Dissipative two-electron transfer: A numerical renormalization group study

Transcription

1 Dissipative two-electro trasfer: A umerical reormalizatio group study Sabie Torow, 1,2 Ralf Bulla, 1,3 Frithjof B. Aders, 4 ad Abraham Nitza 5 1 Theoretische Physik III, Elektroische Korrelatioe ud Magetismus, Istitut für Physik, Uiversität Augsburg, Augsburg, Germay 2 Istitut für Mathematische Physik, TU Brauschweig, Brauschweig, Germay 3 Istitut für Theoretische Physik, Uiversität zu Köl, Köl, Germay 4 Fachbereich Physik, Uiversität Breme, Breme, Germay 5 School of Chemistry, The Sackler Faculty of Exact Scieces, Tel Aviv Uiversity, Tel Aviv 69978, Israel Received 19 March 2008; published 18 July 2008 We ivestigate oequilibrium two-electro trasfer i a model redox system represeted by a two-site exteded Hubbard model ad embedded i a dissipative eviromet. The ifluece of the electro-electro iteractios ad the couplig to a dissipative bosoic bath o the electro trasfer is studied i differet temperature regimes. At high temperatures, Marcus trasfer rates are evaluated, ad at low temperatures, we calculate equilibrium ad oequilibrium populatio probabilities of the door ad acceptor with the operturbative umerical reormalizatio group approach. We obtai the oequilibrium dyamics of the system prepared i a iitial state of two electros at the door site ad idetify coditios uder which the electro trasfer ivolves oe cocerted two-electro step or two sequetial -electro steps. The rates of the sequetial trasfer deped omootoically o the differece betwee the itersite ad o-site Coulomb iteractio, which become reormalized i the presece of the bosoic bath. If this differece is much larger tha the hoppig matrix elemet, the temperature as well as the reorgaizatio eergy, simultaeous trasfer of both electros betwee door ad acceptor ca be observed. DOI: /PhysRevB PACS umber s : a, e, J I. INTRODUCTION Electro trasfer is a key process i chemistry, physics, ad biology 1 4 ecoutered i, e.g., chemical redox processes, charge trasfer i semicoductors, ad the primary steps of photosythesis. I codesed polar eviromets the process ivolves strog couplig to the uderlyig uclear motio ad is usually domiated by the uclear reorgaizatio that accompaies the charge rearragemet. A quatummechaical descriptio of electro trasfer i such a dissipative eviromet is give by the spi-boso model 5,6 ad its variats; this model accouts for the essetial eergetics ad dyamics of the process, such as the omootoic depedece of the trasfer rate o the eergy asymmetry ad the eergy differece betwee the iitial ad fial electroic states. Although stadard descriptios of such processes focus o -electro trasfer, 1,4 6 two-electro trasfer has bee suggested as the domiat mechaism i some bioeergetic processes that occur i proteis, 7,8 trasfer i trasitio-metal complexes, 9,10 electrode reactios, 11 artificial photosythesis ad photoiduced eergy- ad electrotrasfer processes, 12 biological electro-trasfer chais, 13 trasfer i fuel cells, 14 ad i DNA. 15 Further examples are self-exchage reactios such as Tl I /Tl III ad Pt II /Pt IV Ref. 16 ad electro- tuelig i molecular electroic devices. The theoretical descriptio of two-electro-trasfer dyamics differs fudametally from its -electro couterpart. More tha two states have to be cosidered 20,21 ad electro correlatios iduced by the Coulomb repulsio ad the couplig to the eviromet eed to be accouted for. Usually, the o-site Coulomb iteractio i molecules is much larger tha the itersite iteractio However, due to the polarizatio of the local eviromet, the short-rage iteractio may be strogly screeed. The, the itersite iteractio V ca be of the same order or eve exceed the o-site Coulomb iteractio U. 24,25 While U favors a homogeeous charge distributio, the itersite iteractio V iclies spatially ihomogeeous charge accumulatio. Sice the oequilibrium dyamics is govered by the eergy differece U V, the competitio betwee both iteractios strogly iflueces the type of charge-trasfer dyamics. Depedig o the sig of the eergy differece a cocerted two-electro step or two sequetial -electro steps may occur. I this paper, we cosider a system comprised of a door D ad a acceptor A site. They share two electros, which are coupled to a oiteractig bosoic bath. Such a door-acceptor system has four differet states: two doubly occupied door D 2 ad acceptor DA 2 states ad two degeerate states D with oe electro each o the door ad acceptor site with differet spi. Their eergy differece depeds o the differece betwee o-site ad itersite Coulomb repulsio as well as the bias, which we do ot cosider here. The trasitio D 2 DA 2 occurs as a cocerted trasfer of two electros or a ucorrelated sequece of oe-electro-trasfer evets durig which the itermediate D is formed. The trasfer rate of each electro may be differet ad shows a omootoic behavior o the eergy asymmetry betwee the states. I this paper, we are mappig coditios uder which the system performs cocerted twoelectro trasfer or a sequetial -electro process. To this ed we study the oequilibrium dyamics of the dooracceptor system iitially prepared with two electros at the door site. We evaluate the rates for -electro trasitios ad a electro- trasfer i differet regimes of the /2008/78 3 / The America Physical Society

2 TORNOW et al. Coulomb repulsio ad evirometal respose. The occurrece of such a correlated electro- trasfer ca be already uderstood withi a door-acceptor system, decoupled from the eviromet i which the strog o-site Coulomb repulsio exceeds cosiderably the itersite repulsio. We start from a doubly occupied excited door state D 2 compared to the D groud states. Eergy coservatio implies a cocerted electro trasfer. If the trasfermatrix elemet is much smaller tha this eergy differece the trasfer occurs as a tuelig process of a electro i which the itermediate states D are occupied oly virtually aalogous to a superexchage process see, e.g., Ref. 1. I the preset paper, we ivestigate the effect of couplig to a dissipative bosoic eviromet with a total umber of two electros occupyig door ad acceptor sites. These two electros experiece the o-site Coulomb repulsio U whe occupyig the same site ad the Coulomb repulsio V whe occupyig differet sites. I this paper, we restrict ourselves to the simplest case