecronically excied donor sae is likely o be displaced from he poenial energy surface of he ground sae. Oherwise one may no expec any broadening of specra induced by he bah degrees of freedom, since he Franck Condon overlaps of harmonic modes wih small displacemens are small. A shor laser pulse o creae a populaion on he donor poenial energy surface is inroduced. I is assumed ha he pulse duraion ime is shor enough o ignore boh elecron ransfer process from D o A and propagaion of he nuclear wave packe on he donor surface during he pulse duraion ime. his condiion can be me by using a femosecond laser pulse when he ime scales of he elecron ransfer and solven modes are order of subpicosecond o picosecond. If he solvaion ime scale is comparable o he pulse duraion ime, one mus relax hese assumpions. In ha case, we may have o consider he propagaion effec of he nonequilibrium wave packe on he donor surface during he pulse duraion ime. We shall consider his case in anoher paper. he creaed wave packe on he donor poenial energy surface ends oward a new hermal equilibrium sae since he equilibrium posiions of he nuclear degrees of freedom on he donor poenial surface are differen from hose on he ground poenial surface. In he mean ime, he wave packe on he D sae keeps leaking ino he accepor sae by he nonzero FIG. 1. wo cases of poenial surfaces in one-dimensional solvaion coordinae sysem. a and b correspond o and, respecively a deailed discussion on he dimensionaliy parameer is given in Sec. III F. 1. Iniially he saionary nuclear wave packe is in a hermal equilibrium sae on he ground poenial surface. 2. A nonequilibrium wave packe is creaed on he donor surface by a resonan opical pulse. 3. Relaxaion of he nonequilibrium wave packe complees for imes longer han he relaxaion ime. elecron exchange marix elemen. he mos effecive channel in his elecron ransfer process is ha he wave packe on D sae reaches he curve crossing poin beween he poenial surfaces of he donor and accepor saes. his is because in his region of he phase space, he wo saes are isoenergeic and he Franck Condon overlaps are maxima. We will refer o his curve crossing region beween he donor and accepor surfaces as he exi channel. In he case of an underdamped wave packe, which means ha he mean posiion of he wave packe undergoes an oscillaing moion on he harmonic poenial surface of he D sae, one may expec ha he donor sae wave packe ges close o he curve crossing poin periodically. his is he case when he elecron ransfer process is srongly coupled o a few underdamped vibraional modes. 2 On he oher hand, if he elecron ransfer sysem is coupled o a large number of degrees of freedom, even hough each of hem could be underdamped, he superposiion of hese oscillaing feaures makes he ime evoluion of he average posiion of he wave packe overdamped. In any case, regardless of he posiion of he donor sae wave packe, here is nonzero probabiliy beween he donor wave packe and he exi channel. Here he probabiliy is ime-dependen because of he nonsaionariy of he iniial wave packe. Furhermore, is magniude is dependen on he poenial energy surfaces as well as emperaure. Obviously he elecron ransfer rae is proporional o he magniude of he overlap beween he nuclear wave packe and he exi channel in he phase space. Consider he ime immediaely afer a wave packe is creaed on he donor sae. For he sake of simpliciy le us consider a one-dimensional coordinae sysem, where one solvaion coordinae can describe boh opical broadening, relaxaion of he nonequilibrium wave packe, and elecron ransfer. If he cener of he nonequilibrium wave packe is locaed far away from he exi channel see Fig. 1a, we expec o see a small elecron ransfer rae, and vice versa. In his siuaion, he ime-dependen elecron ransfer rae increases in ime unil he wave packe reaches is hermal equilibrium on he donor poenial surface. On he oher J. Chem. Phys., Vol. 13, No. 2, 8 July 1995 Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions
M. Cho and R. J. Silbey: Phooinduced elecron ransfer 597 ha afer he relaxaion ime, he elecron ransfer rae should no depend on ime because we are looking a a wo-level elecron ransfer process where he iniial sae is in a hermal equilibrium sae on he donor sae. In Sec. III, we shall formulae his picure o serve as a reasonable model for he realisic elecron ransfer induced by he opical exciaion. FIG. 2. Model conour plos of hree wo-dimensional poenial surfaces. he q 1 coordinae is chosen o represen he collecive nuclear degrees of freedom associaed wih he elecron ransfer beween D and A noe ha i is parallel wih he line connecing wo minima of D and A, which is shown by a dashed line. he iniial nonequilibrium wave packe shown by hicker ellipsoids is creaed on he donor surface a he origin of (q 1,q 2 ) coordinae sysem and relaxes oward he minimum of he donor surface as shown by a hick dashed arrow. hand, if he iniial wave packe is creaed close o he exi channel see Fig. 1b, closer han he minimum of he donor poenial surface, we expec he elecron ransfer rae decreases in ime unil he wave packe reaches is hermal equilibrium sae. As an example of a wo-dimensional case, in Fig. 2, we draw conour plos of hree harmonic wells associaed wih hree saes. he creaed