The two-state reduction for electron and hole transfer in bridge-mediated electron-transfer reactions

Chemical Physics 176 (1993) 501-520 North-Holland

The two-state reduction for electron and hole transfer in bridge-mediated electron-transfer reactions

Spiros S. Skourtis a, David N. Beratan b and Jose Nelson Onuchic a a Department ofPhysics, University of California, San Diego, La Jolla, CA 92093-0319, USA ’ Department of Chemistry, University of Pittsburgh, Pittsburgh, PA 15260, USA

Received 5 April 1993

We present a unified analysis of the two-state reduction in bridge-mediated electron transfer for both electron and hole transfer. The parameter that characterizes the leading error associated with a two-state reduction is derived in the energy and time domains. A precise definition of the t~nell~ energy is given. We also derive analytically the regimes of validity of the two-state reduction and we interpret it in terms of the time evolution of the purely electronic part of the el~tron-t~sfer probability.

1. Introduction

The theory of electron-transfer (ET) reactions in chemical and biological systems has grown substantially over the years [ 1-5 1. This is partially due to the importance of these reactions in biology, specifically in photo- synthesis and oxidative phosphorylation [ 61. The wealth of experiments on photosynthetic ET reactions has shown that they usually involve the tunnelling of a single electron from a localized donor (D) state to a localized acceptor (A) state both of which are embedded in protein [ 7,8 1. The distance between D and A is large (several angstroms), thus making the direct electronic coupling between them too small to induce ET. Instead ET is induced by the intervening medium between D and A which acts as a “bridge” that creates an effective donor- acceptor (DA) electronic coupling. The bridge consists either of protein or of protein and cofactors, and it provides orbitals that act as virtual states for the tunnelling electron. This mechanism is often referred to as superexchange [ 9 1.

Superexchange ET reactions are usually nonadiabatic [ l-4, lo- 121, and their rate is given by

rate= [2n(H&)*/fi]FC. (1)

FC describes nuclear effects on the ET rate (i.e. Franck-Condon overlap factors and nuclear dissipative widths). H&j is the effective DA electronic coupling provided by the bridge. (H&c)* thus describes the effects of the purely electronic dynamics on the ET rate.

The calculation of HA% involves a two-state (2s) reduction. The Born-Oppenheimer approximation is first implemented so that the D, A and bridge electronic states and energies become functions of the nuclear coordinates. At the nuclear configuration R, where the D and A Born-Oppenheimer surfaces cross the Franck-Condon factors (FC) attain their maximum value. It is at this configuration that the calculation of H&% is carried out. One assumes the validity of the Franck-Condon approximation so that the nuclear coordinates are “frozen” at R,. The resulting electronic Hamiltonian generally contains N+ 2 states where NXP 2 (2 for the D and A states and N for the bridge states). To compute HDA eff this Hamiltonian is reduced to an “effective” two-state donor- acceptor Hamiltonian (PDT). In #$A the D and A energies and the coupling between D and A each contain an additive term that describes the effects of the bridge. The off~iagon~ element of fig! appears in the nonadiabatic rate expression.

0301-0104/93/$06.00 0 1993 Elsevier Science Publishers B.V. AB rights reserved.

502 S.S. Skourtis et al. I Chemical Physics 176 (1993) 501-520

The formal construction of such effective Hamiltonians is called the Lijwdin partitioning technique [ 131 and it was introduced in ET by Larsson [ 14 ] to study the effect of different bridge orbitals on H&c. There has since been a significant effort to compute H ,2.. in different ET systems and this led to ET pathway models [ 15 ] and Green’s function computational techniques [ 16-2 11, as well as quantum-chemical calculations [ 22-26 1. Fur- thermore, ET rates have been measured in several biological and chemical systems [ 271 allowing in some cases to test computations with experiments [ 281.

The 2S reduction requires that the tunnelling energy of a transferring electron (E,,,) be specified. E,,, appears as a parameter in the effective matrix element H &T and more generally in the 2.5 Hamiltonian @&. In actual computations it is usually set equal to the average of the original D and A energies ( (ED+EA)/2) and if necessary it is improved iteratively [ 141.

Since the 2S reduction describes the electronic dynamics in a system with many electronic states in terms of two states, it is an approximation that is valid only in specific regimes. In spite of its wide use in the study of bridge-mediated ET reactions there has not been a rigorous analytical derivation of its regimes of validity and of the error introduced by its implementation. Several authors have addressed different aspects of the 2S approximation. These include its relation to Green’s function perturbation theory and to hole transfer [ 171 as well as the effects of nuclear vibrational and dissipative dynamics [ 29,301. A nonperturbative view of the problem [ 3 1 ] has also been given. However, the question of how to calculate the error associated with the reduction and the closely related question of how to define E,, has not been resolved.

In a previous paper [ 321 we answered these questions for a one-electron system containing an arbitrary number of bridge states and interbridge or bridge-DA couplings. The one-electron model we used describes ET through the conduction band of the bridge. In this paper we generalize our analysis to many-electron systems and to hole transfer through the valence band of the bridge. The parameter that characterizes the validity of the 2S reduction for hole transfer is derived, and we discuss its relation to the leading fractional error associated with the 2S approximation. A precise definition of Et,, in terms of this error is given. In the time domain, the 2S approximation is interpreted in terms of the time evolution of the purely electronic part #I of the ET probability.

In section 2 we review Green’s function formalism as related to ET. In section 3 we present the results of [ 321 where the 2S reduction was studied in a one-electron system that describes ET through the conduction band of the bridge. Sections 4, 5, 6 and 7 are devoted to the many-electron case. In section 4 we discuss the many- electron Hamiltonian and the particle-hole representation. In section 5 the energetics of the valence band are analyzed. Finally, in sections 6 and 7 the 2S reduction is analyzed for hole transfer.

2. Green’s function approach to electron transfer

ET experiments usually involve the excitation of an electron in a donor orbital (D) and the subsequent mea- surement of the probability of ET to an acceptor orbital (A) #2. This probability is the square of the D to A transition amplitude. The D, bridge, and A orbitals and electrons directly involved in the transfer are hereby called the “transfer system” (TRS). It is assumed that prior to the excitation of an electron in D, the TRS consists of N electrons and it is in its lowest energy electronic state I !P$‘) (“ground state”). It is further assumed that the electron excited in D does not belong to the N-electron TRS (i.e. it comes from an orbital that is not directly involved in ET).

Our purpose here is to study the error introduced by the 2s reduction that arises only from the electronic dynamics. There is also error associated with Born-Oppenheimer and Franck-Condon breakdown, and with dissipative effects [ 291. a2 Sometimes the survival probability of D is measured.

S.S. Skourtis et al. I Chemical Physics 176 (1993) 501-SZO 503

Upon insertion of an electron with spin cf in D, the states of the TRS (of now i?+ 1 electrons) become CA,(O) 1 !Pg). The amplitude for a transition to A with spin CI’ after time t is given by 5&*,&t) =

< %7C%&)&7(0) I -Y>. cw and & are electron annihilation and creation operators satisfying the usual fermion anticommutation relations.

