The role of different reorganization energies within the Zusman theory of electron transfer

J. Chem. Phys. 118, 291 (2003); h ps://doi.o g/10.1063/1.1525799 118, 291 © 2003 Ame ican Ins i u e o Physics. The ole o di e en eo ganiza ion ene gies wi hin he Zusman heo y o elec on ans e Ci e as: J. Chem. Phys. 118, 291 (2003); h ps://doi.o g/10.1063/1.1525799 Submi ed: 05 Augus 2002 . Accep ed: 07 Oc obe 2002 . Published Online: 16 Decembe 2002 Jesús Casado-Pascual, Manuel Mo illo, Igo Goychuk, and Pe e Hänggi ARTICLES YOU MAY BE INTERESTED IN Elec on ans e dynamics: Zusman equa ion e sus exac heo y The Jou nal o Chemical Physics 130, 164518 (2009); h ps://doi.o g/10.1063/1.3125003 E ec o ic ion on elec on ans e in biomolecules The Jou nal o Chemical Physics 83, 4491 (1985); h ps://doi.o g/10.1063/1.449017 Dynamic sol en e ec s on ou e -sphe e elec on ans e The Jou nal o Chemical Physics 87, 2090 (1987); h ps://doi.o g/10.1063/1.453184 The ole o di e en eo ganiza ion ene gies wi hin he Zusman heo y o elec on ans e Jesu ´s Casado-Pascual and Manuel Mo illo Fı ´sica Teo ´ ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, Se illa 41080, Spain Igo Goychuk and Pe e Ha ¨nggi Ins i u u ¨ Physik, Uni e si a ¨ Augsbu g, Uni e si a ¨ ss aße 1, D-86135, Augsbu g, Ge many 共Recei ed 5 Augus 2002; accep ed 7 Oc obe 2002兲 We conside he kine ics o elec on ans e eac ions in condensed media wi h di e en eo ganiza ion ene gies o he o wa d and backwa d p ocesses. The s a ing poin o ou analysis is an ex ension o he well-known Zusman equa ions o he case o pa abolic diaba ic cu es wi h di e en cu a u es. Agene alized mas e equa ion o he popula ions as well as o mal exp essions o hei long- ime limi is de i ed. We discuss he condi ions unde which he ime e olu ion o he popula ions o eac an s and p oduc s can be desc ibed a all imes by a single exponen ial law. In he limi o e y small unnel spli ing, a no el a e o mula o he nonadiaba ic ansi ions is ob ained. I gene alizes p e ious esul s de i ed wi hin he con ac app oxima ion. Fo la ge alues o he unnel spli ing, we make use o he consecu i e s ep app oxima ion leading o a a e o mula ha b idges be ween he nonadiaba ic and sol en -con olled adiaba ic egimes. Finally, he analy ical p edic ions o he long- ime popula ions and o he a e cons an a e es ed agains p ecise nume ical solu ions o he s a ing se o pa ial di e en ial equa ions. © 2003 Ame ican Ins i u e o Physics. 关DOI: 10.1063/1.1525799兴 I. INTRODUCTION Elec on ans e eac ions a e o p ime impo ance in many physicochemical and biological p ocesses.1A a e y undamen al le el, an elec on ans e s ep is essen ially an elec on unneling e en in he p esence o a medium 共sol- en 兲. A non-negligible unneling p obabili y equi es eso- nance in he ene gy o localized elec onic s a es. The sol en he mal luc ua ions p o ide he necessa y ene gy o he esonance condi ion. Thus, he kine ics o elec on ans e eac ion equi es an adequa e desc ip ion o he medium he - mal luc ua ions ha media e he elec onic cha ge edis i- bu ion in an elec on ans e .2 In he classical heo y o Ma cus,3Hush,4and Le ich and Dogonadze5 he sol en luc ua ions a e desc ibed by an equilib ium p obabili y law. Thus, he knowledge o he ee ene gy as a unc ion o an app op ia e eac ion coo dina e is all ha is needed o e alua e he a e cons an . Abou 20 yea s ago, Zusman6and Alexand o 7in oduced in o he heo y he idea ha nonequilib ium e ec s associa ed wi h he elaxa ion o sol en luc ua ions could also a ec he a e. In a phenomenological way, Zusman p oposed a se o ou pa ial di e en ial equa ions ha inco po a ed he elax- a ion o he nonequilib ium p obabili y laws o he eac ion coo dina e in each diaba ic s a e and he unneling ansi ions be ween hem. In he o iginal model, he diaba ic s a es a e pa abolic unc ions o he eac ion coo dina e, wi h equal cu a u es o he eac an and p oduc s a es. The cu a u e is ela ed o he eo ganiza ion ene gy in such a way ha equal cu a u es implies ha he alues o he eo ganiza ion ene gy o he o wa d and backwa d eac ions a e iden ical. La e on, Ga g e al.8p esen ed a de i a ion o he Zusman equa ions s a ing om a Hamil onian model. They use ech- niques o unc ional in eg als o ca y ou he elimina ion o he ba h deg ees o eedom om he o al densi y ope a o . P e ious analy ical and nume ical s udies indica e ha he eo ganiza ion ene gies o he di ec and in e se eac- ions migh indeed ha e di e en alues.9–14 I is, he e o e, in e es ing o ex end he classical