nonequilibrium wave packe is shown by hicker ellipsoids. his wodimensional wave packe relaxes oward he minimum of he donor surface as drawn by he dashed arrow. During his relaxaion, here is nonzero leakage of he donor populaion via elecron ransfer mechanism. We can, herefore, expec III. FORMULAION We firs consider he oal Hamilonian of he composie sysem, HH VJ, 1 where H gh g gdh D DAh A A, VEcos Dg*E*cos gd, 2 JAD*DA. Here, for example, h g represens he nuclear Hamionian of he ground sae. he ground sae g is radiaively coupled o he donor sae D by he coupling poenial V(). he cenral frequency of he opical field is, and he ime profile is deermined by E(). is he dipole marix elemen, which could be dependen on coordinaes of he nuclear degrees of freedom. We will keep his coordinae dependence of he dipole marix elemen unil he las sage of our derivaion. he donor and accepor saes are coupled by J, where he coupling srengh is deermined by. We assume ha he bah consiss of harmonic oscillaors coupled o each level linearly, H Dg Ag 1 2 p 2 2 2 x p 2 2 x d / 2 2 2. 3 p 2 2 x a / 2 he energy of he isolaed ground sae is assumed o be zero. Dg and Ag are he energy gaps beween he donor and ground saes and he accepor and ground saes, respecively. When we consider he harmonic oscillaors on he ground sae as a reference, he harmonic modes coupled o he donor and accepor saes are displaced by d and a, respecively. A. Definiions of specral densiies he nuclear Hamilonians, Eq. 3, conains linear coupling erms represening energy flucuaions induced by bah degrees of freedom. Since we only consider linear erms wih respec o he bah harmonic coordinaes, our model Hamilonian includes neiher any phonon-induced exciaion ransfer effecs nor molecular vibraional relaxaion. However, we believe i is sill useful enough o undersand he role of he bah degrees of freedom in he elecron ransfer process when he iniial wave packe creaed by he opical field is in a nonequilibrium siuaion. he magniude of he coupling srengh of bah degrees of freedom is deermined by he displacemens of harmonic modes, for example, d /(2 2 ). hese coupling srenghs are in urn relaed o he ime scale of he relaxaion rae as well as energeics of poenial energy surfaces. We find ha i is useful o define specral densiies represening he coupling srenghs of harmonic modes as Dg d 2 Ag a 2 2 3, 2 3, 3 2. he firs specral densiy, Dg (), is fully responsible o he broadening effec of he opical specra. he las one, AD a d 2 4 J. Chem. Phys., Vol. 13, No. 2, 8 July 1995 Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions
598 M. Cho and R. J. Silbey: Phooinduced elecron ransfer AD (), is associaed wih he equilibrium E process. Because we are considering a nonequilibrium siuaion, i is necessary o include all hree specral densiies as we will show in he following secions. Noe ha in general AD () Ag () Dg (). his unequaliy gives us a hin ha mulidimensionaliy will play a role in he E of he hree-sae sysem. Our goal is evenually o describe he nonequilibrium E rae in erms of he hree specral densiies. B. Perurbaional approach o evaluaion of imedependen populaions In order o obain perurbaion resuls on he nonequilibrium E rae, we shall consider he case of nonadiabaic limi and elemenary ime-dependen perurbaion heory will be used. hroughou his paper, we will reain only erms proporional o 2 and all he higher-order erms wih respec o he elecronic dipole ineracion will be ignored. o calculae he ransiion ampliudes we nex consider he ime-evoluion operaor U, exp i d H e ih exp i dṽj e ih u,, where he Heisenberg operaors Ṽ() and J () in he ineracion picure wih respec o he zeroh-order Hamilonian are Ṽe ih g e ih D Ecos Dg 5 e ih D * e ih g E*cos gd, 6 J e ih A e ih D AD e ih D * e ih A DA. Here u(, ) in Eq. 5 is obviously defined as above. We can expand his evoluion operaor in he ineracion picure as u, exp i dṽj 1i dṽj d dṽṽṽj J ṼJ J. 7 By using he ime-evoluion operaor we can now calculae he ransiion probabiliy in ime. Firs consider he populaion a he donor sae, P D gexp i d H DD exp i d H g gu, DDu, g, where denoes he iniial nuclear wave funcion on he ground sae in he hermal equilibrium. hus he marix elemen of shown above is idenical o race over he hermal bah degrees of freedom. Insering Eq. 7 ino Eq. 8 we find he ime-dependen populaion a he donor sae is, perurbaively, given by P D 1 2 Re d d E*E e ih g * e ih D e ih g e i P D 2 2,O 2 4. Here we invoked he roaing wave approximaion, which assumes ha erms oscillaing wih frequencies of ( Dg ) are ignored because inegrals over hose highly oscillaing funcion is negligibly small. he lowes order conribuion o he populaion of he donor sae is obviously induced by he opical exciaion. he nex higher order erm one should consider is proporional o 2 2, and is magniude is exacly idenical o he lowes order erm for he populaion of he accepor P A () wih opposie sign see Eq. 13. his is because he whole populaion is conserved. Because he accepor is no radiaively coupled o he ground sae, he lowes order erm conribuing o P A () is also proporional o 2 2. Here we should menion ha we have no considered sponaneous loss of donor populaion via radiaive or nonradiaive channels excep for he elecron ransfer process. In oher words, we assume ha he lifeime of he donor sae induced by oher channel is sufficienly long compared o ha induced by he elecron ransfer. Changing he inegraion variable in Eq. 9 o 1, we can rewrie he populaion of he donor sae as P D 1 2 Re d d1 E*E 1 e ih g 1* e ih D 1 e i 1 P D 2 2,O 2 4, 8 9 1 where. If we furher assume ha he pulse duraion ime is sufficienly shor enough o ignore any nuclear dynamics, i.e., we assume ha E()E () where E is a produc of he pulse ampliude and is duraion, hen he populaion of he donor sae is simply given by P D 1 2E 2 2 P D 2 2,O 2 4, 11 where () is a sep funcion. J. Chem. Phys., Vol. 13, No. 2, 8 July 1995 Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions
M. Cho and R. J. Silbey: Phooinduced elecron ransfer 599 Nex consider he populaion of he accepor sae by calculaing he ransiion ampliude, P A gexp i d H A Aexp i d H g. 12 Likewise, insering he ime-evoluion operaor ino he above equaion, he lowes order erm is given by where P A 2 Re d d d d E*E cos cos F,,,O 2 4, 13 F,,, e ih g * e ih D e ih A e ih D e ih g. I is useful o rewrie Eq. 13 by using he following ideniies: d d d d, and d d d d, d d d d. hen Eq. 13 can be wrien as P A 2 Re d d d d E*Ecos cos F,,, 2 Re d d d d E*Ecos cos F,,, 2 Re d d d d E*Ecos cos F,,,O 2 4. 14 In comparison o Eq. 13, here are ime orderings of inegraion variables in Eq. 14. For example, he firs and second erms conain wo consecuive ineracions wih he exernal field o creae a diagonal densiy marix elemen on he donor sae, and he remaining second-order perurbaions by he elecron exchange marix elemens creae populaion on he accepor sae. On he oher hand, he hird erm includes a differen ime ordering. he acions of exernal field perurbaion and elecron exchange perurbaion are alered. herefore, as long as he pulse duraion ime is sufficienly shor compared o he ime scale of he elecron ransfer rae, we can safely ignore he conribuion from he hird erm in Eq. (14). I is worh menioning ha he laer conribuion has a complee analog in he nonlinear four-wave mixing specroscopies known as he coheren arifac. 21 his phenomenon is usually induced when he wo laser pulses overlap in ime so ha here are mixed ime ordering of he field sysem ineracions in he four-wave mixing specroscopy. 22 In erms of four-wave mixing specroscopic language, we use an exernal laser pulse o pump he populaion of he ground sae up o he donor sae, and hen probe he populaion of he donor sae by he second-order ineracion wih he elecron exchange perurbaion. his is herefore more closely relaed o he sponaneous fluorescence measuremen where he probing sep involves acions of he vacuum field operaor which are no conrolled by experimenalis. Likewise, one has no conrol on he acion of elecron exchange perurbaion in our E problem eiher. We now change he inegraion variables in Eq. 14 as 1 and 2 in he firs erm and 1 and 2 in he second erm, respecively, and also le. Equaion 14 can be rewrien as P A 1 2 Re d d2 2d d1 E* 1 Ee i 1F, 1, 2, 1 2 Re d d2 2d d1 E*E 1 e i 1F,, 2, 1 O 2 4. 15 J. Chem. Phys., Vol. 13, No. 2, 8 July 1995 Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions
6 M. Cho and R. J. Silbey: Phooinduced elecron ransfer I should be noed ha he wo ime periods, 1 and 2, are associaed wih he ime evoluions of he off-diagonal densiy marix elemens which are likely o be highly oscillaing wih frequencies deermined by he energy differences beween saes. herefore i is naural o evaluae he inegrals over hese highly oscillaing periods by using he saionary phase approximaion. As menioned before, i is assumed ha he ime profile of he exernal field is effecively a dela funcion, E()E (). Wih his approximaion, we find P A E 2 Re d d2 * e ih D e ih A 2 e ih D 2e ih D O 2 4. 16 As a resul of he ulrafas opical pulse, he same wave packe,, is creaed on he donor sae a ime zero. hen i propagaes on he donor surface for ime. Defining he nonequilibrium wave packes as () e ih D,we find he generalized Fermi Golden rule expression including he nonequilibrium effec. his problem has recenly been discussed by Coalson e al., 19 based on he approximaions of i ignoring nuclear dynamics during he pulse duraion ime and ii ignoring he conribuion from he mixed imeordering erm ha is he hird in Eq. 14. Here we have presened a rigorous basis for he nonequilibrium phooinduced elecron ransfer reacion. C. Populaions and cumulan approximaion We nex calculae ime-dependen populaion of he donor sae. he donor populaion creaed by he opical exciaion was calculaed by considering he second-order erm wih respec o he dipole marix elemen see Eq. 11. As menioned before, he nex higher order erm conribuing o P D (), which is proporional o 2 2, is idenical o he populaion of he accepor sae o his order excep for he opposie sign, i.e., P D ( 2 2,)P A ( 2 2,). herefore he ime-dependen populaion of he donor sae is approximaely given by P D ()P D ( 2,)P A ( 2 2,) O( 2 4 ). Using hese resuls, Eqs. 11 and 16, wefind ha he populaion of he donor sae can be approximaely wrien by exponeniaing he expression o find P D 2E 1 2 exp 2 d k f, 17 where he forward rae kernel represening he ransiion rae per uni ime from donor o accepor is k f 2 Re 2 d2 e ih A 2 e ih D 2. 