where door ad acceptor are each modeled by a molecular orbital. I such a system the differece Ũ=U V is crucial for the dyamics. The couplig to the bosoic bath has two major effects: i the reormalizatio 26,27 of the o-site Coulomb repulsio Ũ to Ũ eff ad ii dephasig as well as dissipatio of the eergy from the door-acceptor system to the bath. The latter leads to the dampig of coheret oscillatios that would otherwise exist betwee the quatum states of the related molecule ad, beyod a characteristic couplig stregth, to icoheret dyamics of the electro-trasfer process. These cosideratios lead us to a dissipative two-site Hubbard model, a miimal model that captures the essetial physics comprisig correlatios betwee electros ad their couplig to the dissipative eviromet. It is discussed i detail i Sec. II. For a compariso to experimetal results, it has to be supplemeted by ab iitio calculatios of the parameters. The equilibrium properties of the model have bee previously studied 28 usig the umerical reormalizatio group NRG, ad the real-time dyamics has bee ivestigated 29,30 usig a Mote Carlo techique at high temperatures where oly icoheret trasfer is preset. I these Mote Carlo calculatios, the effective Coulomb iteractio was chose to be Ũ eff 0 ad o electro- trasfer has bee reported. Two-electro trasfer i a classical bath has bee discussed i Ref. 20 i the framework of three parabolic potetial surfaces for the four states D 2, D, ad DA 2 as a fuctio of a reactio coordiate. A geeralizatio to door-bridge acceptor systems is give i Refs. 8 ad 31. Although two-electro trasfer was observed i some regimes of system parameters i the high-temperature limit, cosiderig a classical bath, it seems reasoable to expect that, at least betwee idetical ceters, electro- tuelig processes are particularly importat at temperatures correspodig to eergies smaller tha the effective eergy differece betwee iitial ad itermediate states Ũ eff. At these temperatures -electro trasfer caot be activated see Sec. VI. Therefore, we focus o the low-temperature regime where the trasfer is domiated by uclear tuelig ad where the bosoic bath has to be treated quatum mechaically. Due to the uclear tuelig the electro-trasfer rate is costat over a wide temperature rage from zero temperature up to temperatures where thermal activatio becomes more importat. 32 I this low-temperature regime, we employ the time-depedet NRG TD-NRG, which covers the whole parameter space from weak to strog dissipatio. The NRG is a accurate approach to calculate thermodyamics ad dyamical properties of quatum impurity models For further details of the NRG, we refer to the recet review 39 o this method. The paper is orgaized as follows: I Sec. II we itroduce the model. Its high-temperature behavior obtaied from the Marcus theory is described i Sec. III. Sectio IV itroduces the NRG method, its extesio to oequilibrium, ad its applicatio to the preset problem. I order to gai a better uderstadig of the oequilibrium dyamics preseted i Sec. VI, we summarize the equilibrium properties of the model i Sec. V. We preset a detailed discussio of the real-time dyamics i Sec. VI. Therei, we focus o the time evolutio of occupatio probabilities of the differet electroic states as the key observables. I particular, whe the dyamics ca be described i terms of rate processes, the depedece of the ad electro- rate o the Coulomb repulsio parameters is aalyzed. A summary of our results is give i Sec. VII. II. MODEL We cosider a model of a two-electro/two-site system coupled to a bosoic bath. It is defied by the Hamiltoia with ad H el =,i=a,d i c i H = H el + H coupl + H b, c i c D c A + c A c D + U c i c i c i c i + V i=a,d 2,,i,j=A,D H coupl =,i=a,d i j g i c i c i 2 b + b, c i c i c j c j, H b = b b, where c i ad c i deote aihilatio ad creatio operators for fermios with spi o the door i=d ad acceptor i=a sites. The Hamiltoia H el correspods to a exteded two-site Hubbard model, with o-site eergies i, hoppig matrix elemet, o-site Coulomb repulsio U, ad a itersite Coulomb repulsio V betwee oe electro o the door ad oe electro o the acceptor. The differece Ũ =U V measures the excess eergy eeded to move a electro betwee the two sites. Such a two-site Hubbard model without couplig to a bosoic bath has bee ivestigated i the cotext of electro trasfer i Ref

3 DISSIPATIVE TWO-ELECTRON TRANSFER: A 0, 0,,, The Hamiltoia H b models the free bosoic bath with boso creatio ad aihilatio operators b ad b, respectively. The electro-boso couplig term, H coupl, has the stadard polaro form with the couplig costat for door ad acceptor give by g D ad g A, respectively. I what follows we set D = A = 2 ad g A= g D =1. The latter choice implies that the polar bath is coupled to the chage i the electroic desity c A c A c D c D, H coupl = c A c A c D c D 2 b + b. 2 This two-site electro-boso Hamiltoia coserves the umber of electros i c i c i ad the square of the total spi S 2 as well as its z compoet S z. The Hilbert space ca therefore be divided ito differet subspaces. I the subspace with oe electro ad S z =1/2, the model is equivalet to the spiboso model. 28 Here, we cosider the subspace with two electros ad S z =0, which is spaed by the states 1 =,0, 2 =,, 3 =,, ad 4 = 0, with the otatio A,D describig the occupatio at the door D ad acceptor A sites. The four-dimesioal basis i the twoelectro subspace is displayed i Fig. 1. We defie the followig observables: dˆ D = 1 1, dˆ A = 4 4, ˆ AB = , which measure the doubly occupacy dˆ D dˆ A o the door acceptor site ad ˆ DA the combied populatio of the states, ad,. Note that i some works the states,0 ad 0, are referred to as localized states. 