Zusman–Alexand o o - mula ion o he case o pa abolas wi h di e en cu a u es. A ew yea s ago, Tang15 discussed such an ex ension. F om he e y beginning in his analysis, Tang made use o he so- called con ac app oxima ion. Namely, he assumed ha un- neling be ween diaba ic cu es akes place s ic ly a hei c ossing poin s, hus neglec ing any delocaliza ion e ec s. E en o diaba ic su aces wi h equal cu a u es, he ac ha unneling ansi ions a e somewha delocalized a ound he c ossing poin is impo an , especially in he in e ed egime.16,17 Delocaliza ion leads o modi ica ions o he a e exp ession wi h espec o he ypical Ma cus–A henius s uc u e. In p e ious wo k,18 we ha e also ound s ong in- dica ions ha he con ac app oxima ion is no adequa e o desc ibe s ongly biased elec on ans e p ocesses. The s a ing poin o his pape is a se o ou pa ial di e en ial equa ions simila o hose used by Tang.15 The s uc u e o such equa ions is he same as in he o iginal Zusman model, bu wi h pa abolic po en ials wi h di e en cu a u es. The equa ions desc ibe he dynamics o diagonal and o -diagonal ma ix elemen s o he educed densi y op- e a o . We accep he e he alidi y o Zusman model. F an suzo 19 has co ec ly poin ed ou he limi a ions o Zus- man equa ions o desc ibe elec onic ans e p ocesses in s ongly pola sol en s. Indeed, i he sol en pola i y is oo s ong, he condi ions unde which Zusman equa ions a e de i ed om a mo e mic oscopic poin o iew20–23 migh be JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 1 1 JANUARY 2003 2910021-9606/2003/118(1)/291/13/$20.00 © 2003 Ame ican Ins i u e o Physics iola ed. This is an in e es ing poin ha we p opose o ad- d ess in he u u e. In he p esen wo k, we will concen a e on he de i a ion o he usual mac oscopic kine ic desc ip- ion o elec on ans e eac ions om a Zusman-like model and on he de e mina ion o sui able exp essions o he a e cons an and he long- ime popula ions o eac an s and p od- uc s. In ou de i a ion, we will no assume he con ac ap- p oxima ion. The scheme o he pape is as ollows: In Sec. II we se up he model and he no a ion. In Sec. III we de i e o mal exp essions o he popula ions on he diaba ic su aces in Laplace space. This is achie ed by using G een unc ions and p ojec ion ope a o echniques on he Zusman equa ions. An al e na i e me hod o solu ion o he Zusman equa ions has been pu o wa d ecen ly by Cao and Jung.24 I is based on he spec al p ope ies o he e olu ion ope a o o he densi y ma ix in he Zusman app oxima ion. As a as we know, his al e na i e has only been applied o diaba ic pa- abolas o equal cu a u es. The long- ime limi is explici ly ob ained. I is also shown ha he popula ions sa is y in e- g odi e en ial equa ions wi h a complica ed ke nel. Unde sui able condi ions which a e discussed in Sec. IV, we p o e ha he popula ions sa is y a single exponen ial elaxa ion law o all ele an ime scales, cha ac e ized by he a e cons an and he long- ime alues o he popula ions. In Sec. V, we p esen app oxima e analy ical exp essions o hose wo kine ic pa ame e s. In Sec. VI we p esen a de ailed compa ison o he analy ical p edic ions and p ecise nume i- cal solu ion o he Zusman equa ions. Finally, we conclude wi h commen s abou he main indings in his wo k. Some o he calcula ions a e e y in ol ed and hey a e p esen ed in he Appendices. II. THE ZUSMAN EQUATIONS FOR DIABATIC POTENTIALS WITH DIFFERENT CURVATURES The basic elemen s o desc ibe elec on ans e p o- cesses a e wo diaba ic elec onic ene gy cu es Vj(x), j ⫽1, 2, and a gene alized one-dimensional eac ion coo di- na e xwi h e ec i e mass m. The elec onic s a es be o e and a e he cha ge ans e will be deno ed as dono , 兩1典, and accep o , 兩2典, espec i ely. The eac ion coo dina e ep esen s a combina ion o he selec ed nuclea modes coupled di ec ly o he elec onic ans e sys em.1The eac ion coo dina e is also coupled o he es o nuclea modes. This coupling in oduces ic ion in he dynamics o he eac ion coo dina e wi h a phenomenological ic ion coe icien ␩ . This is a well es ablished s a ing poin in he mic oscopic ea men o in amolecula elec on ans e .1,8,14 I should be no ed ha ou Vj(x) a e po en ial ene gy cu es. They should no be con used wi h he pa abolic ee ene gy cu es appea ing in al e na i e desc ip ions o elec on ans e , as hose ely- ing on compu e simula ions.10 In he o e damped limi , Zus- man equa ions p o ide an app op ia e desc ip ion o he ime e olu ion o he ma ix elemen s ␳ jk(x, ) ª 具 j,x 兩 ␳ ˆ( ) 兩 x,k 典 o he educed densi y ope a o in he elec- on and eac ion coo dina e Hilbe space. Zusman equa- ions and hei alidi y condi ions ha e been epea edly de- i ed and discussed in he li e a u e.8,19–23 These equa ions ead ⳵ ⳵ ␳ 11共x, 兲⫽L ˆ1 ␳ 11共x, 兲⫹i⌬ 2ប关 ␳ 12共x, 兲⫺ ␳ 21共x, 兲兴,共1兲 ⳵ ⳵ ␳ 22共x, 兲⫽L ˆ2 ␳ 22共x, 兲⫺i⌬ 2ប关 ␳ 12共x, 兲⫺ ␳ 21共x, 兲兴,共2兲 ⳵ ⳵ ␳ 12共x, 兲⫽ 再 L ˆ⫺i ប关V1共x兲⫺V2共x兲兴 冎 ␳ 12共x, 兲 ⫹i⌬ 2ប关 ␳ 11共x, 兲⫺ ␳ 22共x, 兲兴,共3兲 ⳵ ⳵ ␳ 21共x, 兲⫽ 再 L ˆ⫹i ប关V1共x兲⫺V2共x兲兴 冎 ␳ 21共x, 兲 ⫺i⌬ 2ប关 ␳ 11共x, 兲⫺ ␳ 22共x, 兲兴.