18 I is possible o derive his equaion more formally han we have done here, using he parial-ordering procedure POP 23 and runcaing a he second-order cumulan. his leads o he idenical expression for P D (). Here we assume ha he exohermiciy of he elecron ransfer from donor o accepor is large enough o ignore he backward ransiion rae. However as shown by he auhors recenly, 24 i is a sraighforward exercise o include he conribuion from he backward ransfer process. Furhermore, he exponeniaion approximaion we inroduced is exac when we consider a second-order cumulan approximaion o he soluion of he linear sochasic differenial equaion. 24 However we will no pursue his in his paper since here we only focus on he simple case of very large exohermic reacion o make he whole picure as simple as possible. We now invoke he classical Condon approximaion ha he dipole operaor and he elecron exchange operaor, and, respecively, do no depend on he nuclear coordinaes. he forward ransfer rae kernel, Eq. 18, is hen k f 2 Re 2 d2 e ih A 2e ih D 2, 19 where we define () e ih D. If we replace he upper bound of he inegraion over 2 wih, we recover Coalson e al. s resul, ermed he nonequilibrium Golden rule formula Eq. 2-12 in Ref. 19. We see ha our derivaion is useful in exending beyond he shor-pulse approximaion. Rewriing he nuclear Hamilonians, h D and h A,in he ineracion represenaion wih respec o h g we can rewrie he forward ransfer rae kernel as k f 2 Re 2 d2 exp i ds UDg s exp i ds U Ag 2 s exp i 2ds UDg s expi AD 2 i g AD 2, 2 where he zero-cenered difference poenial is defined, in he ineracion picure, U mn se ih g s h m h n h m h n e ih g s for m,ng,d,a. Here he angular bracke represens a hermal average over he nuclear degrees of freedom in he equilibrium ground sae. exp exp denoes posiive negaive ime-ordered exponenial. AD is he energy difference beween he isolaed accepor and donor saes, AD Ag Dg. Because of he sysem bah ineracion, from Eq. 3 he solvaion energies of he donor and accepor saes are d Dg () and d Ag (), respecively. In Eq. 2, he corresponding reorganizaion energy is given by g AD h A h D AD d Ag Dg. 21 I should be noed ha he reorganizaion energy difference in he solvaion energies of he donor and accepor saes, g AD, is evaluaed over he Hamilonian of he ground sae g insead of ha of he donor sae he superscrip g of AD means ha he hermal average is carried ou over he Hamilonian of he ground sae. We may expec ha he reorgani- J. Chem. Phys., Vol. 13, No. 2, 8 July 1995 Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions
M. Cho and R. J. Silbey: Phooinduced elecron ransfer 61 zaion energy approaches o ha evaluaed over he donor Hamilonian as he nonequilibrium populaion creaed on he donor sae by he opical exciaion reaches is equilibrium sae on he donor poenial surface. I now urns ou o be useful o rewrie various reorganizaion energies in erms of specral densiies defined in Eqs. 4, g Dg d Dg, g Ag d Ag, D AD d AD, and g AD g Ag g Dg. 22 Once again we emphasize ha here are wo differen reorganizaion energies, D AD and g AD, associaed wih he elecron ransfer pair. Of he wo, he former appears in he convenional elecron ransfer process since i is evaluaed over he equilibrium disribuion of he donor nuclear degrees of freedom. We nex evaluae he nonlinear correlaion funcion given in Eq. 2 by using he cumulan expansion mehod and runcaing higher order erms han he second. he forward ransfer rae kernel is hen k f 2 Re 2 d2 expi AD 2 i g AD 2 AD 2 where i Im Dg AD Ag expi Im Dg 2 AD 2 Ag 2, mn d dumn U mn l i mn d mn coh/2 1cos i d mn sin, 23 24 where lg when mndg or Ag, and ld when mnad. Noe ha we need o consider all hree specral densiies o compleely describe he forward ransfer rae kernel, whereas he equilibrium elecron ransfer rae is deermined by a single specral densiy, AD (). As can be seen in Eq. 23, he imaginary pars of he double inegraion of correlaion funcions changes he phase of he inegrand in Eq. 23. As he iniial wave packe on he donor sae propagaes oward an equilibrium posiion poenial minimum on he donor sae, he reorganizaion energy associaed wih he elecron ransfer changes in ime. D. Saionary phase approximaion (Laplace mehod) Alhough Eq. 23 can be easily calculaed numerically, we will invoke he saionary phase approximaion o he inegral given in Eq. 23. Since during he ime period of 2 in he inegrand of Eq. 23, he whole inegrand is highly oscillaing, i is suiable o ake shor-ime expansion of he exponen in he inegrand wih respec o 2 o find he saionary phase poin. 24 he inegral, Eq. 23, hen reduces o k f 2 Re 2 d2 exp 1 2 U 2 AD 2 2 i AD g AD Q 2, 25 where he ime-dependen reorganizaion energy Q() is defined in erms of hree specral densiies, Q d Ag AD Dg 1cos. he mean square flucuaion ampliude U 2 AD is given by 26 U 2 AD d AD coh/2 2. 27 U 2 AD is a characerisic quaniy represening he magniude of he bah flucuaion energy. As shown in he definiion, Eq. 27, U 2 AD is srongly dependen on emperaure. When he emperaure is much larger han any harmonic oscillaor energy, he mean square flucuaion ampliude is direcly relaed o he reorganizaion energy as U 2 AD 2k B D AD. his limi is usually referred o he classical or high emperaure limi in he lieraure. 