29 We call them doubly occupied states, while the term localizatio is used below for the self-trappig mechaism. Cosider the 4 4 Hamiltoia matrix i the electroic subspace M ij = i H j i, j=1,...,4. Itroducig the otatio Bosoic Bath ~ U eff FIG. 1. The four states of model Eq. 1 for the symmetric case =0. The eergy differece betwee the doubly occupied door D 2 or acceptor DA 2 ad sigly occupied dooracceptor D depeds o the effective reormalized iteractio Ũ eff defied i Eq Ŷ = b b, Xˆ = b + b, ad shiftig the Hamiltoia by a costat V leads to 4 + Ũ + Xˆ + Ŷ 0 Ŷ Ŷ 0 + Ũ Xˆ + Ŷ, with Ũ=U V. Therefore, the dyamics of the system is govered by the eergy differece Ũ, which replaces the o-site Coulomb repulsio U. If screeig of the local Coulomb repulsio U 24,25 is sufficietly large, Ũ chages its sig ad become effectively attractive. A large itersite Coulomb repulsio V favors a ihomogeeous charge distributio. It is coveiet to rewrite the diagoal matrix elemets of the doubly occupied states i the form ad 1 H 1 = + Ũ eff + b + + b, 4 H 4 = + Ũ eff + b b. Compared with the matrix elemets of states correspodig to D, 2 H 2 = 3 H 3 = b b, 8 we ca easily see that the electro-boso couplig geerates a effective reormalized iteractio, Ũ eff = Ũ 2. The reormalized iteractio Ũ eff determies the eergy differece betwee D 2 DA 2 ad D ad costitutes the oly Coulomb iteractio parameter i the preset model. The reormalizatio stems from a boso-iduced effective electro-electro iteractio, already familiar from the Holstei model. 26 Note that a artificial eergy shift is preset i the -electro subspace spi-boso model ; 6 however, the two states ad are shifted i the same directio, which ca be hadled by resettig the zero of eergy. I aalogy to the spi-boso model, 5,6 the couplig of the electros to the bath degrees of freedom is completely specified by the bath spectral fuctio, J = The spectral fuctio characterizes the bath ad the system-bath couplig ad ca be related to the classical reorgaizatio eergy 6 classical i terms of boso degrees of freedom, which measures the eergy relaxatio that follows a sudde electroic trasitio. The oe-electro trasfer ad

4 TORNOW et al. the correlated two-electro trasfer are associated with reorgaizatio eergies E 1 ad E 2, respectively. For a electro trasfer, e.g., D 2 D the reorgaizatio eergy E 1 is give by 6 E 1 = 2 = 0 d J, 11 ad the correspodig eergy for a correlated two-electro trasfer D 2 DA 2 is E 2 =4E 1. The model we are cosiderig here is completely specified by the parameters,, Ũ, ad ad the bosoic spectral fuctio. I the molecular electro-trasfer problem, the latter fuctio reflects itramolecular vibratios ad the solvet e.g., water or protei or eviromet. Its solvet compoet ca be estimated from the solvet dielectric properties or a classical molecular-dyamics simulatio. I the preset paper, we assume a Ohmic bath model, = J 2 0 c 12 0 otherwise with a cutoff at eergy c. This choice yields the reorgaizatio eergy E 1 =2 c ad the eergy shift Ũ eff =Ũ 2 c. All parameters ad physical quatities are defied i uits of c. Its order of magitude for the itermolecular mode spectrum of a polar solvet is 0.1 ev. III. HIGH-TEMPERATURE LIMIT: MARCUS THEORY I the high-temperature limit, electro trasfer is usually described usig Marcus theory 4 as a rate process withi classical trasitio state theory. Extesios that take ito accout the quatum ature of the uclear motio i the weak electroic couplig limit the so-called oadiabatic limit are also available; 4 however, for simplicity we limit ourselves i what follows to the classical Marcus descriptio. The Marcus electro trasfer rate ca be evaluated for ay amout of trasferred charge: the latter just determies the reormalized potetial surface parameters that eter the rate expressio. Sigle-electro trasitio rates are give by k D 2 D 2 e + Ũ eff E 1 2 /4E 1 T, k D D 2 2 e + Ũ eff + E 1 2 /4E 1 T, k DA 2 D 2 e + Ũ eff E 1 2 /4E 1 T, k D DA 2 2 e + Ũ eff + E 1 2 /4E 1 T I the case U eff secod-order processes are possible that ivolve oly virtual occupatios of the states D, leadig to rates for a electro, k D 2 DA e 2 E 2 2 /4E 2 T, 17 Ũ eff ~ U eff k DA 2 D e 2 + E 2 2 /4E 2 T. 18 Ũ eff The iterplay betwee sequetial ad cocerted two-electro trasfer i the limit of a classical bath with a mode or a reactio coordiate ca be see from these expressios. I the followig we restrict ourselves to the symmetric case =0. Startig with the iitial state D 2, we expect cocerted two-electro trasfer i the Marcus regime whe the rate k D 2 DA 2 is larger tha the rate k D 2 D of the first step of the sequetial process, which is the case whe Ũ eff T ad Ũ eff E 1 as well as E 1 T. I a parameter regio where sequetial trasfer domiates, the rates k D 2 D ad k D DA 2 as well as the correspodig backward rates show a omootoic behavior ad a iverted regime depedet o the effective Coulomb iteractio Ũ eff see Fig. 2. For icoheret trasfer processes which may happe at large temperatures or for a strog couplig to the bosoic bath, a descriptio of the populatio dyamics by kietic equatios determied by the rates is give by ḋ D t = k D 2 D + k D D 2 ṅ DA t = k D DA 2 +2k D 2 D ḋ A t = k DA 2 D + k D 2 DA 2 d D t DA t + k DA 2 D 2 d A t, + k D D 2 DA t d D t +2k DA 2 D d A t, + k DA 2 D 2 d A t + k D DA 2 DA t + k D 2 DA 2 d D t, 19 where d D ad d A are the probabilities to have two electros o the door ad acceptor, respectively. DA is the combied populatio of the states, ad,. For the iitial coditio d D t=0 =1, we obtai, =,. For the ubiased Hamiltoia =0, k DA 2 D =k D 2 D ad must hold. k D DA 2 =k D D 2 W(x) DA 2- DA 2- D - A - FIG. 2. Color olie Potetial surfaces for the differet states of the model i the Marcus theory for Ũ eff 0, =0. The miima of the states, ad, D are set to the origi while those parabolas that correspod to the doubly occupied states,0 D 2 ad 0, DA 2 are shifted. Note that i the case displayed here the trasfer D 2 D is i the iverted regime. x

5 DISSIPATIVE TWO-ELECTRON TRANSFER: A These kietic equatios ca be solved i the hightemperature regime usig the Marcus rates from above. I Sec. VI, we have used these equatios to extract the lowtemperature trasitio rates by fittig the oequilibrium dyamics of d D t, d A t, ad DA t calculated i the icoheret regime with the time-depedet NRG. For t the equilibrium states d A eq, d D eq, ad DA eq are reached, where d D eq = d A eq. It follows that d D eq DA eq = k D DA 2 k DA 2 D, which is accordig to the Marcus rates d D eq DA eq =e Ũ eff/t. Therefore, i the classical limit we arrive at d D cl 0.5 eq =. 