共4兲 He e L ˆja e he Smoluchowski ope a o s desc ibing di usion on each diaba ic po en ial: L ˆj⫽D ⳵ ⳵ x 冋 ⳵ ⳵ x⫹Vj ⬘共x兲 kBT 册 .共5兲 The mac oscopic di usion cons an Dis connec ed wi h he ic ion coe icien ␩ , which is assumed o be iden ical in bo h diaba ic s a es, and he empe a u e Tby he Eins ein ela ion D⫽kBT/ ␩ . The ope a o L ˆ⫽(L ˆ1⫹L ˆ2)/2 desc ibes di usion on he a e age po en ial 关V1(x)⫹V2(x)兴/2. Fi- nally, ⌬deno es he elec onic coupling ma ix elemen , and i cha ac e izes he deg ee o o e lap o he dono and accep- o wa e unc ions. He e, we will ake ⌬ o be independen o he nuclea coo dina es 共Condon app oxima ion兲. In his wo k, we will assume pa abolic diaba ic cu es o he o m Vj共x兲⫽m ␻ j 2 2共x⫺x0 ␦ j,2兲2⫺ ⑀ 0 ␦ j,2 ,共6兲 whe e x0and ⑀ 0a e he ho izon al and e ical shi s, espec- i ely, be ween he minima o he pa abolas 共c . Fig. 1兲. The equencies ␻ jcha ac e ize hei cu a u es, and hey a e FIG. 1. Pa abolic diaba ic su aces wi h di e en cu a u es as a unc ion o he eac ion coo dina e x. No ice ha he numbe o c ossing poin s depends on he alue o he ene gy bias ⑀ 0. 292 J. Chem. Phys., Vol. 118, No. 1, 1 Janua y 2003 Casado-Pascual e al. ela ed o he eo ganiza ion ene gies ␭jby he exp ession ␭j⫽m ␻ j 2x0 2/2. The e o e, he ac ha he cu a u es a e di - e en implies ha hese eo ganiza ion ene gies o he o - wa d and backwa d eac ions a e also di e en . To a oid con usion, we wan o emphasize ha he cu a u es o ou diaba ic po en ial ene gies, ␻ j, can be di e en . I one de- sc ibes elec on ans e p ocesses in e ms o ee ene gy p o iles, hen, as poin ed ou by Tachiya,25 he wo ee en- e gy cu es a e no independen , and hey canno be s ic ly pa abolic when he cu a u es a hei minima a e di e en . Ob iously, ou po en ial ene gies, no being ee ene gies, a e no ied up by such a es ic ion. The di e ence in cu - a u es also yields a di e ence in he phenomenological e- laxa ion imes co esponding o each diaba ic cu e, ␶ j⫽ ␩ m ␻ j 2⫽kBT m ␻ j 2D.共7兲 No ice ha hese elaxa ion imes a e ela ed o he eo gani- za ion ene gies by ␶ 1/ ␶ 2⫽␭2/␭1. Fo la e con enience, we will also in oduce he elaxa ion ime o he o e damped oscilla o on he a e aged po en ial 关V1(x)⫹V2(x)兴/2, ␶ ⫽2kBT mD共 ␻ 1 2⫹ ␻ 2 2兲.共8兲 F om he abo e exp essions, i ollows ha 2/ ␶ ⫽1/ ␶ 1 ⫹1/ ␶ 2. Elec on unneling is mos e ec i e nea he c ossing poin s o he diaba ic cu es, which a e gi en by xj *⫽x0 ␭2⫺␭1 冋 ␭2⫹共⫺1兲j 冑 冉 1⫺ ⑀ 0 ⑀ c 冊 ␭1␭2 册 ,共9兲 whe e ⑀ c⫽␭1␭2/(␭1⫺␭2). Depending on he ela i e al- ues o ⑀ 0and ⑀ c he e can be wo, one, o no c ossing poin s. III. FORMAL SOLUTION OF THE ZUSMAN EQUATIONS In his sec ion we ob ain some exac , hough o mal, analy ical esul s o he ime e olu ion o he popula ions Pj( ). They a e ob ained by in eg a ing he co esponding p obabili y densi ies o e con igu a ion space, i.e., Pj( ) ⫽ 兰 ⫺⬁ ⬁dx ␳ jj(x, ). The i s s ep is o educe he ou Zus- man equa ions o jus wo in eg al equa ions o he diagonal elemen s ␳ jj(x, ). This is achie ed by i s eplacing ␳ 12(x, ) and ␳ 21(x, ) in Eqs. 共1兲and 共2兲by he esul o o mally sol ing he wo o -diagonal equa ions 共3兲and 共4兲. This yields ⳵ ⳵ ␳ jj共x, 兲⫽共⫺1兲j⌬ ប 冕 ⫺⬁ ⬁dx1Im关God共x, 兩 x1兲 ␳ 12共x1,0兲兴 ⫹共⫺1兲j⌬2 2ប2 冕 0 d 1 冕 ⫺⬁ ⬁dx1 ⫻Re关God共x, ⫺ 1 兩 x1兲兴 ⫻关 ␳ 11共x1, 1兲⫺ ␳ 22共x1, 1兲兴⫹L ˆj ␳ jj共x, 兲,共10兲 whe e God(x, 兩 x⬘), he o -diagonal G een unc ion, is he solu ion o he pa ial di e en ial equa ion ⳵ ⳵ God共x, 兩 x⬘兲⫽ 再 L ˆ⫺i ប关V1共x兲⫺V2共x兲兴 冎 God共x, 兩 x⬘兲, 共11兲 wi h ini ial condi ion God共x,0 兩 x⬘兲⫽ ␦ 共x⫺x⬘兲,共12兲 and bounda y condi ions lim x→⫾⬁ God共x, 兩 x⬘兲⫽0. 共13兲 An e alua ion o his G een unc ion o he ha monic po en- ials wi h di e en cu a u es can be ound in Appendix A 1. F om now on, we shall assume ha ␳ 12(x,0)⫽0, so ha he i s e m on he igh -hand side o exp ession 共10兲will no be p esen . This ini ial condi ion desc ibes he usual si ua ion in which he elec onic cohe ences be ween dono and accep- o s a es a e ini ially neglec ed. Nex , o mal solu ion o Eq. 