12 When he mean square roo flucuaion ampliude of he coupled bah degrees of freedom U 2 AD is much larger han he ime scale of Q(), we can approximaely replace he inegraion limi wih insead of in Eq. 25. In his case, we find k f 2 U AD 2 exp AD g AD Q 2 2U AD 2. 28 his resul is excepionally simple. Noe ha Eq. 28 is precisely of he same form wih he Marcus expression for an equilibrium elecron ransfer rae excep ha he reorganizaion energy is replaced wih -dependen one, 26 so we shall refer his resul as nonequilibrium generalizaion of he Marcus E rae consan. We also emphasize ha Eq. 28 includes he complicaed mulidimensional naure of he solvaion coordinae sysem via ime-dependen funcion Q() which is in urn deermined by he hree specral densiies. hus we believe ha we have accomplished our goal of expressing he nonequilibrium E rae kernel in erms of specral densiies. If we relax he dela-funcion approximaion o an opical pulse envelope, we expec o see he ime-dependen change of he Gaussian widh in Eq. 28. his is analogous o he ime-dependen change of he ime-resolved fluorescence widh in liquid. We will discuss his case elsewhere. We nex invesigae some limiing cases of Eq. 28 o give some insighs and o check is consisency wih oher known resuls. Consider he case when is very small bu larger han he inverse of U 2 AD so ha Q() is negligibly small. In his case, he iniial wave packe involved in he elecron ransfer process does no have enough ime o sig- J. Chem. Phys., Vol. 13, No. 2, 8 July 1995 Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions
62 M. Cho and R. J. Silbey: Phooinduced elecron ransfer nificanly propagae on he donor surface, hence he corresponding reorganizaion energy should be equal o g AD his is he solvaion energy difference when he nuclear configuraion of he bah coordinaes is idenical o ha of he equilibrium disribuion on he poenial surface of he ground sae. As we allow he wave packe propagae on he donor poenial energy surface for ime, he reorganizaion energy changes in ime by g AD Q(). Finally for a long ime, Q() approaches o an asympoic value: lim Q g AD D AD. In his long ime limi, he relevan reorganizaion energy reduces o D AD, which is he solvaion energy difference evaluaed over he equilibrium disribuion of he bah degrees of freedom on he donor surface no he poenial surface of he ground sae. his can be undersood simply: For such a long ime, he wavepacke creaed on he donor surface reaches hermal equilibrium on he donor surface so ha here is a consan rae of leaking of he donor populaion ino he accepor surface by he second-order elecron ransfer mechanism. hus we have shown ha he generalized resul, Eq. 28, reaches he known equilibrium value Marcus expression afer he relaxaion ime on he donor surface. Finally i should be noed ha as emperaure decreases he saionary phase approximaion used in his secion becomes invalid, since he shor-ime approximaion by a Gaussian see Eq. 25 is no reliable. In case of he low emperaure regime, we should use Eq. 23 o calculae he nonequilibrium E rae kernel insead of Eq. 28, even hough Eq. 23 conains an undesirable addiional inegraion. E. Furher simplificaion: wo orhogonal solvaion coordinaes Our resuls, Eqs. 17 wih 28, are valid regardless of he shapes and magniudes of he hree specral densiies. In oher words, hose resuls can be applied o mulidimensional siuaions by insering appropriae forms for he hree specral densiies. We canno, in general, obain wo orhogonal solvaion coordinaes o fully describe he hree-sae sysem, since here is no simple way o include he complicaed cross correlaion effec. However i will sill be useful o exrac wo orhogonal coordinaes o simplify he whole picure. Paricularly, when we assume ha he funcional forms of he hree specral densiies are he same, we may find wo orhogonal solvaion coordinaes represening mulidimensional bah flucuaions. his is because he poenial energy surfaces associaed wih he hree saes can be described by wo-dimensional harmonic wells wih same curvaures. We explain he procedure of finding wo orhogonal coordinaes below. When we used ineracion represenaions of he donor and accepor nuclear Hamilonians in Eq. 2, we considered hree ime-dependen difference poenials, U Dg (),U AD (), and U Ag (), which are all collecive coordinaes. Among he hree variables, U AD () is direcly relaed o he elecron ransfer beween D and A, so we will choose U AD () as he primary coordinae, q 1. Now here are wo remaining variables, U Dg () and U Ag (), bu from he definiion U Ag ()U AD ()U Dg (), only one is independen. herefore, we can choose eiher U Dg () oru Ag () as he oher coordinae. We ake U Dg (), since his variable is direcly responsible for he specroscopic broadening effecs. Alhough here are wo represenaive coordinaes, U AD () and U Dg (), hese wo are no orhogonal o each oher. hus we use Schmid orhogonalizaion mehod o obain wo orhogonal coordinaes. We consider q 1 U AD, q 2 U Dg U AD. 29 Here we should deermine saisfying q 1 q 2. By insering U Ag ()U AD ()U Dg () and using he fac ha U AD U Dg 2(U 1 2 Ag U 2 AD U 2 Dg ), we find U AD 2 U 2 Dg U 2 Ag 2U 2. 