20 e Ũ/T +1 With the help of the kietic equatios we ca describe cocerted two-electro trasfer, a purely sequetial electro trasfer as well as a combied process which shows first a trasfer which is followed by a -electro trasfer. As log as the -electro-trasfer rates are small k D 2 D k D 2 DA 2 ad d D eq = d A eq 0.5, the state D is oly weakly populated ad DA t is costat ad close to zero. The dyamics is domiated by a electro- trasfer. The combied process is expected if d D eq = d A eq 0.5. First the populatio d A rises quickly while DA stays close to zero. Later a slow icrease i DA to its equilibrium is observed. For k D 2 D k D 2 DA 2 the trasfer is purely sequetial.... (a)... (c) J( )... c (b) 1 t t 1 t m t m m m+1 m+2 IV. LOW-TEMPERATURE LIMIT: THE NUMERICAL RENORMALIZATION GROUP At low temperature, the quatum geeralizatio of the Marcus theory replaces the classical eviromet by a bath of oiteractig bosoic degrees of freedom. Very early o, the oadiabatic weak-couplig limit was ivestigated. 41 The strog-couplig limit of such a model has bee addressed usig the oiteractig blib approximatio NIBA, 5 path-itegral methods, 6 ad recetly also the umerical reormalizatio group, which we employ i this paper. Origially the NRG was iveted by Wilso for a fermioic bath to solve the Kodo problem. 36,37 The fermioic NRG is a stadard ad very powerful tool to ivestigate complex quatum impurity problems. 39 The method was recetly exteded to treat quatum impurities coupled to a bosoic bath, 38,42 to a combiatio of fermioic ad bosoic baths, 43 ad to the calculatio of real-time dyamics out of equilibrium The operturbative NRG approach has bee successfully applied to arbitrary electro-bath couplig stregths. 38,42 44 A. Equilibrium NRG E m E m+1 E m+2 E m+3 FIG. 3. Color olie Scheme of the bosoic NRG. a The bosoic eergy cotiuum is discretized o a logarithmic mesh usig a parameter 1. Oly a bosoic mode i each iterval +1 c, c visualized by the circles couples directly to the electroic subsystem. b This discretized model is mapped exactly oto a tight-bidig chai via a uitary trasformatio Refs. 36 ad 39 : oly the first chai site couples directly to the dooracceptor system. The hoppig t betwee eighborig bosoic sites decreases expoetially with the distace from the door-acceptor system, i.e., t. The eergy spectrum of the Hamiltoia is calculated by successively applyig the reormalizatio group trasformatio 21, diagoalizig the ew Hamiltoia ad rescalig the spectrum as depicted schematically i pael c for the sequece of Hamiltoias H m to H m+3. After each iteratio oly, the N s eigestates of site m+1 with the lowest eergies are kept. This trucatio is depicted by a horizotal dashed lie. The umerical reormalizatio group achieves the separatio of eergy scales by logarithmic discretizatio of the eergy cotiuum ito itervals m+1 c, m c, m N 0, defiig the discretizatio parameter 1. Oly oe mode of each iterval couples directly to the quatum impurity, idicated by the circles i Fig. 3 a. This discrete represetatio of the cotiuum is mapped oto a semi-ifiite tight-bidig chai usig a exact uitary trasformatio. Hereby, the quatum impurity couples oly to the very first chai site as depicted i Fig. 3 b. The tight-bidig parameters t likig cosecutive sites of the chai m ad m+1 fall off expoetially as t m m. Each bosoic chai site is viewed as represetative of a eergy shell sice its eergy w m also decreases as w m m establishig a eergy hierarchy. Both esure that mode couplig ca oly occur be

6 TORNOW et al. twee eighborig eergy shells, which is essetial for the applicatio of the reormalizatio group procedure. To this ed, the reormalizatio group trasformatio R H reads H m+1 = R H m = H m + m+1 t m a m a m+1 + t m a m+1 a m + w m a m+1 a m+1, 21 where H m is the Hamiltoia of a fiite chai up to the m, as depicted i Fig. 3 b. The aihilatio creatio operators of site m are deoted by a m a m ad w m labels the eergy of the bosoic mode of site m. Note that the rescalig of the Hamiltoia H m by esures the ivariace of the eergy spectrum of fixed-poit Hamiltoias uder the RG trasformatio R H m. The RG trasformatio 21 is used to set up ad iteratively diagoalize the sequece of Hamiltoias H.Ithe first step, oly the electroic door-acceptor system couplig to the bosoic site m=0 is cosidered. It turs out to be sufficiet 38,39,42 to iclude oly the N b lowest-lyig bosoic states, where N b takes typical values of The reaso for that is quite subtle: the couplig betwee differet sites decays expoetially ad is restricted to earesteighbor couplig by costructio, both essetial for the RG procedure. I each successive step: i a fiite umber of N b bosoic states of the ext site m+1 are added, ii the Hamiltoia matrices are diagoalized, ad iii oly the lowest N s states are retaied i each iteratio. The discardig of higheergy states is justified by the Boltzmaia form of the equilibrium desity operator whe simultaeously the temperature is lowered i each iteratio step to the order T m m w c. To illustrate the procedure, the lowest-lyig eergies of the Hamiltoia H m to H m+3 are schematically depicted i pael c of Fig. 3. We typically use N b 8 ad keep about N s =100 states after each iteratio usig a discretizatio parameter =2. Deotig the set of low-lyig eigestates by r N ad the correspodig eigevalues E r N O 1 at iteratio N, the equilibrium desity matrix 0 is give 39 by 0 = 1 Z N e N E r r NN r, r 22 where Z N = r e E r N ad are of the order O 1, such that T N =w c N /. The thermodyamic expectatio value of each local observable Ô is accessible at each temperature T N by the trace Ô eq =Tr 0 Ô = 1 Z N e E r NN r Ô r N. 23 r The procedure described above turs out to be very accurate because the coupligs t m betwee the bosoic sites alog the chai are fallig off expoetially so that the rest of the semi-ifiite chai cotributes oly perturbatively 36,39 at each iteratio m, while cotributios from the discarded high-eergy states are expoetially suppressed by the Boltzma factor. B. Time depedet NRG While the equilibrium properties are fully determied by the eergy spectrum of the Hamiltoia, the oequilibrium dyamics requires two coditios: the iitial coditio ecoded i the may-body desity operator 0 ad the Hamiltoia H f that govers its time-evolutio. For a timeidepedet Hamiltoia, the desity operator evolves accordig to ˆ t 0 =e ihft 0 e ihft. All time-depedet expectatio values Ô t are give by Ô t =Tr t Ô =Tr e ihft 0 e ihft Ô. 24 We obtai the desity operator 0 from a idepedet NRG ru usig a suitable iitial Hamiltoia H i. By choosig appropriate parameters i H i, we prepare the system such that for the calculatios preseted