The transition amplitude SAaP,DV( t) is related to the zero-temperature, real-time Green’s function [ 331

GA&&~) = - + ( ~~l~~~~~~(~)~~(O)~l%> f (2)

where rt”(~~~.(t)C’b,(O))=~~~.(t)~~(O) for t>O, and -&,(0)cA,,#(t) for t<O. G..,,,,,DU(t)= -i&a,,Do( t) /li for t> 0, so we restrict attention to positive times.

GAa,,Da( t) is the Fourier transform of a time-independent Green’s function G*,-,,,,(E):

GA~‘,D~( t ) = YIE s ,,,exp(-~~l~)G,,,,(E).

--oD

G Aa,,D#)=< Y%“Icf,,,G(E)&l %‘>, where

(3)

G(E) = 1

~-(~-I~~)+i~’ (4)

8$’ is the energy of 1 !Pg>, and q is an infinitesimal positive number. The poles of GA,P,D,(E) are the electron affinities of the TRS. They are zz’j,= 8:” - &j’, where the c$‘+’ denote the eigenenergies of the (N+ l)- electron TRS. The residues of GAo,,na(E) at .$,, are given by Res[GAo~,oa(E)]EaJdn= ( !P$‘]c?,_,,, 1 !Pf*‘)( Yr+* It”$,,l !?$?), where the 1 Yf+’ ) are the (N+ 1 )-electron eigenstates of the TRS. In the discussion that follows we will often make use of the residue expansion of G,,,,&E):

In terms of G Aa,,D& t) the ET probability is given by

P ~~‘,~~(t)--~‘lG~b,,~dlf) 12. (6)

In this paper Green’s function method is used to analyze the 2S reduction in terms of the time evolution of the ET probability (eq. (6) ). Our work differs from the common application of the Ltiwdin partitioning technique. In terms of the (N+ 1 )-electron TRS, the Lowdin technique is used to approximately solve the eigenvalue equation (E-A) I !Pr+l) = 0 for only two of the eigenstates/energies of J?. We study 1 / (E-f?) (eq. (4) ) and its Fourier transform, and consider all of the eigenstateslenergies in a self-consistent manner. Furthermore, for the rn~y~lect~n case we partition according to hole number, not orbital type.

We expect that an ET reaction is describable as an effective 25 system if the time evolution of the electronic part of the ET probability (eq. (6 ) ) resembles a single sinusoidal oscillation (fig. 1) . Mathematically, this occurs when two residues of GAee,Da(E) are of much larger magnitude than the rest. This pair dominates in the residue expansion of G?Ad,De( t) (eq. (5) ) giving rise to a large (in amplitude) sinusoidal oscillation in PM,D,( t). If the two residues correspond to energies 8:” that are “close” #3 to the energies of the donor and acceptor states (&, 1 Yg ), tf&, 1 !P$‘) ), the dominant oscillation is between these two states. The effective 2S system can then be thought of as an effective donor-acceptor system,

s3 The meaning of “close” is made precise in the following sections.

504 S.S. Skourtis et al. / Chemical Physics 176 (1993) 501-520

I p,(t)

c

(timaL,

Fig. 1. Time evolution of exact ET probability in the 2S limit. P& 1) resembles a single sinusoidal oscillation, i.e. it can be approximately described as the sum of one large sinusoidal osciha- tion and of many smaller oscillations.

3. One-electron case

In a recent paper 132 J, the 2S limit was analyzed for a one-electron TRS. The model used describes ET through the conduction band of a bridge (‘Iigs. 2a, 2b). A tint-binding Hamiltonian was used, given by

&&~i-A,+P. (7)

AD, and & are the H~iltonians for the DA and bridge subsystems in figs. 2a, 2b, where W 1

H~A=ED~D)(DI+EA~A)(AI, (81

and

@x%-T (&#t~,)(4JI)f 2 ( ~bnmI&) <&I)= C (~~~l&)(B~t). (9) n,m m

ID> , I A) and J b,) denote localized spatial orbitals of the donor, acceptor, and bridge respectively. &,nnr are tbe couplings between the different 1 b,,), whereas 1 B,) denote the delocalized bridge eigenstates. Pdescribes the coupling between DA and bridge:

p= I: (uD,iD>(b,i+VAnIA)(b,Ifh.c.)= c (BD,tD>(B,/+rSAmIA)(B,I+h.c.). (10) n m

In ref. [ 32 ] it was assumed that En> EA and that the conduction band of the bridge lies above the DA band (i.e. En1 > ED). However, we could also have chosen &a,!&, without affecting any of the arguments.

For a one-electron system the ground state of the TRS prior to the insertion of an electron in D is a vacuum state without electrons (i.e. 1 Y$’ ) = 10 > , &g = 0 since N= 0). Since the Hamiltonian in eq. ( 7) conserves spin, it follows that GAa”,oo(t) =&,GAn(t), G A~,Dc(E)=&oGA~(E)t where G&t)= -i(Aje-~f’*(D)/Zi, and G~(~)=(All/{~-#+i~){D).The~sidueexp~sion (eq. (5)) thenbecomes

GAD(~)= - i z exp( -i&t,%) Res[G&E) fES6.. n

(11)

M In ref. [32j Z& also includes a direct DA coupling u ,,b. The model is thus more general than fg 2a. However in the present paper we assume that Q, =O. This is not a necessary assumption for the arguments that fohow but it is a good approximation for long-distance electron transfer. The present discussion can be generalized to vAD # 0 by setting E ,=E, and E..,=E_, where E,, E_ correspond to the higher and lower eigenenergies of the DA subsystem (see ref. [ 32 ] ). For this reason we will often refer to En, EA more generally as the DA “band”.

505

‘Ng+2-

"shiftmd" btidge b&nd

=I31 - E3- LII_LI_---___________----~~

DA-bridge coupling

!

EB1- =,,a

% v _c_-II____--l--- _-_--c-_

‘A -

E2- %a 1 'shiftad"_

Da bssld

&A EJ_- hi

Fig. 2. (a) Model used to describe ET from 1 D} to [A) through theconductionbandofbridge (eqs. (7)~(lO)).The lb,), I&,} are localized bridge states with energies &, I&,,. z+,~~ denote the couplings between these states. (b) Same model as in (a), where the isolated bridge (i.e. uncoupled from ID>, /A) ), has been diagonaIized (see eq. (9) ). IS,> (Ebm ) are the bridge eigenstates (energies), and &,&&) denote the electronic couplings between these etgenstates and 1 D ) ( 1 A) ) . (c) The “shifted” DA and bridge bands ( ff,, 82 and ff3-&+2), arising from the zeroth order DA and bridge bands (En, I& and En,-&,). E,= indicates the position of &, &a with respect to tbe bridge.