3 AD his quaniy is fully deermined by he mean square flucuaion ampliudes. In order o obain Eq. 3, we ry o make q 1 () and q 2 () orhogonal o each oher a equal ime,. Here he approximaion we inroduce is ha q 1 ()q 2 () for all and, which we shall refer as orhogonalizaion approximaion. In erms of he wo orhogonal coordinaes we can rewrie he hree difference poenials as U Ag (1)q 1 q 2, U Dg q 1 q 2, and U AD q 1. Insering hese ino Eq. 23 and using he orhogonal propery of he wo coordinaes q 1 and q 2, he ime-dependen rae kernel can be recas in he form k f 2 Re 2 d2 expi AD 2 i g AD 2 AD 2 2i Im AD AD 2. 31 By inroducing he orhogonaliy approximaion discussed above he resuling formula becomes dependen on one correlaion funcion, AD (). Furhermore by applying he saionary phase approximaion o he inegral and replacing he inegraion limi wih infiniy, we find ha he nonequilibrium rae kernel becomes where k f 2 U AD 2 exp AD g AD 2Q AD 2 2U 2, AD Q AD d AD 1cos. 32 33 As can be seen in Eq. 32, he facor plays a criical role in deermining he magniude of he nonequilibrium rae kernel. As we will show in he following secion, his facor is a sensiive funcion of he dimensionaliy of he solvaion coordinaes. As briefly discussed in he Inroducion, since we have o deal wih hree saes, we have found ha he mulidimensional solvaion coordinae sysem can be reduced o a wo- J. Chem. Phys., Vol. 13, No. 2, 8 July 1995 Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions
M. Cho and R. J. Silbey: Phooinduced elecron ransfer 63 dimensional one by using he Schmid orhogonalizaion mehod. We nex show he imporance of he dimensionaliy of he solvaion coordinae sysem. F. Dimensionaliy of solvaion coordinae sysem Alhough we showed ha he wo orhogonal coordinaes can be exraced from he general mulidimensional coordinae sysem, we have no discussed he possible implicaion of he wo dimensionaliy of he solvaion coordinae sysem. One of he quesions arising immediaely is ha how large he flucuaion ampliudes of he wo solvaion coordinaes we chose are. Paricularly, hese quaniies are imporan in deermining he probabiliy of finding nuclear configuraion maximizing he Franck Condon overlap for E. As we shall show laer i is useful o inroduce a dimensional parameer,, defined by U 2 Ag U 2 AD U 2 Dg 2U 2 AD U 2 Dg cos. 34 his relaionship among he mean square flucuaion ampliudes can be viewed as a riangle wih hree sides of U 2 Ag, U 2 AD, and U 2 Dg. he angle beween he wo sides, U 2 Dg and U 2 AD, is he dimensionaliy parameer. he widh of he q 1 disribuion is deermined by he square roo of mean square flucuaion ampliude of U AD, ha is o say, q 2 1 U 2 AD. he mean square flucuaion ampliude of q 2 is wrien, in erms of he dimensionaliy parameer, by q 2 2 U 2 Dg sin. 35 Alhough only U 2 Dg appears in Eq. 35, he square roo of he mean square flucuaion ampliude of q 2 is deermined by all hree quaniies, U 2 Dg, U 2 AD, and U 2 Ag. In case when or equivalenly /2, q 2 2 is solely deermined by U 2 Dg, herefore he wo flucuaing coordinaes, U Dg () and U AD (), are orhogonal o each oher. Consequenly, flucuaions of U Dg (), which is associaed wih opical broadening effec and relaxaion of he nonequilibrium wave packe on he donor surface, do no affec he elecron ransfer rae associaed wih flucuaions of U AD (). ha is, nonequilibrium preparaion of he nuclear wave packe on he donor surface does no affec o he E rae a all. On he oher hand, when is equal o or, he solvaion coordinae sysem is in one-dimensional siuaion, ha is o say, we need only one solvaion coordinae in order o describe boh he opical exciaion and he elecron ransfer. In his one-dimensional limi, he flucuaion ampliude along he q 2 axis is zero, as can be seen in Eq. 35, so ha we can jusify ha a one-dimensional solvaion coordinae can fully describe boh opical ransiion and elecron ransfer. he difference beween he wo cases, and, is he alignmen of he poenial energy surfaces in he one-dimensional solvaion coordinae sysem see Fig. 1. As can be seen in Eq. 35, he mean square flucuaion ampliude of q 2 is deermined by a projecion of he mean square flucuaion ampliude of U Dg (). his can be reinerpreed by noing ha he projecion of U 2 Dg ono he primary coordinae, q 1, is deermined by U 2 Dg cos. herefore, when /2, here is no projecion of he U Dg () ono FIG. 3. wo cases of poenial surfaces in he one-dimensional solvaion coordinae sysem. a and b correspond o and, respecively. See figure capion of Fig. 1 for he sepwise explanaion. he solvaion coordinae associaed wih E so ha here is no effec of flucuaions of U Dg () on E rae kernel. On he oher hand, when or, here are maximum influences of nonequilibrium preparaion on he E process. IV. NUMERICAL CALCULAIONS In his secion we presen numerical calculaions of he ime-dependen populaions and rae kernel for various siuaions. In order o calculae hese quaniies we need o define he specral densiies. he specral densiies are assumed o be he following form: 3 exp/ c. 36 s associaed wih each specral densiy are deermined by he classical reorganizaion energies we use. For he sake of simpliciy we will consider he classical high emperaure limi only where he mean square flucuaion ampliude is proporional o he classical reorganizaion energy by U 2 AD 2 D AD k B for example. herefore, from Eq. 34, here is a relaionship among he hree classical reorganizaion energies, g Ag D AD g Dg 2 D g AD Dg cos. 37 We will consider ha he wo among he hree reorganizaion