i this paper the two electros are located o the door site ad the acceptor site is empty. I geeral, the iitial desity operator 0 cotais states, which are most likely superpositios of excited states of H f. For the calculatio of the real-time dyamics of electrotrasfer reactios, it is therefore ot sufficiet to take ito accout oly the retaied states of the Hamiltoia H f obtaied from a NRG procedure. The recetly developed TD- NRG Refs. 33 ad 34 circumvets this problem by icludig cotributios from all states. It turs out that the set of all discarded states elimiated durig the NRG procedure form a complete basis set 33,34 of the Wilso chai, which is also a approximate eigebasis of the Hamiltoia. Usig this complete basis, it was show 33,34 that Eq. 24 trasforms ito the cetral equatio of the TD-NRG for the temperature T N, N Ô t = m=0 tru r,s e i E r m m E s t O m r,s red s,r m, 25 where O m r,s = r;m Ô s;m are the matrix elemets of ay operator Ô of the electroic subsystem at iteratio m, ad E m r,e m s are the eigeeergies of the eigestates r;m ad s;m of H f m. At each iteratio m, the chai is formally partitioed ito a system part o which the Hamiltoia H m acts exclusively ad a eviromet part formed by the bosoic sites m +1 to N. Tracig out these evirometal degrees of freedom e yields the reduced desity matrix, 33,34 red m = s,e;m 0 r,e;m, s,r e 26 at iteratio m, where 0 is give by Eq. 22 usig H i. The restricted sum tru r,s i Eq. 25 implies that at least oe of the states r ad s is discarded at iteratio m. Excitatios ivolvig oly kept states cotribute at a later iteratio ad must be excluded from the sum. As a cosequece, all eergy shells m cotribute to the time evolutio: the short-time dyamics is govered by the high-eergy states, while the log-time behavior is determied by the low-lyig excitatios. Dephasig ad dissipatio are ecoded i the phase factors e i E r m m E s t as well as the reduced desity matrix red s,r m

7 DISSIPATIVE TWO-ELECTRON TRANSFER: A Discretizig the bath cotiuum will lead to fiite-size oscillatios of the real-time dyamics aroud the cotiuum solutio ad deviatios of expectatio values from the true equilibrium at log-time scales. I order to separate the uphysical fiite-size oscillatios from the true cotiuum behavior, we average over differet bath discretizatio schemes usig Oliveira s z averagig for details see Refs. 34 ad 45. We average over eight differet bath discretizatios i our calculatio. V. EQUILIBRIUM PROPERTIES I order to gai a better uderstadig of the oequilibrium dyamics preseted i Sec. VI, we briefly summarize the equilibrium properties of the model give by Eq. 1. It has bee aalyzed i Ref. 28, where self-trappig localizatio i the ad two-electro subspace was foud. We start with the phase diagram of the two-site model, as show i Fig. 4. Oly for =0 a quatum phase trasitio of Kosterlitz-Thouless type separates a localized phase for c from a delocalized phase for c. We plot the phase boudaries betwee localized ad delocalized phases i the -Ũ plae, both for - ad two-electro subspaces gray ad black lies i Fig. 4, respectively. For the -electro subspace, the Coulomb repulsio is irrelevat, ad the phase boudary does ot deped o Ũ. The value of the critical couplig stregth, c, is idetical to those of the correspodig spi-boso model. The critical value 5,42 of c depeds o the tuelig rate ad reaches c =1 for 0. The phase boudary for the two-electro subspace does deped o Ũ, which has drastic cosequeces for the electro-trasfer process. Imagie that, by a suitable choice of parameters, the system is placed betwee the two phase boudaries above the -electro gray lie ad below the two-electro phase boudary black lie i the area idicated by I i Fig. 4. The the system would be i the localized phase i the -electro subspace. However, oe additioal secod electro immediately places the system i the delocalized phase, ad oe or eve both electros ca be trasferred. Similarly, a secod electro added to the system i the parameter regime of area II shows the opposite behavior: both electros get localized although a electro could be trasferred. Note the differet values of c s eve for Ũ=0 i the ad the two-electro subspace: the couplig of the door ad/or acceptor system to the bath iduces a effective attractive Coulomb iteractio Ũ eff = 2 c betwee the electros. O the localized side of the trasitio, the electro tuelig is reormalized to zero so that a electro trasfer is clearly abset i this regime. This statemet holds oly for Ohmic dissipatio, o which we focus here; deep i the sub-ohmic regime, coheret oscillatios have bee recetly observed eve i the localized phase see Ref. 44. Figure 5 shows results for the double occupatio probability as a fuctio of the electro-bath couplig for differet Ũ calculated with the equilibrium NRG. For the symmetric model cosidered here, the equilibrium probabilities for the FIG. 4. Zero-temperature phase diagram of the model Eq. 1 for =0 ad =0.1 c. The critical dissipatio stregth c is plotted as a fuctio of Ũ i the two-particle subspace black lie ad i the -particle subspace gray lie, respectively. double occupatio o door ad acceptor sites are equal: dˆ A eq = dˆ D eq d eq usig the observables defied i Eq. 3. The probability of havig two electros at differet sites is give by ˆ DA eq =1 2 d eq. The average double occupacy d eq decreases with icreasig effective Coulomb repulsio Ũ ad icreases with icreasig. This ca uderstood i terms of the effective Coulomb iteractio Ũ eff =Ũ 2 c, reormalized due to the couplig to the bosoic bath. The delocalizatio/localizatio phase trasitio occurs whe d eq 0.5, as ca be see by comparig Figs. 4 ad 5. For Ũ eff 0 ad Ũ eff, we are able to project out the D excited states. The our model maps o a spi-boso model 2 with a effective hoppig /Ũ eff betwee the states D 2 ad DA 2. The dyamics will be govered by electro s if D 2 or DA 2 are the iitial states. The double occupacy d eq is calculated aalytically for =0 ad arbitrary ad Ũ. For T 0, d eq approaches FIG. 5. Low-temperature equilibrium probability for double occupacy of door ad acceptor d eq for =0.1 c ad =0 as a fuctio of for Ũ= c,0, ad c. I the limit of =0 the depedece of d eq o ad Ũ is give aalytically i Eq