ResIGdE>Ln= (A 1 ul, > ( u’, I D) , and 1 F,,), 8, are the spatial one-electron eigenstates and eigenenergies

of the TRS. The ET probability is given by P,.,& t ) = fi ’ I GA&t ) I z. The question posed in ref. [32] is what it means ta have an effective 2s system with a specific E,,, We

answered this question first in the energy domain and then self-consistently in the time domain. G~~~~) was written in a form appropriate for 2s reduction by pa~~tioniug the Hilbert space into DA and bridge subspaces, giving

506 S.S. Skourtis et ai. /Chemical Physics I76 (1993) 501-520

G~(E)== [E-H&-f,(E)] [E-Hz(E) ] -Hg$(E)H,s(E) *

The terms HEff(E), ( i andj stand for D or A), are

H~ff(E)=~~~~+~~~(E), SW = <i I ~~~~(E)~~l~) . (13)

(2 is the projection operator for the bridge: &= Xc, 1 B,) (B, 1. l&(E) = (E-&&&)-’ is the bridge Green’s function.

The secular equation (SE ) of fi is given by

f(E) = [E-Hff(E)] [E-Hz(E)] -~~~(E)~~~(~) =0 . (14)

Its roots 8, (eigenenergies of the TRS) are the poies of G&E). Eqs. (12) and (14) resemble the form of Green’s function and SE of a pure 2s system (see Appendix A).

In the energy domain the 2s approximation corresponds to the adoption of a 2s Hamiltonian

(15)

where the h,(E) are given in eq. ( 13 ) . In ref. [ 32 ] we showed that tin% (E& arises from a particular expansion off(E), followed by the neglect of certain terms in the expansion. The resulting SE is as-like and it corresponds to #$,(E,,,). The expansion parameter can be identified with the leading error introduced by the 2S reduction and as such it characterizes its validity. This parameter is expressed in terms of the eigenstates of #gA(E,,) and the lowest eigenstate of the bridge conduction band. It can thus be implemented in any computation of an effective 2s system to calculate the error introduced by the approximation.

In particular, eq. ( 14) can be written as

f(E)=&(E)-- (E-ED)~AA(E)-(E-EA)~DD(E)+~DD(E)~~A(E)-~AD(E)~DA(E)=O,

where

(16)

(17)

and&(E)= (E-ED)(E--E,) is the characteristic polynomial ofthe DA subsystem (#nA in eq. (8)). If two roots dpI, CC” off(E) =O are “well separated” from the other roots &-B N+2 and are “close” to each other, then c&, tfipz can be obtained to a good approximation by an effective 25 SE. If S+$ and $-c&+.~ correspond to the “shifted” DA and bridge bands (see fig. 2c for the meaning of “shifted” bands), this 2s equation comes from an effective donor-acceptor Hamihonian as in eq. ( 15). The meaning of “well separated” and “close” to each other is defined by the condition: I( &‘11,2-Etun)/(EBl -Et,,) 1 -CCC 1. I&, is the lowest eigenenergy of the bridge. In the present context Et, can be thought of as any energy that identities the position of the “shifted” DA band (&‘r, c&) with respect to the bridge band (fig. 2~).

To obtain a SE whose eigenene~es $, 6; satisfy this condition, we expands in eq. ( 16) in powers of (E-E,,)/(EBm--E,,) and keep only terms that are zeroth order in these parameters. For each h,(E) inf(E) we get

ME) =&j(L,) + (E-E,,&M+,(E,,,) +... (18)

(&=d/dE). Neglecting (E-E,,,,)&h,(E,,,) and higher order terms gives

f(E) =.X&E) - (~-E~)~~tE~~) - (E-E~)~~~(~~~~)

+h,tE,,,)h,(E,,)-h,(E,,)h,tE,,,)=O. (19)

S.S. Skmrtis et ai. /Chemical Physics 176 (1993) Sol-520 507

This is the SE of #~~(E~~~) in eq. ( 15). Therefore, the parameter that characterizes the validity of a 25 reduction involving Ad ( Etnn ) is

~::~d(E,~~)= 1 [~,1.2(Etun)--Etunl/(EB,-Et”n)l 7

where M 8’,,Z (E,,, ) are the eigenenergies of fig, ( Et,,) :

4,2VLJ = f (4, +EA) + f [hxA%d +b~(Gun) 1

If both r, (E,,,), Y,(E,,) e 1, the 2S reduction is a very good approximation. If at least one of 2”i (E,,),

Y’2(E~,,,)kl, the 25 reduction fails since the expansion off(E) in (E-E~~~)/(~~~-E~“~) fails. In any 2s

reduction one should find the optimal E,, that minimizes simultaneously Yi and Y, (see ref. [ 32 ] ). These arguments can be extended to the time domain (Appendix B). If Y, (E,,,), Y,(E,,) << 1, then in the

residue expansion of GAD ( t ) (eq. ( 11) ), both Res [ GAD(E) lE= h and Res [ GAD (E) lE= h are of much larger magnitude than Res [ G,(E) ] EP Qg_m+2. The adoption of Z?$ ( Et,,) corresponds to setting

Res[G*~(E)lE=b:=i11AD(~,un)l[~2(Etun)-~,(Etua)l, (22)

and neglecting all the other residues in GAD ( t ) , The error 6.R i,* ( E;un ) associated with eq. (22) contains terms such as #6 I( s$“,~ -E,, )&hm (.I&, ) / ( dp, - 8, ) 1. The neglected residues (which correspond to the “shifted” bridge eigenenergies in fig. 2c: &s - g;li+* =EB,,J, are given by ResIG,,(E)lE=E,,=BAmamD/tEBn-E~“n)(EBnr- E,,,~.Since~~(E,,~,~~(E~,,~~l,both~~1,~(Etun~andRes[G~~~E~l,,~~~ResCG~~(~~l~~n.z~eq~ (221, and the error introduced in the time domain is small. This means that in the 25 limit the ET probability resembles a single sinusoidal oscillation, and &$A(Et,,) provides a single sinusoidal fit to this probability. Further- more, E,,, is a fitting parameter (see fig. 3 ). An example of this limit for a three state TRS is given in Appendix L.

The characterization of a 25 reduction in terms of Y,,JE,,,) allows us to distinguish between the simple perturbative limit ( 1 h,(E,,,) 1 <EB, -Et”,), and the more restrictive 25 limit (h&E,,,) eEB, -Etu, and c$~ - S; K ( EB, - &I ) /2). The former is characterized by weak coupling between DA and bridge, whereas the

IB The superscript “co&” indicates that the bridge-mediated DA coupling considered here is due to the conduction band. yb More terms containing dEADA, dEhDD, and d,h M must be added to SR,,2 for an accurate description of the error.

Exact fit: 2 Sin W(ntunit

t

Fig. 3. In the 2S limit, (r,fE&, rz(Et,)< t), the use of H$$(&,,,) corresponds to a fitting of the exact ET probability by a single sinusoidal form: PO(EtW) sin*[o(&)t]. P&?&J= 12h~(Etyn)l[B2(Etu.)-~~1(E,,)lIZ, and WLI) = [Lg2(E1”.)-~,(Etu0)l/2J2. h&E,,) and 4(&,& 4(-%) are given in eqs. ( 17), (2 1). E,, can be interpreted as a fitting parameter. Away from this limit the exact P&C) does not resemble a single sinusoidal form and HfA(Em) cannot be used to describe the ET dynamics.