共10兲in e ms o he diago- nal G een unc ions Gd (j)(x, 兩 x⬘) leads o ␳ jj共x, 兲⫽共⫺1兲j 冕 0 d 1 冕 ⫺⬁ ⬁dx1aj共x, ⫺ 1 兩 x1兲 ⫻关 ␳ 11共x1, 1兲⫺ ␳ 22共x1, 1兲兴 ⫹ 冕 ⫺⬁ ⬁dx1Gd (j)共x, 兩 x1兲 ␳ jj共x1,0兲.共14兲 The p opaga o s aj(x, 兩 x⬘) in Eq. 共14兲a e gi en in e ms o he G een unc ions by aj共x, 兩 x⬘兲⫽⌬2 2ប2 冕 0 d ⬘ 冕 ⫺⬁ ⬁dx⬙Gd (j)共x, ⫺ ⬘ 兩 x⬙兲Re关God共x⬙, ⬘ 兩 x⬘兲兴.共15兲 The diagonal G een unc ions desc ibe he di usi e mo ion on he diaba ic cu es Vj(x). They a e he solu ion o he pa ial di e en ial equa ions ⳵ ⳵ Gd (j)共x, 兩 x⬘兲⫽L ˆjGd (j)共x, 兩 x⬘兲,共16兲 wi h he same ype o ini ial and bounda y condi ions as God(x, 兩 x⬘). Explici exp essions o Gd (j)(x, 兩 x⬘) can be ound in Appendix A 2. Typically, a eac ion s a s om a si ua ion whe e he sol en is he mally equilib a ed. Thus, we will es ic ou s udy o ini ial condi ions o he diagonal e ms o he o m ␳ jj(x,0)⫽gj(x)Pj(0), whe e gj共x兲⫽exp关⫺Vj共x兲/kBT兴 冕 ⫺⬁ ⬁dx⬘exp关⫺Vj共x⬘兲/kBT兴 共17兲 a e he equilib ium dis ibu ions on each o he diaba ic cu es, and Pj(0) a e he ini ial condi ions o he popula- ions. Ob iously, wi h hese ini ial condi ions, he second e m on he igh -hand side o exp ession 共14兲 educes o gj(x)Pj(0). 293J. Chem. Phys., Vol. 118, No. 1, 1 Janua y 2003 Elec on ans e wi h di e en eo ganiza ion ene gies Fo la e con enience, we will ew i e Eq. 共14兲in ma ix no a ion by in oducing he one-column ec o %(x, ) wi h componen s %j(x, )⫽ ␳ jj(x, ), and he 2⫻2 ma ices U, A(x, 兩 x⬘), and g(x) wi h ma ix elemen s Ujk⫽(⫺1)j⫹k, Ajk(x, 兩 x⬘)⫽aj(x, 兩 x⬘) ␦ j,k, and gjk(x)⫽gj(x) ␦ j,k, espec- i ely. The solu ion o Eq. 共14兲is bes sough by using he Laplace ans o m ˜ (s)⫽ 兰 0 ⬁d ( )exp(⫺s ). Thus, one inds % ˜ 共x,s兲⫽s⫺1g共x兲P共0兲⫺ 冕 ⫺⬁ ⬁dx1A ˜ 共x,s 兩 x1兲U% ˜ 共x1,s兲.共18兲 We will use s anda d p ojec ion ope a o echniques o ge an exp ession o he Laplace ans o m o he popula- ions P ˜ (s). We de ine he p ojec ion ope a o s ⌸and Qas ⌸F共x兲⫽F储共x兲⫽g共x兲 冕 ⫺⬁ ⬁dx⬘F共x⬘兲,共19兲 QF共x兲⫽F共x兲⫽F共x兲⫺F 储 共x兲,共20兲 F(x) being an a bi a y one-column ec o o 2⫻2 ma ix depending on x. By ac ing wi h ⌸on Eq. 共18兲and in eg a ing o e x, one ob ains a e some simpli ica ions P共0兲⫽ 冋 sI⫹ 冕 ⫺⬁ ⬁dx1K ˜ 共x1,s兲Ug共x1兲 册 P ˜ 共s兲 ⫹ 冕 ⫺⬁ ⬁dx1K ˜ 共x1,s兲U% ˜ 共x1,s兲,共21兲 whe e Iis he 2⫻2 uni ma ix and K ˜ 共x,s兲⫽⌬2 2ប2 冕 ⫺⬁ ⬁dx⬘Re关G ˜ od共x⬘,s 兩 x兲兴.共22兲 Using Eq. 共A16兲in Eq. 共22兲, we ob ain K ˜ 共x,s兲⫽⌬2 2ប2 冕 0 ⬁d Re 兵 exp关⫺s ⫺c共 ,x兲兴 其 ,共23兲 wi h c( ,x) gi en by Eq. 共A11兲. The ac ion o Qon Eq. 共18兲 leads o % ˜ 共x,s兲⫽⫺ 冕 ⫺⬁ ⬁dx1A ˜ 共x,s 兩 x1兲U关g共x1兲P ˜ 共s兲⫹% ˜ 共x1,s兲兴, 共24兲 whe e A ˜ 共x,s 兩 x⬘兲⫽⌬2 2ប2g共x兲 冕 ⫺⬁ ⬁dx⬙J ˜ 共x,s 兩 x⬙兲 ⫻Re关G ˜ od共x⬙,s 兩 x⬘兲兴,共25兲 wi h J ˜ (x,s 兩 x⬘) being he 2⫻2 diagonal ma ix wi h ma ix elemen s J ˜ jk共x,s 兩 x⬘兲⫽ ␦ j,k 兵 关gj共x兲兴⫺1G ˜ d (j)共x,s 兩 x⬘兲⫺s⫺1 其 .共26兲 A o mal solu ion o % ˜ (x,s) is ob ained by sol ing i e a- i ely he in eg al equa ion 共24兲. Subs i u ion o he esul in Eq. 共21兲leads o P共0兲⫽关sI⫹Uk ˜ 共s兲兴P ˜ 共s兲,共27兲 whe e k ˜ (s) is he diagonal ma ix ob ained by summing he se ies k ˜ 共s兲⫽兺 n⫽1 ⬁ k ˜ (n)共s兲.共28兲 The e ms o he se ies k ˜ (n)(s) a e gi en by k ˜ (1)共s兲⫽ 冕 ⫺⬁ ⬁dx1K ˜ 共x1,s兲g共x1兲共29兲 and k ˜ (n)共s兲⫽共⫺1兲n⫺1 冕 ⫺⬁ ⬁dx1••• 冕 ⫺⬁ ⬁dxnK ˜ 共x1,s兲 ⫻ 兿 j⫽2 n T 关A ˜ 共xj⫺1,s 兩 xj兲兴g共xn兲共30兲 o n⭓2. Acco ding o Eq. 共27兲, he Laplace ans o m o he popula ions can be exp essed in e ms o he diagonal ma ix k ˜ (s)as P ˜ j共s兲⫽Pj共0兲⫹s⫺1关 ␦ j,1k ˜ 22共s兲⫹ ␦ j,2k ˜ 11共s兲兴 s⫹k ˜ 11共s兲⫹k ˜ 22共s兲.共31兲 A e ca ying ou he in e se Laplace ans o m o he abo e exp ession, one ob ains he ime e olu ion o he popula- ions, Pj( ). I ollows om exp ession 共31兲 ha he long- ime limi o he popula ions is gi en by Pj共⬁兲ªlim →⫹⬁ Pj共 兲⫽lim s→0⫹ sP ˜ j共s兲 ⫽ ␦ j,1k ˜ 22共0兲⫹ ␦ j,2k ˜ 11共0兲 k ˜ 11共0兲⫹k ˜ 22共0兲.共32兲 No ice ha hese alues a e independen o he ini ial condi- ions Pj(0). Finally, ea anging Eq. 共27兲and ca ying ou he in e se Laplace ans o m, we ind ha he popula ions Pj( ) sa is y he se o gene alized mas e equa ions d d P1共 兲⫽⫺ 冕 0 d ⬘关k11共 ⫺ ⬘兲P1共 ⬘兲⫺k22共 ⫺ ⬘兲P2共 ⬘兲兴, 共33兲 d d P2共 兲⫽⫺ 冕 0 d ⬘关k22共 ⫺ ⬘兲P2共 ⬘兲⫺k11共 ⫺ ⬘兲P1共 ⬘兲兴, whe e kjj( ) is he in e se Laplace ans o m o k ˜ jj(s). F om he abo e se o equa ions, i ollows immedia ely he con- se a ion o p obabili y, i.e., P1( )⫹P2( )⫽1. Thus, he se 共33兲 educes o a single in eg odi e en ial equa ion o , say, he popula ion P1( ). This equa ion can be con enien ly w i en as d d P1共 兲⫽⫺ 冕 0 d ⬘关k11共 ⬘兲⫹k22共 ⬘兲兴P1共 ⫺ ⬘兲 ⫹ 冕 0 d ⬘k22共 ⬘兲.共34兲 Then, a e solu ion o Eq. 