8 TORNOW et al. (a) FIG. 6. Color olie Equilibrium probability for double occupacy of door ad acceptor d eq as a fuctio of temperature T for =0.1 c, Ũ eff =0.1 c, =0.04 circles, dashed lie ad or =0.1 c, Ũ eff = c, =0.04 squares, solid lie. For compariso the high-temperature result Eq. 20 is show for Ũ eff =0.1 c dashed lie ad Ũ eff = c solid lie. d eq = 4 2, 27 Ũ Ũ + Ũ while i the opposite limit, T, we obtai d eq The low-temperature limit 27 is icluded as ed-poits of the curves i Fig. 5. Let us ow tur to the temperature depedece of d eq. Figure 6 shows results for temperatures betwee T =0.004 c ad T=0.2 c for several choices of model parameters. Our calculatios imply a idepedet check of the correct t behavior i Sec. VI. Additioally, we ca make coectio to the high-temperature results of Sec. III. For temperatures T Ũ eff, the double occupacy d eq is costat as expected from quatum statistics but deviates drastically from the predictios of the Marcus theory give by Eq. 20. The double occupacy d D eq calculated with the NRG approaches the value 0.5/ 1+e Ũ eff/t for Ũ eff T. This result idicates that for Ũ eff T Marcus theory is ot applicable while low-temperature methods like the NRG are valid. VI. NONEQUILIBRIUM DYNAMICS We employ the time-depedet NRG to evaluate the lowtemperature time evolutio of the local occupacies usig Eq. 25 ad ivestigate the ifluece of differet Coulomb iteractios Ũ, -electro hoppig matrix elemets, coupligs betwee the electroic system to the bosoic bath, ad temperatures T betwee T= c ad T =0.125 c. The door ad/or acceptor subsystem is iitially prepared i a state with the two electros placed o the door site ad evolves accordig to Hamiltoia 1. We calculate the time-depedet expectatio values d D t = dˆ D t, d A t = dˆ A t ad DA t = ˆ DA t usig Eq. 25. These expectatio values are related at ay time by the completeess relatio d D t +d A t + DA t =1. The time evolutio of DA t serves as criterio to distiguish betwee direct two-electro trasfer ad two cosecutive oe-electro steps. If DA t (b) (c) FIG. 7. Low-temperature populatio probabilities P t =d D t thick black lie, d A t thi black lie, ad DA t gray lie as fuctios of time. The parameters are Ũ= c, =0.1 c, =0, ad T= c. The couplig icreases from the upper pael =0 Ũ eff = c, to the middle pael =0.04 Ũ eff = 1.08 c, ad to the lower pael =0.16 Ũ eff = 1.32 c. remais close to zero or stays costat throughout the electro-trasfer process, the two states D are oly virtually occupied, ad the cocerted two-electro trasfer is observed. A sigificat icrease i DA t as a fuctio of time is take as a idicatio of step-by-step -electro trasfer. I the absece of the electro-boso couplig =0, the dyamics is fully determied by the dyamics of the four eigestates of H el. I the limit Ũ, we obtai d D;A t Ũ 2 cos Ũt 1 Ũ 2 2 cos 4 2 Ũ t,

9 DISSIPATIVE TWO-ELECTRON TRANSFER: A FIG. 8. Low-temperature populatio probabilities P t =d D t thick black lie, d A t thi black lie, ad DA t gray lie as fuctios of time. The parameters are Ũ= c, =0.1 c, =0, ad T= c. The couplig to the bosoic bath icreases from pael a =0 Ũ eff = c,to pael b =0.02 Ũ eff =0.96 c, to pael c =0.52 Ũ eff = 0.04 c, ad to pael d =0.55 Ũ eff = 0.1 c. DA t 4 2 while i the limit of Ũ=0 Ũ Ũ 2 cos Ũt, d D;A t = cos 4 t 4 8 cos 2 t, DA t = cos 4 t FIG. 9. Low-temperature populatio probabilities P t =d D t black lie ad DA t gray lie as fuctios of time for =0.04 full lie ad =0.36 dashed lie. The effective eergy differece betwee the states D 2 ad D is kept costat Ũ eff = c. The other parameters are =0.1 c ad =0. A fiite value of the couplig, 0, gives rise to dampig of those coheret oscillatios. Furthermore, the Coulomb iteractio is reormalized to Ũ eff =Ũ 2 c. For Ũ eff 0, the states D 2 ad DA 2 are eergetically favored. The two itermediate states D are oly virtually occupied for Ũ eff,t, similar to the superexchage process. 1 This regime ca be described by a spi-boso model with a effective iterstate couplig eff 4 2 /Ũ eff. The spi-boso model has three dyamical regimes. 5 For smaller tha some characteristic value, it exhibits damped coheret oscillatios betwee the two states. If is larger tha this value the oscillatios disappear ad the kietics is domiated by a relaxatio process. Here, rates ca be defied ad the populatio probabilities ca be fitted with the kietic equatio 19. For a further icrease i, the electroic system shows the oset of localizatio for T 0 ad does ot evolve toward the other acceptor site. I Fig. 7 we plot the low-temperature populatio probabilities d D t, d A t ad DA t for Ũ= c, =0.1 c, ad T= c ad differet coupligs. For =0 upper pael the oscillatios have two frequecies see Eq. 28. The electro oscillates from door to acceptor with the small frequecy 4 2 /Ũ, whereas the fast oscillatios with frequecy Ũ characterize the virtual occupatio of the highlyig states D. A icrease i leads to dampig of the oscillatios middle pael ad relaxatio lower pael. At about =0.3 the electro gets self-trapped ad the system shows a phase trasitio to the localized phase at T=0 see Fig. 4. The cofiguratio D is see ot to be ivolved i the dyamics as D is very small ad without ascedig slope. Sice Ũ eff the state D caot be populated as log Ũ eff T. A more complicated behavior is expected withi the four accessible electroic states whe Ũ eff 0. I this case the delocalized states D have the lowest eergy, ad sequetial trasfer is required to reach the equilibrium state. Pair trasfer occurs o a smaller time scale. Thus, a combied ad sequetial trasfer o two differet time scales govers the dyamics for these parameters. The four paels i Fig. 8 depict the time evolutio of the occupatio probabilities d D t, d A t, ad DA t for Ũ= c