508 S.S. Sburtis el al. / Chemical Physics 176 (1993) 501-520

latter also requires that the DA bandwidth is small with respect to its separation from the bridge band. When the width of the DA band is greater than its ~p~ation from the bridge band ( &-- &I > (EB, - &a )/2), the 2s approximation fails even if the coupling between DA and bridge is very weak. This is because r,,z(E,,,) k 1 for at least one of the &&I&,) in eq. (2 1). In this case the 2s reduction is not a good approximation even if perturbation theory is. A single energy such as E,,, cannot be used to define the position of the DA band with respect to the bridge band (fig 2~).

4. Hole transfer

The above treatment is generalizable to hole transfer. We imagine a TRS with N/2 bridge spatial orbitals 1 b,) (fig. 4), where initially the total number of electrons is N (N is even). An additional electron is subs~uently inserted in D. We derive the conditions under which the time evolution of the ET from D to A in the (IV+ l)- electron system is describable by an effective 2S, DA Hamiltonian.

4.1. Many-electron Hamiltonians and notation

We use a Hubbard-like Hamiltonian where the sole source of electron correlation is the on-site electron- electron repulsion. However, our arguments are largely independent of the type of Hamiltonian used to describe electron correlations and tunneling. The important parameters are the relative energies of the DA and bridge bands and the electronic couplings between them. The TRS H~iltonian is written as

(23)

where I?,,* and I& denote the isolated (i.e. uncoupled from each other) DA and bridge subsystems and P denotes the coupling between them.

I?,,* and I& are given by

(24)

The &,rb are one-electron Hamiltonians describing single-electron orbital energies. The I? describe the effect of electron- electron repulsion when two electrons occupy the same orbital. pB is the Hamiltonian for the coupling between the different bridge spatial orbitals { 1 b,) }.

The fiOorb are given by

A orb(DA) = 1 I&fi~c,+J&A~al , 0 (25)

EA; IA>

Ebn;lbn'

Fig. 4. Model used to describe hole-mediated ET. Energies and interstate couplings of localized one-electron D, A, and bridge spatial orbitals. Compare with fig. 2a.

S.S. Skourtis et al. / Chemical Physics 176 (1993) 501-520 509

where A. 1o‘ = c?,cji, ( i = {D, A, b,} ) are occupation number operators #‘, and E,,, EA, Ebn are the energies of the spatial orbitals 1 D) , ) A) and 1 b,) respectively (fig. 4). The 6 are

0rJ, = Ur& AD, + u, fi& A& ) cr,= 1 Ubnfibntftbd 3 (26) n

where U, is the electron-electron repulsion between two electrons in the spatial orbital I i) . FB and Pare given

by

Vb,,,,, is the coupling between 1 b,) and I b,) , zy,,, is the coupling between I D ) and I b,) , and VA, is the analogous quantity for I A).

Since it is assumed that the direct coupling between D and A is negligible, the ET from D to A is solely mediated through the creation and propagation of bridge holes. It is therefore convenient to use a notation that labels the states of the total TRS (DA+ bridge) by the number of bridge holes they contain. The ET can be described as a series of transitions between states with differing number of bridge holes resulting in the net transfer of an electron from D to A. We therefore rewrite the total Hamiltonian fi (eq. (23) ) as

A=&+P, where I& =&A +Z& .

So fi,, describes the isolated bridge and DA subsystems.

(28)

For the N-electron system, we denote by lM(N, h) ) any single Slater determinant state of the spin orbitals I i) 8 I a) ( i= {D, A, b,} ) containing h DA electrons and h bridge holes. For example, one of the states l M(N, 1) > is given by IDt, h1, htl, . . . . bNIZtl) = CL, cl,, ..J?J~,~~ IO), where IO) is the vacuum state with no electrons. The N-electron eigenstates of @, with h bridge holes are denoted by I K( N, h) ) . In general, I K(N, h ) ) # I M( IV, h) ) since FB (eq. (27) ) couples the different I M(N, h) ) . The only exception arises for the state with zero bridge holes

lNO)=lht~..., b,,,tl)=~6,,...C~~,,,,, IO> >

which is an eigenstate of Ijo with eigenenergy

(29)

E:= c (2&n+ ‘%,) . (30)

For the (N+ 1 )-electron system the single-determinant states and the eigenstates of fi0 with h bridge holes are denoted by ) M( IV+ 1, h) ) and I K( N+ 1, h) ) respectively. The states I M(N+ 1,O) ) with zero bridge holes are eigenstates of &. By virtue of the orthonormality of the spin orbitals I i) @ 1 a) we have (M’ (N’, h’ ) I M(N, h)) =dM’M&N&,,, and (K’(N’, h’) IK(N, h)) =&KdwN&h.

4.2. Particle-hole representation

In general the total number of valence electrons in the bridge is much greater than the number of DA electrons, i.e. N X. h. It is thus convenient to use particle-hole representation (see also ref. [ 5 ] ). The vacuum state of the system is taken to be 1 N, 0) (eq. (29) ), instead of I 0). Two sets of operators are defined that act on IN, 0). The bridge hole operators

&,o=~b,,-0, ~bmo=&.s, (31)

and the particle operators &A)o, &(AjO. In this representation fibA (eq. (24) ) remains the same, and

*’ It is assumed that et,, c,,, satisfy the fermion anticommutation relations.

510 S.S. Skourtis et al. /Chemical Physics I76 (1993) 501-520

Z&, =fEf+fi$,") , where figh) =fi$&Bj + 6hh) + T&“) , (32)

and Et is given in eq. (30). The superscript (h) indicates that only the hole operators &,,,, &, appear in the Hamiltonian. The different terms in eq. (32) are

where A I,“) = a?Ji, are hole occupation number operators. Similarly,

p= C (v~,~b,dh-~+v~n~haa~“--o+h.c.) %a

(34)

( Pcreates DA particle-bridge hole pairs). Apart from the additive constant fEt, I& has the same Hubbard-like form in particle-hole representation as

in the original electron representation, although the orbital energies and tunnelling terms are different. The bridge energies are measured with respect to the energy of the full bridge, Et.

The states ) M(N) h’ ) ) , 1 K(N’ , h’ ) ) discussed in section 4.1 are products of DA and bridge states. In particle-hole representation

(35)

lM(N+l, It)) = II(DA, h+ 1))~ Im(B, h)> = ~~~A~~...&+,~~~ dJ,_,...&+. IN, 0) . (36) VP

h+l h

I /(DA, h’ ) ) stands for any &-electron Slater determinant of spin orbitals ID) @ I a) and/or I A) @ I a). ) m(B, h) ) denotes any h-hole Slater determinant of spin orbitals I b,) @ I o) #‘. The I Z) are eigenstates of E.&,* in eq. ( 24) and we denote their eigenenergies by E& and E&J ’ ) respectively. The I K’ (IV’, h’ ) ) are given by

lK(N, h)> = IW’A, h)>@ IQ& h)> = ‘%,,,,...~~,A,, &(B,h) IN, 0) , (37)

h

and

lK(N+l, h)> = Il(DA, h+ l)>@ IW, h)> = ‘%,A,,...&,,,~ &(B,h) IN, 0) . L

h+l

(38)

I k(B, h) ) denotes any h-hole eigenstate of the bridge (Hamiltonian fir, or AAh) in eqs. (24), (32) ). It is in general a linear combination of the single-determinant states I m (B, h) ) , i.e.