共34兲, he e olu ion o P2( ) ol- lows immedia ely. Up o now, he o mal esul s ha we ha e ob ained a e exac . We ha e only assumed he con e gence o he se ies 共28兲and a speci ic amily o ini ial condi ions o he densi- 294 J. Chem. Phys., Vol. 118, No. 1, 1 Janua y 2003 Casado-Pascual e al. ies ␳ jk(x, ). The esul 共33兲is impo an . The popula ions on he dono and accep o diaba ic cu es sa is y i s o de in eg odi e en ial equa ions wi h a con olu ion s uc u e. The con olu ion ke nels, kjj( ), a e a he complica ed. The nex impo an ask is o analyze unde which condi ions can he solu ions o Eq. 共33兲be p ope ly app oxima ed by a single exponen ial ime e olu ion. IV. VALIDITY CONDITIONS FOR THE RATE REGIME In kine ics, he ime e olu ion o he popula ions on he dono , P1( ), and accep o , P2( ), diaba ic cu es is usually gi en by Pj共 兲⫽Pj共⬁兲⫹关Pj共0兲⫺Pj共⬁兲兴e⫺⌫ ,共35兲 whe e ⌫is he o al a e cons an . In his sec ion we analyze he condi ions unde which his a e desc ip ion o he ime e olu ion o he popula ions can be ob ained om he Zus- man equa ions. The s a ing poin is he in eg odi e en ial equa ion 共34兲. In his equa ion one can dis inguish wo di - e en clea -cu ime scales. The i s one, ␶ a, is associa ed wi h he elaxa ion ime o he ke nels kjj( ). As all he ime dependence o kjj( ) is h ough he diagonal and o - diagonal G een unc ions, ␶ adepends mainly on he elax- a ion imes o hese G een unc ions. The second one is gi en by ␶ b⫽关k ˜ 11(0)⫹k ˜ 22(0)兴⫺1and, as we will see below, i is associa ed wi h he elaxa ion ime o he popula ions. In gene al, ␶ bdepends on he elaxa ion imes o he G een unc ions and also on he cha ac e is ic unneling ime scale ប/⌬. In he nex sec ion, we will ob ain app oxima e exp es- sions o ␶ b ⫺1. In o de o s udy he alidi y condi ions o he a e e- gime, i is con enien o exp ess Eq. 共34兲in dimensionless o m as d dY P1共Y; ␨ 兲⫽⫺ 冕 0 Y/ ␨ dy⬘关 ␬ 11共y⬘兲⫹ ␬ 22共y⬘兲兴P1共Y⫺ ␨ y⬘; ␨ 兲 ⫹ 冕 0 Y/ ␨ dy⬘ ␬ 22共y⬘兲.共36兲 He e we ha e in oduced he dimensionless quan i ies Y⫽ / ␶ b,y⬘⫽ ⬘/ ␶ a, ␨ ⫽ ␶ a/ ␶ b, and ␬ jj(y⬘) ⫽ ␶ a ␶ bkjj( ␶ ay⬘), and we ha e indica ed explici ly he de- pendence o P1(Y; ␨ ) on he pa ame e ␨ . No ice ha he dimensionless in eg a ion a iable y⬘has been chosen so ha he con ibu ions o he in eg and o alues o y⬘much la ge han uni y can be sa ely neglec ed. Le us assume ha we a e in a egime in which ␶ a Ⰶ ␶ b. To ind he leading-o de app oxima ion o he solu ion o Eq. 共36兲as ␨ →0⫹, i.e., P1(Y;0)⫽lim ␨ →0⫹P1(Y; ␨ ), we ake he limi ␨ →0⫹in Eq. 共36兲while keeping Y⫽0 ixed. The esul is d dY P1共Y;0兲⫽⫺ 冕 0 ⬁dy⬘关 ␬ 11共y⬘兲⫹ ␬ 22共y⬘兲兴P1共Y;0兲 ⫹ 冕 0 ⬁dy⬘ ␬ 22共y⬘兲⫽⫺P1共Y;0兲⫹P1共⬁兲, 共37兲 whe e we ha e aken in o accoun ha he alues o he in eg and o y⬘much la ge han uni y a e negligible. The solu ion o Eq. 共37兲is P1(Y;0)⫽Ce⫺Y⫹P1(⬁), Cbeing an unknown cons an o in eg a ion. This cons an is de e - mined by making use o he ini ial condi ion, i.e., limY→0⫹P1(Y;0)⫽C⫹P1(⬁)⫽P1(0). The e o e, a e go- ing back o he o iginal a iable , we ind ha he leading- o de app oxima ion o he solu ion o Eq. 共36兲as ␨ →0⫹is gi en by Eq. 共35兲wi h j⫽1 and ⌫⫽ ␶ b ⫺1. F om he conse - a ion o p obabili y, i ollows immedia ely ha , in his limi , he popula ion P2( ) is also o he o m 共35兲. In conclusion, we ha e p o ed ha , when ␶ aⰆ ␶ b, he alues o he popula ions ob ained om he Zusman equa- ions can be p ope ly app oxima ed by he a e exp ession 共35兲wi h o al a e ⌫⫽k ˜ 11共0兲⫹k ˜ 22共0兲.共38兲 No ice ha , when ␶ aⰆ ␶ b, Eq. 共35兲desc ibes p ope ly he elaxa ion o he popula ions o all he ele an ime scales associa ed wi h Pj( ), e en o sho imes. When he con- di ion ␶ aⰆ ␶ bis iola ed, hen he disc epancies be ween he p edic ions o Eq. 共35兲and hose o he Zusman equa ions migh be ele an a all imes, as we will see la e , when we compa e wi h nume ical solu ion o Zusman equa ion 共see Sec. VI兲. V. ANALYTICAL EXPRESSIONS FOR THE LONG-TIME POPULATIONS AND THE TOTAL RATE CONSTANT F om now on, we will assume ha a a e desc ip ion o he ime e olu ion o he popula ions is app op ia e. In ha case, he pa ame e s Pj(⬁) and ⌫can be exp essed in e ms o he diagonal ma ix k ˜ (0), acco ding o Eqs. 共32兲and 共38兲. The e alua ion o his ma ix en ails he summa ion o he se ies 共28兲. In o de o do so, one has o eso o app oxi- ma ions. The na u e o he app oxima ions is dic a ed by he ela i e alues o he unneling equency and he cha ac e - is ic equencies o he sol en dynamics. A. The nonadiaba ic limi I he cha ac e is ic unneling equency ⌬/បis e y small ela i e o he elaxa ion equencies associa ed wi h he G een unc ions, hen unneling becomes he limi ing s ep mechanism o he a e p ocess. In his nonadiaba ic e- gime, he ma ix elemen s o k ˜ (0) can be app oxima ed as k ˜ jj共0兲⬇kNA (j)ª⌬2lim ⌬→0 k ˜ jj共0兲 ⌬2⫽k ˜ jj (1)共0兲.