10 TORNOW et al. FIG. 10. Low-temperature populatio probabilities P t =d D t thick black lie, d A t thi black lie, ad DA t gray lie as fuctios of time. The parameters are Ũ=0, =0.1 c, =0, ad T= c. a =0 Ũ eff =0, b =0.01 Ũ eff = 0.02 c, c =0.04 Ũ eff = 0.08 c, ad d =0.1 Ũ eff = 0.2 c. ad =0.1 c ad four differet values of : =0,0.02, 0.52,0.54. The udamped coheret oscillatios of pael a decay expoetially for small dampig depicted i pael b. Icreasig further yields a fiite populatio of the states D : sequetial trasfer becomes the mai process, as show i paels c ad d. The crossover from a combied trasfer ad slow -electro trasfer pael b to purely sequetial trasfer pael c ad pael d with a complex dyamics is due to a combied effect of dissipatio ad decrease i the effective eergy differece betwee the relevat states. A eve larger leads to a egative Ũ eff ad a very slow trasfer util the oset of localizatio at c, which is ot show here. To separate the ifluece of dissipatio from the reormalizatio of Ũ due to the couplig to the bosoic bath, we plot d D t ad DA t for a costat effective Ũ eff = c ad differet couplig i Fig. 9. The dyamics chages from trasfer with a slow icrease of the occupacy at =0.04 due to the low-lyig states D to icoheret relaxatio ad sequetial trasfer for =0.36. As log as E 1 =2 c Ũ eff, trasfer is observed o a short-time scale. For E 1 Ũ eff oly oe electro is trasferred ad the system relaxes rapidly ito its equilibrium state D without ay short-time-cocerted trasfer. I Fig. 10 the evolutio of the dyamics is show for Ũ =0 ad icreasig. The doubly occupied states are the groud states of the door/acceptor system for fiite sice Ũ eff = 2 c 0. With icreasig, the amplitude of coheret oscillatios acquire a small dampig. I additio, trasfer is favored ad DA t decreases. The simple damped oscillatios are replaced by a much more complex dyamics comprisig of strogly reormalized oscillatio frequecy ad a strog dampig for =0.04. At about =0.36 ot show here the critical couplig c is reached ad the system is localized. Next we study the effect of chagig Ũ at costat system-bath couplig =0.04 Fig. 11, =0.16 Fig. 12, ad =0.36 =0.36 Fig. 13 ad =0.1 c. I the lower dampig case Fig. 11, the trasfer is reflected by damped electro- oscillatios for Ũ= c i Fig. 11 a. Icreasig Ũ= 0.5 c i Fig. 11 b leads to a icrease i the populatio probability of D ad to a chage i the fast oscillatios whit a approximate frequecy of Ũ eff. Whe Ũ becomes positive Ũ=0.5 c Fig. 11 c, the -electro trasfer becomes fast ad the mai process uless Ũ eff is ot too large. I fact at Ũ= c the rate from D 2 to D becomes smaller Fig. 11 d ad additioal electro- trasfer is observed. The graphs Figs. 11 a ad 11 d ca be uderstood i terms of Eq. 28 sice /Ũ 1 ad =0.04 is small. By the weak couplig to the eviromet, is slightly reduced, ad the oscillatio amplitude decays expoetially. The differece betwee the two paels a ad d arises from i Ũ eff =Ũ 2 c istead of the Ũ eterig Eq. 28 ad ii from the dissipatio which favors the relaxatio ito the ew thermodyamic groud state: while the oscillatio frequecies are roughly the same for Ũ = c, the delocalized states have a lower eergy i Fig. 11 d so that DA t has to icrease to its ew equilibrium value. The approximatios made i Eq. 28 do ot hold ay loger for the parameters i Figs. 11 b ad 11 c. The electroic dyamics is govered by additioal frequecies ad becomes more complex. However, the results ca still be aalyzed ad uderstood withi the aalytical results of d D t, d A t, ad DA t for 0. Whe the couplig is icreased to =0.16, a differet picture emerges. Very high-frequecy oscillatios with a small amplitude are superimposed o a slowly decayig d D t depicted i the upper pael of Fig. 12. Averagig over those oscillatios, we ca fit the populatio probabilities to the kietic equatio 19. By this procedure, we extract the

11 DISSIPATIVE TWO-ELECTRON TRANSFER: A (a) (b) FIG. 12. Upper pael: Low-temperature populatio probabilities P t =d D t thick black lie, d A t thi black lie, ad DA t gray lie as fuctios of time. Ũ= 0.9 c ad 1.5 c from bottom to top for d D as well as from top to bottom for d A. Lower pael: Electro rate k for the trasfer from D 2 DA 2 as a fuctio of Ũ eff. The parameters for both paels are =0.16, T c, =0.1 c, ad =0. Iset: Eergy levels of states D 2, D, ad DA 2. (c) (d) FIG. 11. Low-temperature populatio probabilities P t =d D t thick black lie, d A t thi black lie, ad DA t gray lie as fuctios of time. The parameters are =0.04 ad Ũ= c pael a, Ũ= 0.5 c pael b, Ũ=0.5 c pael c, ad Ũ= c pael d. pheomeological rates as a fuctio of Ũ eff for fixed =0.16. As show i the lower pael of Fig. 12 the cocerted trasfer rate k D 2 DA 2 icreases with icreasig Ũ eff Ũ eff 0. This was expected from the rate Eq. 17 i the classical limit. The trasfer is foud to be icoheret ad sequetial i the higher damped case =0.36 for ot too large Ũ. The populatio probabilities are show for Ũ=0.6 c,1.7 c, ad 2.5 c i the upper pael of Fig. 13. By fittig the curves with the help of the kietic equatios, Eq. 19, we obtai the rate of the -electro trasfer D 2 to D, which is a omootoic fuctio of Ũ eff with a maximum at Ũ eff =E c see lower pael. It is plotted together with the Marcus rate at T=0.008 c. For varyig temperatures we foud that the fitted rate is approximately costat for temperatures T c i the cosidered parameter space. Although the qualitative behavior is captured by the Marcus rate the asymmetric shape of the NRG result is more realistic i the uclear tuelig regime. As Ũ icreases further the sequetial trasfer becomes egligible i the iverted regio. As a matter of fact, a icreasig value of Ũ shifts the system away from the phase-trasitio lie deeper ito the delocalized phase as ca be see i the equilibrium phase diagram of Fig. 4. Here, the dyamics is domiated by coheret oscillatios with a very small frequecy, displayed for Ũ=5 c i the middle pael of Fig. 13. Fially, the effect of temperature is studied i Fig. 14 where Ũ= 0.01 c, =0.001 c. The temperature is varied