IW, h)> =&hj IN 0) 3 where d&&h) = :c csFs tain_,...aim_,.) . (39)

h

The sum is over all possible combinations F, of h-hole creation operators. We denote its eigenenergy under l?kh) by E&h,‘.

Therefore, each eigenstate of the uncoupled DA-bridge system with h bridge holes has been written as a

x6 The states in eqs. (35)) (36) may differ by at most a sign from the 1 M( N’ , h’ ) ) defined in section 4.1, since the ordering of the C?t is different in the original representation. Therefore, the equal signs in eqs. (35), (36) do not mean strict equality. However, as long as one representation is used consistently throughout a calculation no problem arises.

S.S. ~~~$t~s et ai. /Chemical Physics I76 (1993) 501-520 511

product of an eigenstate of the DA subsystem with h or h+ 1 electrons, and an eigenstate of the bridge with h bridge holes (eqs. (37 ), (38 ) ) . The eigenenergy of such a state under k,-, is the sum of the eigenenergies of the product states,

&I IQN h)) = (.&Y+E&*,) INN h) > 3 J%(h) =J%%+w . (40)

and

~~I~(N+l,h))=(E~+E~~f)tK(N+l,h)), I$&\ =~~~~I)+~~~) . (41)

ti in eq. (34) couples states with h bridge holes to states with h + 1 or h - 1 bridge holes. For the (N+ l)- electron system the matrix elements of fi (for h > 1) are denoted by

v~~=(K’(N+l,h)lCrlK(N+l,h-1)). (42)

5. The ground state of the N-electron system

To compute G ACs,b,(E), GAD(t) (eqs. (2), (4)), it is necessary to know the initial and final many-electron states involved in the electron transfer. They have been denoted by & 1 !Pt ) and ej,,. / !Pf ) where I Yg> is the electronic ground state of the TRS prior to the insertion of an additional electron in spin orbital I D) @ I o) .

Up to now the form of I Yf) has not been specified. In general this state is different for different ET systems. For example, it may consist of a full or nearly full bridge with empty D and A, or it may be that D and A contain electrons which couple to the bridge forming a complicated I !P$’ ) .

Here we restrict attention to cases where D and A are initially empty and the bridge is full (fig. 5a). Within our model this is equivalent to assuming that the N-electron ground state of fl is equal to IN, 0) in eq. (29). However IN, 0) is an eigenstate of &, (eq. (28) ), not &. Pcouples IN, 0) (zero bridge holes) to states with one bridge hole (i.e. 1 M(N, 1) ) and I K(N, 1) ) ). Therefore, if we are to set 1 !Pz > = 1 N, 0 >, the following inequality must be satisfied for all 1 K( N, 1) > :

ED ; ID>

ED ; ID>

+ EA; I=

EA; iA>

-b Ibl> t ! -+ lbW?>

’ i lb,>

(at Ebn; Ibn >

Ibl> I b N/2 >

lb,,,>

tb) &bn; lb,>

Fig. 5. (a) “Ground state” / Ff> of ~~lectron-t~nsfer system (i.e. prior to the insertion of an electron in D), for the case of weak coupling between DA and bridge. / Y$‘vf;‘> = IN, 0), where IN, 0) is given in eq. (29). (b) Initial donor state of the (N+ t )-electron- transfer system fan electron has been inserted in D). The donor state is &, / P$‘> c &, j N, 0), consistent with the assumption in (a).

(E&r) given in eq. (40) ). This inequality comes from a perturbative calculation of I !#) where we set IK(N, l))=II(DA, l))@lk(B, 1)) sothat

(44)

When eq. (43) is satisfied 1 Y$) = IN, 0). This means that one can choose the ground state of the N-electron system to be a single Slater determinant of the full bridge if the donor-acceptor band is very weakly coupled to the one-hole bridge band (eq. (43) ). That is, if the couplings between the full bridge and the one-hole eigenstates of the system are small with respect to the energy difference between the donor (acceptor) energies and the one-hole bridge eigenenergies. What does this mean in terms of the energies of U,, Ej, and ZQ?

In eq. (43 ), Efih = ED or EA since [ l( DA, 1) ) corresponds to any one-electron state of the DA subsystem. E&L) is the eigenenergy of any one-hole eigenstate 1 k( B, 1) ) of AAh) (eq. ( 32) ). The matrix elements of fiBh) for the one-hole states I m (N, 1) ) = &, I N, 0) are

<N,~l~~~~~Bh~~dmoI~,O~=-~,~,~~~~n+~bn~~nm+~~nrn~~-~~nm~l. (45)

The bridge Hamiltonian has a simple tight-binding form with orbital energies - (&,, i- U,) and hopping terms - vbmm. The hole states && 1 N, 0) thus behave as single particles, tunnelling between the different bridge spatial orbitals.

Suppose that the bridge has only nearest neighbor interactions vb n,,,=&,andthatE&=E6, U,=U,.IfN/2is large, the one-hole band of the bridge can be calculated approximately by use of periodic boundary conditions, i.e. I bl ) = I bN/*+ 1 ) . The hole eigenstates are given by

Ik(B, l))=bflN, 0) , where &= & g, exp(iM&%, , (46)

and the eigenenergies are

E~)=-~(E~+U~)~2~*cos(k~~)]. (47)

k= 2x/( N&/Z) and I0 is the distance between I b,) and 1 b,,, 1 ), If it is further assumed that D and A only couple to I bl > and I b&, i.e. the Only COUphgS in P (eq. (34) ) are VD1= VA,/, = I’, then the matrix element between IN,O) and IK(N, 1)) ineq. (43)is

Then IN, 0) is an accurate ground state if

(48)

(49)

since (Eb+ U,+ 2~) is the highest eigenenergy of the one-hole bridge band. Eq. ( 49 ) is a specific example of the more general inequaiity (43 ) . It can be a useful criterion for determining

whether the ground state of the ET system prior to the insertion of an additional electron in D can be constructed solely from the bridge spatial orbitals.

6. Partitioning according to number of bridge holes

As in the one-electron case a closed expression for G Aap,Do( E) can be obtained by repeated use of the partitioning technique. The Hamiltonian (eqs. (23), (24) ) does not change the spin of the electron, so GAa,,no( t) is

S.S. Skourtis et al. /Chemical Physics 176 (1993) 501-520 513

zero for (I’ $0. One can therefore choose the spin of the electron inserted in D to be either t or 4, and then consider either GAt,Dt (t) or Gnr,ul(t ). Suppose we pick GA,,,,, (t). If in addition we assume that 1 !#) = IN, 0) as discussed in the previous section (fig. 5a), then the initial state of the (N+ 1 )-electron TRS is c#,, IN, 0) (fig. 5b). This state has (I&~, - fitotalz ) = 1 and it does not couple to any states with (&, - Ltir ) # I @‘. So when considering the time evolution of c?&, 1 N, 0) we need only restrict attention to the subspace of (N+ 1 )-electron states with ( fimW, - A,Wr > = 1.