共39兲 A e inse ing he exp essions 共17兲and 共23兲, wi h s⫽0, in o Eq. 共29兲and in eg a ing o e x1, we ind kNA (j)⫽ ⌬2⌳j 1/2 ប2 冕 0 ⬁d Re 兵 Nj共 兲exp关Rj共 兲兴 其 ,共40兲 whe e 295J. Chem. Phys., Vol. 118, No. 1, 1 Janua y 2003 Elec on ans e wi h di e en eo ganiza ion ene gies Nj共 兲⫽ ␣ 1/2exp关共1⫺ ␣ 兲 /共2 ␶ 兲兴 兵 共 ␣ ⫹1兲关2⌳j⫹共 ␣ ⫺1兲/2兴⫹共 ␣ ⫺1兲关2⌳j⫺共 ␣ ⫹1兲/2兴exp关⫺2 ␣ / ␶ 兴 其 1/2 ,共41兲 Rj共 兲⫽i 冋 ␭1␭2共⌳1⫺⌳2⫹i ␹ 兲 ប共␭1⫹␭2兲 ␣ 2⫺ ⑀ 0 ប 册 ⫹4 ␶ ␭1 2␭2 2 ប␭j共␭1⫹␭2兲2 ␣ 3 ⫻ 再 关i共2⌳j⫹1兲共⫺1兲j⫺1⫹ ␹ 兴 ␣ sinh共 ␣ / ␶ 兲 4⌳j ␣ cosh共 ␣ / ␶ 兲⫹关4⌳j⫹ ␣ 2⫺1兴sinh共 ␣ / ␶ 兲 ⫹2⌳j关2i共⫺1兲j⫺1⫹ ␹ 兴关cosh共 ␣ / ␶ 兲⫺1兴 4⌳j ␣ cosh共 ␣ / ␶ 兲⫹关4⌳j⫹ ␣ 2⫺1兴sinh共 ␣ / ␶ 兲 冎 ,共42兲 and he dimensionless pa ame e s ⌳j, ␹ , and ␣ a e de ined in Eqs. 共A12兲–共A14兲. The emaining ime in eg al in Eq. 共40兲 can be calcula ed by a nume ical quad a u e. The exp essions o he long- ime popula ions and he a e cons an in he nonadiaba ic limi , Pj (NA)(⬁) and ⌫NA , a e ob ained by e- placing k ˜ jj(0) wi h kNA (j)in Eqs. 共32兲and 共38兲. To he bes o ou knowledge, Eqs. 共40兲–共42兲ha e ne e been de i ed p e- iously in he li e a u e. These exp essions o he nonadia- ba ic pa ame e s a e one o he main esul s in his pape . They cons i u e a gene aliza ion o he ypical golden ule a e exp essions o he case o diaba ic pa abolas wi h di e - en cu a u es. No ice ha we a e no assuming ha he un- neling ansi ions a e exac ly localized a he c ossing poin s o he pa abolas as i is assumed in he so-called con ac app oxima ion.15,18 Ac ually, ou exp essions e lec he de- localiza ion induced by he sol en dynamics. This delocal- iza ion migh be impo an . As we ha e p e iously shown,18 de ailed compa ison o he esul s ob ained wi h he a e o - mulas, wi h and wi hou he con ac app oxima ion, and he esul s o he nume ical solu ion o he Zusman equa ions indica es ha he a e exp essions wi h he con ac app oxi- ma ion become in alid o la ge alues o he ene gy bias, ⑀ 0. Fo he case o equal cu a u es, ␭1⫽␭2⫽␭, Eq. 共40兲 educes o he known esul 21 kNA (j)⫽⌬2 2ប2 冕 0 ⬁d exp 冋 2␭kBT ␶ 2 ប2 冉 1⫺e⫺ / ␶ ⫺ ␶ 冊 册 ⫻cos 冋 ␭ ␶ ប共1⫺e⫺ / ␶ 兲⫹共⫺1兲j ⑀ 0 ប 册 .共43兲 In his case, he ime in eg al can be e alua ed explici ly in e ms o he comple e 关⌫(x)兴and incomple e 关⌫(x,y)兴 Gamma unc ions26 as kNA (j)⫽⌬2 ␶ 2ប2Re 兵 ebb⫺aj关⌫共aj兲⫺⌫共aj,b兲兴 其 ,共44兲 whe e we ha e in oduced he dimensionless pa ame e s aj ⫽ ␶ 关2␭kBT ␶ /ប⫺(⫺1)ji ⑀ 0兴/បand b⫽ ␶ ␭关2kBT ␶ /ប⫹i兴/ប. By he ela ion ␥ (a,x)ª⌫(a)⫺⌫(a,x)⫽a⫺1xae⫺xM(1,1 ⫹a,x),27 whe e M(a,b,c) is he Kumme ’s unc ion, ou Eq. 共44兲is equi alen o Eq. 共3.13兲in Re . 19. I should be no iced ha , as F an suzo 19 has poin ed ou , Eq. 共43兲can lead o nonphysical p edic ions, such as iola ion o he de ailed balance and nega i e alues o he a es in s ongly pola sol en s wi h la ge eo ganiza ion en- e gies. As we discuss in Sec. VI, in he case o di e en eo ganiza ion ene gies and ⑀ 0⬎ ⑀ c, we ha e also obse ed de ia ions be ween he alues o he long- ime popula ions p edic ed by Eqs. 共32兲and 共40兲–共42兲and hose expec ed om s a is ical he modynamical conside a ions in he semi- classical limi . These anomalous esul s, which a e in insic o he Zusman equa ions, can be used as a nume ical c i e- ion in o de o es he alidi y o he Zusman desc ip ion o ET eac ions. The exp essions o he nonadiaba ic a e cons an s 共40兲–共42兲关o Eq. 共43兲in he case o equal cu a u es兴sim- pli y conside ably i one makes use o he con ac app oxi- ma ion. Wi hin his app oxima ion, one assumes ha he elec onic ansi ions ake place p ecisely a he c ossing poin s, so ha , he unc ion K ˜ (x,0) in Eq. 共29兲can be ap- p oxima ed by K ˜ 共x,0兲⯝ ␲ ⌬2 2ប ␦ 关V1共x兲⫺V2共x兲兴.共45兲 Then, he nonadiaba ic a e cons an , o ⑀ 0⬍ ⑀ c, can be ex- p essed as15,18 kNA (1)⫽⌬2 4ប 冑 ␲ kBT␭2共1⫺ ⑀ 0/ ⑀ c兲 再 exp 冋 ⫺共␭2⫺ ⑀ 0兲2 4␭⫹共 ⑀ 0兲kBT 册 ⫹exp 冋 ⫺共␭2⫺ ⑀ 0兲2 4␭⫺共 ⑀ 0兲kBT 册 冎 ,共46兲 kNA (2)⫽ 冑 ␭2 ␭1exp 冉 ⫺ ⑀ 0 kBT 冊 kNA (1) ,共47兲 whe e we ha e de ined he auxilia y, bias-dependen quan i- ies ␭⫾共 ⑀ 0兲⫽关␭2⫾ 冑 共1⫺ ⑀ 0/ ⑀ c兲␭1␭2兴2 4␭1.共48兲 This con ac app oxima ion plays an essen ial ole in Tang’s analysis o he Zusman equa ions.15 In he case o equal cu - a u es, he nonadiaba ic a e cons an s in Eqs. 共46兲and 共47兲 educe o he celeb a ed Ma cus–Le ich–Dogonadze a e3,5 and, he e o e, hey can be conside ed as i s na u al gene ali- za ion o he case o di e en cu a u es. B. The consecu i e s ep app oxima ion In o de o go beyond he nonadiaba ic limi , we need o e alua e he se ies in Eq. 