12 TORNOW et al. (a) FIG. 14. Populatio probability DA of the state D as a fuctio of time t for temperatures betwee T 0.02 c ad T =0.125 c. The parameters are Ũ= 0.01 c, =0.001 c, ad =0.03. (b) (c) FIG. 13. Upper pael: Low-temperature populatio probabilities P t =d D t thick black lie, d A t thi black lie, ad DA t gray lie as fuctios of time. For d A from top to bottom Ũ =0.6 c, 1.7 c, ad 2.5 c. The other parameters are =0.36, T c, =0.1 c, ad =0. Middle pael: Low-temperature populatio probabilities with U=5 c. Lower pael: Sigle-electro rate k for the trasfer from D 2 D deduced by fittig the populatio probabilities with the kietic equatios Eq. 19 squares ad Marcus rate Eq. 13 full lie with T=0.008 c as a fuctio of the o-site Coulomb repulsio. The Marcus rate is ormalized so that both curves have the same maximal rate. Iset: Eergy levels of states D 2, D, ad DA 2. from c to c. For T= c to T 0.02 c, the populatio probability is temperature idepedet. As log as Ũ eff T, trasfer is observed the probability of D stays costat.ast Ũ eff the states D are see to cotribute ad are thermally populated. VII. SUMMARY AND CONCLUSION I this paper, we have studied the electro-trasfer properties of two excess electros i a redox system modeled as a dissipative two-site Hubbard model a model which ca be viewed as the simplest geeralizatio of the spi-boso model to iclude may-particle effects. These may-particle effects are due to o-site ad itersite Coulomb iteractios, U ad V, respectively, as well as the effective iteractios iduced by the couplig to a commo bosoic bath. These iteractio parameters ca be calculated by ab iitio methods for a specific system see, for example, Refs. 22 ad 24. I our two-site model oly the differece Ũ=U V eters the dyamics. I the presece of a bosoic bath, the effective eergy Ũ is reormalized to Ũ eff =Ũ 2 c. A effective attractive iteractio Ũ eff 0 favors the localizatio of two electros o the same site; a repulsive Ũ eff 0 favors the distributio of electros o differet sites. The itricate correlated dyamics of two electros depeds o the activatio eergy. Therefore, the trasfer characteristics i the ubiased case depeds strogly o the effective o-site Coulomb repulsio Ũ eff. Three rates have to be cosidered: the forward ad backward rates betwee the doubly occupied states D 2,DA 2 ad the two itermediate degeerate states D as well as the direct rate betwee D 2 ad DA 2. How these rates deped o Ũ eff is summarized i Table I. We have performed calculatios for the probabilities P t of doubly ad sigly occupied door ad acceptor states usig the time-depedet umerical reormalizatio group method. 33,34 This iformatio helps us to idetify coditios uder which the systems performs a cocerted two-electro trasfer, b ucorrelated sequetial -electro trasfer, or c fast cocerted two-electro trasfer followed by a -electro trasfer. With the time-depedet NRG method we ca describe the crossover from damped coheret oscillatios to icoheret relaxatio as well as to localizatio at T 0. The temperatures are chose to be 0.1 c T c. For larger temperatures, whe the bosoic bath ca be treated classically, the Marcus rates are applicable. For Ũ eff,e 1,T cocerted electro trasfer occurs i both methods: i the uclear tuelig regime withi the NRG as well as i the limit of a classical bath withi the

13 DISSIPATIVE TWO-ELECTRON TRANSFER: A TABLE I. Summary of the results. The effective Coulomb repulsio is defied by Ũ eff =U V 2 c. The correspodig reorgaizatio eergy is E 1 =2 c ; the bias is =0. Startig with two electros o the door the system performs either a sequetial -electro trasfer D 2 D DA 2 or a trasfer D 2 DA 2 depedig o Ũ eff. Ũ eff, T Ũ eff Ũ eff E 1 Ũ eff E 1 Ũ eff, Ũ eff E 1, T Ũ eff Ũ eff E 1 Ũ eff E 1 Ũ eff, Ũ eff E 1, T Ũ eff Sigle-electro trasfer Sigle-electro trasfer Ũ eff 0: Sigle-electro trasfer k D 2 D faster ad k D D 2 slower with icreasig Ũ eff Sigle-electro trasfer k D 2 D ad k D D 2 slower with icreasig Ũ eff Electro- trasfer i additio slow -electro trasfer Ũ eff 0: Sigle-electro trasfer k D D 2 faster ad k D 2 D slower with icreasig Ũ eff Sigle-electro trasfer k D D 2 ad k D 2 D slower with icreasig Ũ eff Electro- trasfer Marcus theory. As log as T Ũ eff, however, thermal activatio is abset ad uclear tuelig is the mai process. Oly a full quatum-mechaical calculatio yields the correct relaxatio rates which are govered by quatum-fluctuatio, dephasig, ad eergy exchage with the eviromet. For small / Ũ eff we foud a effective hoppig via virtual populatio of the low-lyig or high-lyig states D. Whe the equilibrium probability for the states D is fiite, a slow -electro trasfer accompaies the faster trasfer. I cotrast to the -electro trasfer with a frequecy of the order, the frequecy of the trasfer is of the order 4 2 / Ũ eff. The cocerted trasfer becomes more ucorrelated ad sequetial at short times at high temperatures T Ũ eff, icreasig couplig to the bosoic bath E 1 Ũ eff or larger -electro hoppig Ũ eff. The sequetial trasfer rate is omootoic with icreasig Ũ eff. At first, the trasitio rate from D 2 to the delocalized states D icreases for small Ũ eff 0, reaches a maximum for Ũ eff =E 1 before it decreases agai. The rate for the cosecutive process D DA 2, however, decreases with icreasig Ũ. For a egative effective Coulomb matrix elemet Ũ eff, the trasfer rate of the secod process D DA 2 is maximal for Ũ eff = E 1. I this parameter regime, we expect that the secod electro follows very shortly after the first electro was trasferred. The trasfer kietics of more tha two excess charges i, for example, biochemical reactio schemes or molecular electroics applicatios is cotrolled by the molecule specific Coulomb iteractio ad its polar eviromet. Our study reveals the coditios for cocerted two-electro trasfer ad sequetial -electro trasfer. Cocerted two-electro trasfer is expected i compouds where the differece of the itersite Coulomb repulsio ad effective o-site repulsio are much larger tha the -electro hoppig ad larger tha the temperature ad reorgaizatio eergy. Furthermore, we have show that the omootoic characteristic of sequetial -electro trasfer strogly depeds o the Coulomb iteractio. A further study will iclude the ifluece of a fiite-eergy differece betwee the door ad acceptor site. We will also report o the ifluece of Coulomb repulsio ad may-particle effects o the lograge charge trasfer usig a loger Hubbard chai as bridge betwee door ad acceptor ceters. ACKNOWLEDGMENTS S.T. is grateful to the School of Chemistry of Tel Aviv Uiversity ad to the Racah Istitute of Physics of the Hebrew Uiversity of Jerusalem for the kid hospitality durig her stay ad partial support Tel Aviv Uiversity. This research was supported by the Germa Sciece Foudatio DFG through Grat No. SFB 484 S.T. ad R.B. ad through Grats No. AN 275/5-1 ad No. 275/6-1 F.B.A., by the Natioal Sciece Foudatio uder Grat No. NFS PHYS F.B.A., by the Germa-Israel Foudatio A.N., by the Israel Sciece Foudatio A.N., ad by the U.S.-Israel Biatioal Sciece Foudatio A.N.. F.B.A. ackowledges supercomputer support by the NIC Forschugszetrum Jülich uder Project No. HHB000. We ackowledge helpful discussios with A. Schiller ad D. Vollhardt