To compute G,,,,, (E), this subspace is further partitioned into groups of states with an equal number of holes in the bridge. This partitioning is different from the one-electron case where the Hilbert space was divided into DA and bridge subspaces. The projection operators for the states with zero and h bridge holes are

IM(N+l, h)), IK(N+l,h)) are given in eqs. (36), (38). Only those ]M) and IE) having (A, talT - &,Calr ) =I 1 are included in & By virtue of the orthonormality of these states, p&h = 0, &,&, = 0 for h # h' , andP2=p, &i=&.

From eq. (4) and the ass~ption that 1 !?$‘> = IN, 0), (i.e. dg=Eg), it follows that G,,,,(E) is the (N, 0 I QAt CT+&, 1 N, 0) matrix element of

G(E)= 1 E-(#6"h'+f)' (52)

where

A6”h’ &* +Ap , (53)

(fin,, , I?$/‘) given in eqs. ( 24 ) , ( 32 ) ) . To obtain an expression for GAt,nt (E) we substitute f= P+ C iI I &, into eq. ( 52 ) (more conveniently expressed as (E- [I?&*) + p] ) e (E) = f) . The result is

~(E-l?~h')P

i

-mL

-!wr", Qw+#&Wl -&Q2

0

i

i%?(E)P 1'. &?(E)o, X

&?;E)f '.:.' fj3&,g

By use of the first column components of (54), we get

[E-1F3Ar6"h)B-l(~)~]~~(E)f3=P.

SW) = WI 1 1 &IV,

ME)-‘-$I vQ2 92UW-‘-d2~~ 3t?3UQ&3 v02

&2 @iI*

(54)

(55)

(56)

where&(E) (h=l-3),isgivenby

x9 ( > = <pi 1 y> denotes average.

514 S.S. Skourtis et al. / Chemical Physics 176 (1993) 501-520

(57)

& (E) is the particle-hole Green’s function operator for the uncoupled DA-bridge system, with h + 1 DA electrons and h bridge holes.

The (iV, 0 1 &‘r,, CL, 1 N, 0) and (N, 0 1 CA, &, 1 iV, 0) matrix elements of ( 55 ) are used to obtain the following expression for GAt,ut (E):

G At,Dt tE) = [E-H&-f

H%D, (~‘3 DUDE I [E-fC!!At(E)l -H%At(E)Hi”Dt(E) ’

(58)

The terms H:$ (E), (i and j stand for D or A), are

H~~~(E)=(N,01~~:,Ei6eh’~ftlN,0)+h,,,t(E) 3 (59)

where (iV, 0 I c,,figh) cjf 1 N, 0) = EiS,, and

h,,,(E)=(N,Ol~,,,~(E)~~~IN,O). (60)

G At,Dt (E) in eq. (58) is identical in form to GAn( E) for the one-electron system (eq. ( 12) >. The SE of fi- fEt is given by

f(E)= [E--f@~t (~91 [E-H~~A,(E)I-H~A,(E)H~~D~(E)=O. (61)

The roots off(E) = 0 are the electron affinities of the TRS, C& = @‘+ ’ -Et. The derivation of an effective 2S, DA Hamiltonian for the many-electron system is now completely analogous

to the one-electron case. The SE is rewritten as

f(E)=.&(E)- (E-ED)~A~A~(E)-(E-EA)~D~D~(E)+~D~D~ (E)~A~A~(E)-~A~D~(E)~D~A~(E)=O, (62)

where f&(E) = (E-ED) (E - EA) . It is assumed that the electron affinities &r = #+I - E$‘, d2 = 13” ’ -Et

arising from the shifted zero-hole band #lo are “well separated” from the one-, two-, and three-hole bands, and are also “close” to each other. That is, I (dl,2-E,u,) / (Et,, - Eclosest ) I c 1, where E,, is any energy between &r and &* and Eclosest is the energy Eh ?I+,‘) that is closest to ED or EA. To obtain an approximate SE with roots Z& dZ satisfying this condition, we expand the hit,? (E) in eq. ( 62 ) in powers of (E-E,, ) / (E,, - Eclosest ), as in eq. ( 18 ). Keeping only zeroth order terms gives

f(E)=&(E)- (E-ED)~A~A~(E~U,)-(E-EA)~D~D~(E~,,)

+h D~DT(~,,,)~,,,,(E,,,)-~,,D,(~,,)~D,,,(E,,,)=O,

where h,,,, (&,, ) = <N 0 I et;., ~(~5,~ > cJT I N 0 > . Eq. (63) is the SE of

(63)

PgA, ( Et,,) = ( EDh~~,~~(E~)

tun EAh~~~T~f )) .

tlln

The parameter that characterizes the validity of this 2S reduction is

(64)

(65)

*lo This is the analogue of the shifted DA band in fig. 2c. In the present case the perturbation of E,,, E., arises from the valence band. &‘+I, q+’ correspond to eigenenergies of states with predominantly &, ) N, 0) and cl, 1 N, 0) character.

S.S. Skourtis et al. / Chemical Physics 176 (1993) 501-520 515

where #I I a’, 2 (Et,,,) = 8, ,2 (E,, ) - E$’ are the eigenenergies of k&, (Et,,). This is analogous to rT2d (E,,) for the one-electron case #I’. However, the effective matrix elements &, (E,,,) of #$& (&,,) are more complicated than the corresponding matrix elements in the one-electron case (eq. ( 15 ) ). For the present model where the maximum number of bridge holes is 3 (fig. 5b), each h,, (E,,) = (IV, 0 I ~~~~~E~~)~~~ IN, 0) contains one-, two- and three-hole cont~butions (as can be seen from the expression for L?(E) in eqs. (56), (57) ).

If the D and A spatial orbitals are very weakly coupled to the bridge (so it is valid to expand the kitit{&,,) up to first order in V/ (E,, - E$\ ) ) , the ET from D to A is mediated by the propagation of a single bridge hole. To lowest order in V/ (Et,, -Ez&f ), the effective DA matrix element is

(66)

V#$o VBt are given in eq. (42). &I is the projection operator for the states with one bridge hole and & (E) is the one-hole Green’s function defined in eq. (57). However, as the strength of the coupling increases and there is deviation from the 2s limit, two- and three-hole tunnelling also participates in the ET reaction. The effective DA matrix element is not adequately described by eq. ( 66 ) . It is necessary to include the contributions of&(E) and&(E) ins(E) (eq. (56)).