共28兲. We will do his by ex ending he consecu i e s ep app oxima ion20,21 o he case o di e - 296 J. Chem. Phys., Vol. 118, No. 1, 1 Janua y 2003 Casado-Pascual e al. en eo ganiza ion ene gies o he o wa d and backwa d eac ions. In his app oxima ion, he e ms o he se ies a e simpli ied a e disen angling he dynamical e ec s associ- a ed wi h di usion om hose elying on unneling. We will assume ha he unc ion J ˜ (x,0 兩 x⬙) in Eq. 共26兲 a ies in x⬙ wi h a cha ac e is ic scale much la ge han he wid h o he in e al a ound x⬘whe e G ˜ od(x⬙,0 兩 x⬘) di e s app eciably om ze o. Then, acco ding o Eqs. 共22兲and 共25兲, one can app oxima e A ˜ 共x,0 兩 x⬘兲⬇g共x兲J ˜ 共x,0 兩 x⬘兲K ˜ 共x⬘,0兲.共49兲 Wi h his simpli ied exp ession o A ˜ (x,0 兩 x⬘), he exac ex- p ession in Eq. 共30兲 o he e ms k ˜ (n)(0) in he se ies expan- sion can be app oxima ed by k ˜ (n)共0兲⬇共⫺1兲n⫺1 冕 ⫺⬁ ⬁dx1... 冕 ⫺⬁ ⬁dxnK ˜ 共xn,0兲g共xn兲 ⫻ 兿 j⫽2 n T 关K ˜ 共xj⫺1,0兲g共xj⫺1兲J ˜ 共xj⫺1,0 兩 xj兲兴共50兲 o n⭓2. Hence o h, we will assume ha he e a e wo c ossing poin s, i.e., ⑀ 0⬍ ⑀ c. Then, as i can be checked by nume ical in eg a ion o Eq. 共23兲wi h s⫽0, he unc ion K ˜ (x,0) shows peaks o simila heigh s and wid hs cen e ed a he c ossing poin s, a leas when hey a e well sepa a ed. Assuming ha he cha ac e is ic scale o a ia ion o he unc ions J ˜ (x,0 兩 x⬘) and g(x) a e also much la ge han he wid hs o hose peaks, we inally ob ain ha , o n⭓2, he ma ices k ˜ (n)(0) can be well app oxima ed by k ˜ (n)共0兲⬇ ␯ (n)k ˜ (1)共0兲,共51兲 whe e ␯ (n)⫽共⫺1兲n⫺1兺 j1⫽1 2 ••• 兺 jn⫽1 2 j1••• jn ⫻ 兿 l⫽2 n T 关k ˜ (1)共0兲J ˜ 共xjl⫺1 *,0 兩 xjl *兲兴.共52兲 In he abo e exp ession, xj * ep esen s he coo dina e o he j h c ossing poin gi en by Eq. 共9兲, and he coe icien s j⫽g1共xj *兲 g1共x1 *兲⫹g1共x2 *兲 ⫽g2共xj *兲 g2共x1 *兲⫹g2共x2 *兲共53兲 deno e he equilib ium weigh s o he wo c ossing poin s con ibu ions. Then, acco ding o Eq. 共28兲, we ind ha k ˜ 共0兲⬇ ␯ k ˜ (1)共0兲,共54兲 whe e ␯ is he esul o summing up he se ies ␯ ⫽兺 n⫽1 ⬁ ␯ (n),共55兲 wi h ␯ (1)⫽1. In Appendix B we ca y ou he summa ion o his se ies explici ly 关c . Eq. 共B7兲兴. The aces appea ing in Eq. 共B7兲共 he coe icien s ⌶l,m) can be exp essed in e ms o he nonadiaba ic a e cons an s, kNA (j),as T 关k ˜ (1)共0兲J ˜ 共xl *,0 兩 xm *兲兴⫽兺 n⫽1 2kNA (n) kDlm (n).共56兲 He e, acco ding o Eqs. 共26兲and 共A18兲, we ha e de ined he coe icien s kDlm (j)as 1 kDlm (j)ªlim s→0⫹ J ˜ jj共xl *,s 兩 xm *兲⫽ ␶ j 冕 0 ⬁dz 再 exp 冋 ␭j 2kBT 冉 共yl⫹ym⫺2 ␦ j,2兲2 ez⫹1⫺共yl⫺ym兲2 ez⫺1 冊 册 共1⫺e⫺2z兲1/2 ⫺1 冎 ,共57兲 whe e we ha e exp essed he ime in eg al in dimensionless uni s, and we ha e in oduced he dimensionless coo dina es o he c ossing poin s yl⫽xl */x0. The coe icien s kDlm (j)a ise om he di usional dynamics along he diaba ic su aces. The diagonal e ms kDll (j)can be exp essed h ough he gene - alized hype geome ic unc ions,26 2F2(a,b;c,d;z), as21,18 1 kDll (j)⫽ ␶ j 冋 ln2⫹ 2Eal (j) kBT2F2 冉 1,1; 3 2,2; Eal (j) kBT 冊 册 ,共58兲 whe e Eal (j)a e he ac i a ion ene gies measu ed om he bo om o he diaba ic po en ial Vj(x) o he c ossing poin xl *, i.e., Eal (j)⫽Vj(xl *)⫹ ⑀ 0 ␦ j,2 . Finally, aking in o accoun Eqs. 共54兲,共56兲, and 共B7兲,we can conclude ha , in he consecu i e s ep app oxima ion 共CSA兲, he ma ix elemen s o k ˜ (0) can be app oxima ed as k ˜ jj共0兲⬇kCSA (j) ª 1⫹ 1 2兺 n⫽1 2 兺 l⫽1 2 兺 m⫽1 2 共⫺1兲l⫹mkNA (n) kDlm (n) 兿 l⫽1 2 冋 1⫹ l兺 n⫽1 2kNA (n) kDll (n) 册 ⫺ 1 2 冋 兺 n⫽1 2kNA (n) kD12 (n) 册 2kNA (j). 共59兲 The exp essions o he long- ime popula ions and he a e cons an in he consecu i e s ep app oxima ion, Pj (CSA)(⬁) and ⌫CSA , a e ob ained by eplacing k ˜ jj(0) wi h kCSA (j)in Eqs. 共32兲and 共38兲. No ice ha he equilib ium popula ions in he consecu i e s ep app oxima ion coincide wi h hose ob ained 297J. Chem. Phys., Vol. 118, No. 1, 1 Janua y 2003 Elec on ans e wi h di e en eo ganiza ion ene gies wi hin he nonadiaba ic limi , namely, Pj (CSA)(⬁) ⫽Pj (NA)(⬁). This can be easily seen a e subs i u ion o Eq. 共59兲in o Eq. 共32兲. As we ha e p e iously analyzed,18 i he c ossing poin x2 *is much highe in ene gy han x1 *, hen one can eplace in Eq. 共59兲 1⇒1 and 2⇒0. In his case, exp ession 共59兲sim- pli ies conside ably o kCSA (j)⬇kNA (j) 1⫹kNA (1)/kD11 (1) ⫹kNA (2)/kD11 (2) .共60兲 The abo e o mula has he same s uc u e as he one used in he li e a u e o equal cu a u es.20,21 Equa ion 共59兲and i s simpli ied e sion, Eq. 共60兲, desc ibe in a uni ied way he di e en a e egimes, anging om nonadiaba ic o sol en con olled adiaba ic eac ions, depending upon he ela i e alues o he sys em pa ame e s cha ac e izing unneling and di usion. The de i a ion o Eq. 