energies are given and he remaining one is deermined by he relaionship given above. We firs calculae he ime-dependen rae kernel, Eq. 28 wih specral densiies of he same form defined in Eq. 36 wih c 1 cm 1. he elecron exchange marix elemen is assumed o be 3 cm 1, hroughou he numerical g calculaions. wo reorganizaion energies, Dg and D AD are 1 and 2 cm 1, respecively. he remaining reorganizaion energy, g Ag, should be calculaed from Eq. 37 when we specify he dimensionaliy of he sysem. he energy gap ( AD ) beween he isolaed accepor and donor is 5 cm 1. Since AD D AD, poenial surfaces of he elecron ransfer pair are in he so-called invered regime see Fig. 3. We now presen he ime-dependen rae kernel, Eq. 28, in Fig. 4a. he dashed curve corresponds o he case of J. Chem. Phys., Vol. 13, No. 2, 8 July 1995 Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions
64 M. Cho and R. J. Silbey: Phooinduced elecron ransfer FIG. 4. a ime-dependen E rae kernels, Eq. 28, are calculaed for hree cases when dashed curve, /2 solid curve, and doed curve. b Donor populaions are calculaed for dashed curve, /2 solid curve, and doed curve. c Semilogarihmic plos of he donor populaions are shown. he specral densiies are given by Eq. 36 wih c 1 cm 1. s associaed wih each specral densiy are deermined by he classical reorganizaion energies from he definiions in Eq. 22. AD 2 cm 1 and Dg 1 cm 1. Ag is deermined by using he relaionship among he hree classical reorganizaion energies see Eq. 37. In his case, Ag for, /2, and are 35.6, 12, and 294.4 cm 1, respecively. AD 5 cm 1. 3 K. see Fig. 3a. Since his is a one-dimensional case, we can draw he hree poenial surfaces wih respec o a single coordinae. he nonequilibrium nuclear wave packe creaed on he donor surface ends o relax down o he minimum of he donor surface. he disance beween he cener of he nonequilibrium wave packe and he exi channel curve crossing poin beween he donor and accepor surfaces becomes closer as ime increases i should be noed ha he disance we have menioned here should be considered as he magniude of he ime-dependen reorganizaion energy, so i should no be confused wih he real disance. herefore we expec ha he ime-dependen rae kernel increases in ime and reaches is limiing value shown by he plaeau in Fig. 4a. On he oher hand, when, even hough his is also a one-dimensional case, he posiion of he minimum of he ground sae is differen from ha for see Fig. 3b. Again he cener of he nonequilibrium wave packe propagaes on he donor surface o reach an equilibrium value. For a shor ime.8 ps, he cener of he nonequilibrium wave packe becomes closer o he exi channel. Afer i reaches he cener of he exi channel, he disance beween he cener of he nonequilibrium wave packe and he exi channel increases so ha he rae kernel decreases oward a limiing value see doed curves in Fig. 4a. As he hird example, we consider he case of /2. We canno draw hree poenial surfaces as Fig. 3, because he wo dimensionaliy of poenial surfaces are imporan refer o Fig. 2, even hough Fig. 2 is no precisely he relevan one. As we menioned before, if /2, he wo solvaion coordinaes associaed wih opical ransiion as well as relaxaion of he nonequilibrium wave packe and elecron ransfer are orhogonal so ha here is no effec of nonequilibrium preparaion of he donor populaion on he elecron ransfer rae. herefore, he rae kernel, in his case, is a consan and idenical o he equilibrium value solid curve in Fig. 4a. Wih he ime-dependen rae kernels calculaed above shown in Fig. 4a, we calculae he donor populaions in ime. We normalize he iniial donor populaion o be uniy. he solid curve in Fig. 4b is corresponding o he case of /2. Since he E rae kernel is consan, he donor populaion is a simple exponenial funcion wih respec o ime. If we assume ha he relaxaion of he nonequilibrium wavepacke can be ignored, we expec ha he donor populaion decays as he solid curve. he dashed curve is corresponding o he case of. Because he magniude of he rae kernel is small compared o he equilibrium value as shown by he dashed curve in Fig. 4a, he donor populaion decreases very slowly for shor ime 2 ps. As ime increases, he exponenial decaying paern is recovered. For, he donor populaion doed curve in Fig. 4b decays faser han he equilibrium case or he case of /2. his can be undersood ha he associaed rae kernel doed curve in Fig. 4a becomes larger han he equilibrium value afer 5 fs. Finally we presen, in Fig. 4c, semilogarihmic plos of donor populaions o show nonexponenial decaying paerns when he nuclear wave packe is in nonequilibrium siuaion. In Fig. 5, we presen similar numerical resuls excep g ha he classical reorganizaion energies, Dg and D AD, are 2 and 2 cm 1, respecively. Since AD D AD, his corresponds o he normal regime see Fig. 1. In Fig. 5a, he ime-dependen rae kernels for, /2, and, are shown by dashed, solid, and doed curves, respecively. As can be seen in he poenial surfaces in Fig. 1a, wecan undersand he ime dependence of he rae kernels for and. In boh cases, he limiing values approach o he equilibrium value, which is also idenical o he rae kernel for /2. Donor populaions in ime for he hree cases are shown in Fig. 5b. Also we find srong nonexponenial paerns for shor ime less han 2 ps. Semilogarihmic plos are shown in Fig. 5c. Paricularly, he decaying paern dashed curve in Fig. 5c of can be viewed as a double exponenial wih boh fas and slow ones. We noe ha his behavior iniially he donor populaion decays quickly and hen follows a slow exponenial decay has no been seen in Coalson e al. s compuer simulaion sudies. his is because hey did no fully consider mulidimensional aspec of he poenial surfaces explicily. he slow par of he decay is J. Chem. Phys., Vol. 13, No. 2, 8 July 1995 Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions
M. Cho and R. J. Silbey: Phooinduced elecron ransfer 65 our long-ime exponenial decay consan wih respec o he isolaed free energy gaps. As shown conclusively in our model calculaions, he shor-ime nonexponenial decay of he donor populaion can be aribued o he nonequilibrium naure of he iniial preparaion of donor populaion by an ulrafas opical pulse. Boh paerns, quick decay followed by slow exponenial decay as well as slow decay followed by fas exponenial decay, can be found in he phooinduced E and are dependen on he mulidimensional poenial surfaces. V. SUMMARY FIG. 5. Similar plos are shown excep ha he wo reorganizaion energies AD and AD are assumed o be 2 and 2 cm 1, respecively. Consequenly, he remaining reorganizaion energies, Ag, for, /2, and are 935.1, 22, and 3464.9 cm 1, respecively. All he oher parameers are he same wih Fig. 4. a ime-dependen E rae kernels, using Eq. 28, are calculaed when dashed curve, /2 solid curve, and doed curve. b Donor populaions are calculaed when dashed curve, /2 solid curve, and doed curve. c Semilogarihmic plos of he donor populaions are shown. Dashed, solid and doed curves correspond o, /2, and, respecively. herefore relaed o he equilibrium E rae consan, whereas he fas par of he decay is induced by he relaxaion of he nonequilibrium wave packe on he donor surface. In his secion, alhough we presened very limied cases in he wide parameer space, we found ha he resuling decaying paerns of donor populaions in ime are shown o be very sensiive wih respec o he nonequilibrium naures of he phooinduced iniiaion of he E in condensed media. Since he opical exciaion sep can be sudied by various linear and nonlinear specroscopic measuremens, one can ge deailed informaion on he mean square flucuaion ampliude, U 2 Dg, which is in urn relaed o he classical reorganizaion energy, g Dg. More specifically, he classical reorganizaion energy is approximaely equal o he wice of he fluorescence Sokes shif. hus half of he inpus we need can be obained via specroscopic measuremens. Sill we are suffering from he lack of direc experimenal mehods o obain he mean square flucuaion ampliude direcly associaed wih he elecron ransfer. hese parameers can be, in principle, obained by ploing equilibrium rae consans In his paper, we presened a heoreical descripion of he phooinduced E process in condensed media. Assuming ha he bah consiss of harmonic oscillaors and ha hose harmonic modes are linearly coupled o he hree saes, we formulaed he ime-dependen E rae kernel including nonequilibrium feaures of opical preparaion of donor populaion. We assumed ha he ulrafas laser pulse is shor enough o ignore any elecron ransfer during he pulse duraion ime. In his case we can obain a generalized expression for he nonadiabaic E rae kernel ha is idenical o ha discussed by Coalson e al. 19 We furher applied a saionary phase approximaion wihin he cumulan mehod. We finally found a simple and ineresing generalizaion of he Marcus expression of E rae kernel, where he nonequilibrium naure is included via he ime-dependen reorganizaion energy. In order o fully undersand he nonequilibrium naure of he phooinduced E, we discussed mulidimensional aspecs of he solvaion coordinae sysem. In some limiing case we found ha wo orhogonal coordinaes can be obained by using he Schmid orhogonalizaion mehod. We believe ha here are many direcions o exend our work. One is o consider a finie pulse duraion effec. In his case we canno simply ignore he fully coheren conribuion, where he ime ordering is mixed. Furhermore, he propagaion of he iniial wave packe on he donor surface during he pulse duraion ime could be imporan if he relaxaion ime is comparable o he ime scale of he pulse duraion ime. Anoher generalizaion is o develop eiher a fully phase space picure e.g., Wigner disribuion funcion wih wo orhogonal coordinaes and wo conjugae momena, or coordinae space picure wih wo coordinaes. hen, he E rae kernel should be given by an inegral of he disribuion funcions over he phase space or over coordinae space. he picure emerging from hese procedures show clearly ha he condiional probabiliy of finding he overlap beween he disribuion funcions associaed wih he nonequilibrium wave packe and wih he exi channel is direcly proporional o he ime-dependen E rae kernel we discussed in his paper. Finally, in his paper, we have no specifically sudied he effecs of molecular vibraions on he dynamics. Formally, we can proceed in much he same spiri as above o ake hese effecs ino accoun. J. Chem. Phys., Vol. 13, No. 2, 8 July 1995 Downloaded 27 Oc 212 o 18.189.115.229. Redisribuion subjec o AIP license or copyrigh; see hp://jcp.aip.org/abou/righs_and_permissions