7. Discussion

We have derived parameters that characterize any 2s reduction in bridge-mediated ET through the valence and conduction bands of the bridge. These parameters, (Yr?” and r;t”, in eqs. (20), (65) ), can be thought of as the leading error associated with the description of electron and hole-propagated ET by an effective 2s Ham- iltonian (such as tin% ( Et,,), @iAt (Et,,) in eqs. ( 15 ), (64) ). If both Y, and Y, are much less than unity such a description is valid. If at least one of rl, TZ is greater than unity the 2s reduction is not a good approximation. We have interpreted the 2s approximation in terms of the time evolution of the ET probability, and have also computed the 2s error in the time domain. When the 2s reduction is a good approximation the effective donor- acceptor matrix element that enters the rate expression is the off-diagonal matrix element of the effective 25 Hamiltonian.

In the limits where the 2s reduction fails (Y, and/or Y2 2 1) the time evolution of the electronic part of the ET probability does not resemble a single sinusoidal oscillation. So trying to approximate this probability by a single sinusoidal form {i.e. adopting an effective 2s Ham~tonian) does not make sense. In this case one cannot simply associate the effective 2s matrix element with the oscillation frequency of the electron between donor and acceptor. This is because other oscillations involving transitions to bridge states may become important_ For example, when the donor, acceptor, and one bridge state become resonant, it is not clear that the electron will predominantly move from donor to acceptor. It may actually first move to the resonant bridge state then come back and try to move to the acceptor and actually go to the donor, etc.

Another regime where the 2s approximation is not valid arises when the energy difference between donor and acceptor is similar to or greater than the energy difference between donor (or acceptor) and bridge. It is then not possible to define a 2s Hamiltonian with a single E,, even in the limit of very weak DA-bridge coupling (i.e. perturbative limit). Since the 2s reduction is usually carried out at the crossing point between donor and acceptor Born-Oppenheimer surfaces, this is not a problem as long as the coupling to the bridge is sufficiently weak. However, as one moves away from the crossing point and the donor-acceptor energy gap increases, the 2s reduction ceases to be a valid approximation although perturbation theory may still be a good approximation.

“I The superscript “val” indicates that the b~~e-rn~at~ DA coupling is due to the valence band of the bridge. *‘* The calculation of the error associated with the 2s reduction in the time domain is also analogous to the one-electron case (Appendix BI.

516 S.S. Skourtis et al. / Chemical Physics I76 (1993) 501-520

In general, the different regimes of validity and failure of the 25 approximation can be characterized by r1,2 (see also ref. f 32 ] ). For example, as the DA band approaches one of the bridge bands rl,2-+co, indicating the failure of the 2s approximation. Fu~he~o~, in any 2s reduction, the compu~tion of ~~~d(E~~~) and r;$ ( Et& gives a measure of the relative cont~butions of electron and hole propa~tion in an ET reaction. For very long distance ET in proteins we expect that r3(E,,,) - 10e6 and I~~~~(~~~~)/~~~(~~~ j < 1 (i.e. the 2s reduction is a very good approximation and hole propagation dominates). For many of the artificial ET compounds that are currently being studied by ab initio and semi-empirical methods the values of these parameters could be very different.

It should be emphasized that the present analysis is generalizable to models that describe electron propagation through the conduction band of the bridge and simultaneous hole propagation through the valence band. In this case the matrix elements of the effective 2s Hamiltonian will have both electron and hole cont~butions and may exhibit electron-hole interference. Furthermore, these matrix elements can always be written as sums over electron and hole “Green’s function pathways” if the localized orbitals of the bridge 1 b,) are used as a basis (see figs. 2a and 4).

By partitioning according to the number of bridge holes, we have reduced the problem of electron transfer through a filled bridge to one of hole propagation through an isolated bridge that may involve multibole contributions(desc~bedbythe~~(~) ineqs. (56), (57)).

In the regime of very weak DA-bridge coupling, hutAt (E,,,) has only a one-hole propagation cont~bution (eq. (66) ), and the 25 reduction is a good approximation. The imputation of hDtAt (E;,,) as well as the complete 2s reduction is then an effective one-particle problem. If there is stronger coupling, hDtAt(Etuo) contains many-hole contributions and the 25 reduction is a worse approximation. In this case care must be taken in the description of the electronic state of the transfer system prior to the excitation of an electron in the donor (section 5 ) . For long-distance ET in proteins the maximum number of holes in the bridge (which in our model is 3), is very small with respect to the total number of bridge electrons. It is therefore expected that an “effective” one-hole description is a good approximation.

8. Conclusion

The main goal of this work has been to understand the two-state reduction for electron and hole transfer, both in the energy and time domains. Although we have chosen tint-binding and Hubbard-like Ham~tonians to describe the electronic dynamics, our results are largely independent of the many-electron H~iltonian considered. That is, our derivation of the expressions for the leading error introduced by a two-state reduction (in both energy and time domains) does not depend on the parameters of the Hamiltonian used. Rather these expressions are functions of the energies of the isolated (i.e. uncoupled from each other) donor-acceptor and bridge bands, and of the electronic couplings between these bands. For this reason these results are useful in any quantum-chemical calculation of the electronic energies and couplings of an electron-transfer system [ 5,22- 261. They may be used to judge whether the electron-transfer dynamics in this system can be adequately described by an effective two-state Hamiltonian, and to calculate the error associated with this Hamiltonian. Fur- thermore, they provide an alternative (see also ref. 134 J), systematic way of judging whether the effective matrix elements of this Hamiltoni~ can be computed using independent-panicle models,

Acknowledgement

We thank Jeff Regan and Nick Socci for helpful discussions. Work in San Diego was supported by the National Science Foundation (Grant No. ~CB-9018768). JNO is a Beckman Young Investigator and is partially in residence at the Instituto de Fisica e Q&mica de S&%o Carlos, Universidade de Sgo Paula, 13560, Sso Carlos, SP,

S.S. Swurtis et al. / Chemical Physics 176 (1993) XII-520 517

Brazil during summers. Tbe work in Pittsbur~ was supported by the DOE Advanced Industrial Concepts Di- vision and a NSF National Young Investigator Award.

For a pure 2s system described by the Hamiltonian #I3 &s =Eal~~<~l+E~lB>~Plf(II,,/~)~BI+ h.c. ), the Green’s function G,(E) is given by

H, Gm(E)= (E-E,)(E-Ep)- j&J’-

(67)

The SE of &, is j&(E) =O, where j&(E) = (E-Ea) (E-EB) - lHsul 2. Its roots (eigenenergies of &s> are given by E, =: (E, +Ep) /2 i ,/[ (E, - E,) /2 ] 2 + I Hm I 2. The residues of Green’s function are

Res [G,&E)l,+ =TH&(E+ -E-1, (681

so that

G,(t)= - fE HaE_ [exp(-~~~/~~-exp~-~_~/~~] +-

(see eqs. ( 5), ( 11) ). Therefore, the transition probability from 1 a} to I /3> is purely sinusoidal as a function of time:

Appendix B

We compute the residues of Gm(E) (eqs. (12), (13)) under the assumption that X’,(E,,,), r2(Eti,) +z 1 (eq. (20)). We use theformulaRes[GAD(E)]E,_L= [P(E)/&~(E)]~=~,,, wherep(E)=?z&E) andf(E) is given by eq. ( 16). Expanding h,(E) in p(E), &f(E) as in eq. ( 18 ), and keeping only terms of magnitude hii( gives ResIG&E)],,, x rh,(E,,,)lI~(Erun)-$(Etun)l, where 4, 4 are given in eq. (21).