共59兲is one o he main esul s o his pape , and as a as we know, i has ne e been ob ained be o e. A ew yea s ago, Tang15 a i ed o an exp ession somewha simila o Eq. 共59兲. A de ailed compa ison o ou wo k and ha o Tang e eals, none heless, some impo an di e ences. Fi s , Tang neglec s he o -diagonal di usion e ms, kD12 (j), p esen in ou Eq. 共59兲. Second, he nonadia- ba ic a e cons an s appea ing in Tang’s exp ession a e he ones ob ained wi hin he con ac app oxima ion 关c . Eqs. 共46兲–共48兲兴. VI. COMPARISON WITH NUMERICAL RESULTS AND DISCUSSION In his sec ion we shall compa e ou analy ical esul s wi h hose p o ided by nume ical in eg a ion o he Zusman equa ions. The la e has been ca ied ou using he s anda d nume ical algo i hm g oup ou ine D03PCF on a LINUX PC wi h an In el 800 MHz p ocesso . In he nume ical p oce- du e, a i icial abso bing bounda y condi ions ha e been p ope ly supe imposed a away om he eac ion egion, in o de o model he na u al bounda y condi ions, ␳ ij(x, ) →0a x→⫾⬁. Such a modeling did no a ec he quali y o he nume ics, which was con olled by he nume ical conse - a ion o he o al p obabili y P1( )⫹P2( )⫽1 on he whole ime scale. Namely, he de ia ion o he o al p obabili y om uni y did no exceed 3⫻10⫺7 o he mesh o 1500 space poin s and he single ime s ep accu acy pa ame e o 10⫺7. We ha e adjus ed bo h he numbe o mesh poin s and he ime accu acy in o de o achie e con e gence o he esul s wi hin he wid h o he plo ed cu es. Depending upon he alues o he pa ame e s, he calcula ion o a elax- a ion cu e in ol ing 100 ime poin s ook om abou se - e al seconds o abou hal an hou . The long- ime popula ion, P1(⬁), and he a e cons an , ⌫, ha e been ex ac ed om he nume ical P1( ) making use o a nonlinea , single- exponen ial i ing p ocedu e in GNUPLOT. The ollowing se o pa ame e alues is kep ixed in he calcula ions: ␭1⫽800 cm⫺1,␭2⫽200 cm⫺1,T⫽300 K. The o he pa ame e s, ⑀ 0,⌬, and ␶ , ha e been a ied. S ongly di e en alues o he eo ganiza ion ene gies ␭1and ␭2 ha e been chosen on pu pose, in o de o demons a e he quali y o ou analy ical esul s. In ealis ic si ua ions, he di e ence be ween he eo ganiza ion ene gies may no be so d ama ically la ge. Fo example, i was ound in Re . 10 ha he eac ion o p ima y cha ge sepa a ion in he bac e ial pho osyn he ic cen e imme sed in a nonpola lipid mem- b ane occu s wi h ␭1⬇1.45 kcal/mol⬇507.22 cm⫺1,␭2 ⬇1.55 kcal/mol⬇542.20 cm⫺1. In such a case, we expec ou app oxima e esul s o wo k e en be e o a simila se o he emaining pa ame e s. In Fig. 2 we show wo ypical nume ical e olu ions o P1( ) and hei co esponding single-exponen ial i ing cu es o a ixed alue ⌬⫽20 cm⫺1and wo di e en alues o he elaxa ion ime 共a兲 ␶ ⫽0.2 ps and 共b兲 ␶ ⫽2 ps. Figu e 2共a兲demons a es ha he e olu ion is single exponen ial o a e y good deg ee. The inc ease o ␶ by one o de o magni- ude 关c . Fig. 2共b兲兴 in oduces isible de ia ions om he s ic ly exponen ial beha io . In he ollowing, we es ic ou analysis o he case ⌬⭐10 cm⫺1and ␶ ⭐2.5 ps in o de o ensu e he s ic ly exponen ial cha ac e o he e olu ion. In Figs. 3, 4, and 5 we depic he nume ical and analy i- cal esul s o a ixed alue ␶ ⫽1 ps and h ee di e en alues o he unneling ma ix elemen , ⌬⫽1, 5, and 10 cm⫺1, e- spec i ely. In he h ee igu es we ind an excellen ag ee- men be ween he nume ics and ou analy ical heo y. In pa - icula , o weak unneling, ⌬⫽1cm ⫺1, he ans e is nonadiaba ic and he nume ical ans e a e ⌫is pe ec ly ep oduced by he nonadiaba ic a e exp ession, Eqs. 共38兲 and 共40兲–共42兲, in he whole ange o he elec onic ene gy bias ⑀ 0关c . Fig. 3共a兲兴. When ⌬inc eases, he nonadiaba ic a e exp ession s a s o ail 关c . Figs. 4共a兲and 5共a兲兴, espe- cially in he icini y o he decoupling poin ⑀ 0⫽ ⑀ co he wo diaba ic ene gy su aces ( ⑀ c⬇266 cm⫺1 o he p esen pa ame e s兲. He e, he adiaba ic co ec ions due o he slug- gish dynamics o he eac ion coo dina e become inc eas- ingly impo an as he nonadiaba ic unneling ge s d as ically accele a ed. Howe e , he nume ical esul s a e s ill p e y well ep oduced by he consecu i e s ep a e gi en in Eqs. 共38兲,共59兲,共40兲–共42兲,共53兲, and 共57兲. This ag eemen holds only in he ange ⑀ 0⬍ ⑀ c, since o ⑀ 0⭓ ⑀ c he consecu i e FIG. 2. Compa ison be ween he nume ical esul s o he e olu ion o he dono popula ion, P1( ), 共solid lines兲and hei single-exponen ial i ing cu es 共dashed lines兲 o wo di e en alues o he elaxa ion ime ␶ . The pa ame e alues a e ␭1⫽800 cm⫺1,␭2⫽200 cm⫺1,⌬⫽20 cm⫺1,T ⫽300 K, 共a兲 ␶ ⫽0.2 ps and 共b兲 ␶ ⫽2ps. 298 J. Chem. Phys., Vol. 118, No. 1, 1 Janua y 2003 Casado-Pascual e al.