These residues are 2S-like (see Apkndix A). The leading error in this c~culation (associated with the neglected

terms (E-E,,&.M&%,) 1 is approximately 8E&L) N I(~,z -Et,n)dEhADtEk,)l(S-~)l (with additional terms involving dEhDA(Etu,), dEhAA(Etul), and d&&Et,,,,) ).

The residues at the eigenenergies close to the bridge eigenenergies are obtained by rewriting G&E) as GAD(E) =P(E)/.#XE), where

J7.4dE) p(E)= (E-E,)(E-E,) ’

and

f(E) =ncEl _ ALA nhDD(E) + J%xAA(E) flkm&W)

E-E, - E-E, (E-E,)(E-EA) - (E-E,)(E-E,) ’

(69)

(701

*I3 Here by “Hamiltonian” we mean the more general operator I?--&$‘. Therefore 1 a) and /&I may be many-electron states, and the “energies” E,, Ep correspond to electron affinities.


In eqs. (69), (70),

II(E)= $, (E-E,,) , Ifh,(E)= m$, ~~~~~~ n (E-EBn) (71) n+m

(72)

where (iy)=D,A).AssumingthatTI(E,,),rZ(Etun)<<1,andexpandingp(E),S(E) in (E-E&/(E,,- E,,,), (~~-E~~)/(E~*-E~~), (EA-E,,,)/(E,,-E,,f,gives

p(E)* m$, E.;c> B

E tun Bm IItDE,, npm (E-EBn) (73)

and

(74)

K,,( Et& are the energy shifts of the bridge eigenstates I B,) induced by their coupling to I D) and I A) - In general 1 Km,( E,,) 1 +z i h,(Etun) I, i.e. the pe~urbation of 1 D) and I A) due to the bridge is much greater than the pe~urbation of the bridge due to 1 D) and I A). We can then neglect the K~~(E~~) so that f(E) =lf(E) (eq. (7 1) ). The residue at the eigenenergy &a EB,, is then given by Res[G,D(E)]E=a,;z:

~(_E,,)/~,~(EB,)=BA,P,D/~EB~- x!L,)(EB,,,-&~). Since r,(E,,,), Tz(E,,,)~I, both 6RdEtun) and Res[GAD(E)lE~:EB,<<Res[GAD(E)lE=d,,2 ineq. (22).

Appendix C

As an illustration of the 2s limit, we consider the symmetric system

~"~,,=~~A+EB~B><BI+~PID>(~I+~~A>(BI+~.~.), (75)

where~DA=EO(~D)(D~+~A)(A~)+(8~D)(A~+h.c.).ItisaSSumedthatEB~E~andii~O.Forthissystem GAD(t) is given by eq. ( 11) with n= 1-3. Jr-S; are the eigenenergies of &. We solve for &r-S; and Res[GAD(E)]E=h_bj,andexpandthesolutionsinpowersof8/(E,-E+).E+andE_ (E+>E_)aretheeige- nenergies of &,,A (“DA band”). They are given by E, - -E. + i7 and E_ =Eo - 17. EB is the energy of the bridge (“bridge band”). The expansion in /?/ (EB -E+ ) is valid (converges) if 1 PI < I EB - E+ I . A necessary condition for convergence is: I EB - E+ I > 0 (the bridge and DA bands are not resonant ) . The result of the expansion is

d$=E_, Res[GAD(E)]E=dh=-~,

to all orders, and

(76)

&+E+-2&, B +

Res[G,&E)]s=~=; -

4aEi+2&, RW%dE)lE=~= (77) B- +

to first and second order. In eq. (77) we implicitly assumed that I /31 -K 1 EB-E+ I since we have only retained terms up to second order. This is the limit of very weak coupling to the bridge. If in addition E, - E _ << EB - E_ ,

S.S. Skourtis et al. I Chemical Physics 176 (1993) 501420 519

expanding in j?/ (Ee - E, ) is approximately equivalent to expanding in j3/ ( EB -Et,,) where E_ s E,, 5 E+, since~/(E~-E+)~[l+(~~-~_~/(E~-E_)]~/(~~-~_~.

Beyond first order, the expansion in /3/ (EB-E+ ) is equivalent to an expansion in h(E+ )/ (Es-E+ ), where

h(E+)= - jJ2(EB-E+ ). In the new notation, the eigenenergies and residues in eq. (77) have been calculated to zeroth and first order in h (E, ) / (EB - E, ) respectively. Furthermore, the limit of very weak coupling to the bridge discussed above may be written as

Ih( <EB-E+ and E+ -E_ <<EB-E_ , (78)

whereas the (weaker) convergence limit is 1 h (E, ) 1-c EB - E, and E, -E_ <EB-E_. Substituting eqs. (76), ( 77 ) into eq. ( 11) and collecting terms with coefficients of the same order in h (E, ) / ( EB - E+ ), gives

G~(f)=G~~(f)+G~~(f)+... , (79)

where

GA%(f)= &[exp(_E.$)-exp(_itE++~(E+‘lf)],

@$0(t)=_ $B i[E+ +Zh(E+)lt i[Ea-2h(E+)]f

B -I- fi fi )I* (80)

(81)

TheETprobab~ity~~(~)=~*lG~(~)12is

PAD(t)=P~~(t)+P~~(t)+... ,

whereP~~(t)=R2~G~(t)~2,andP~(f)=2~2Re[G~(t)*G~~(f)~.Wehave

PA$j(f)=sin20,f, P&j(f)=-2s ( -sin2w,fSf cosm2f-+ coso,f) , B- +

(82)

(83)

whereo,=[(E+-E_)/2+h(E+)]/fi,02=[EB-E+-4h(E+)]/h,andWg=[EB-E_-2h(E+)]/~. So PAD(f) consists of a dominant sinusoidal oscillation of unit amplitude and frequency ol, and of much

smaller sinusoidal oscillations of amplitude - 2h (E, ) / (Ea -E, ) and frequencies oI-w3. The large oscillation arises from G&(t) (eq. (80) ) which is the Green’s function of

E,+h(E+) D+h(E+) p’(E+)=(F+h(E+)

(84)

Z@$!( E, ) is thus meaningful only in a limit such as eq. (78), where it describes the largest amplitude (zeroth order in h (E, ) / (&-E+ ) ) probability oscillation between D and A. It neglects smaller oscillations whose amplitudes are of order h (E, ) / ( EB - E+ ) or higher.

E, is an initial upper bound for E,, and it is a very good approximation if the conditions in eq. (78) hold. For weaker inequality conditions the initial E,, must be chosen between E_ and E, and/or improved iteratively. For example if 1 h (E, ) I <l&-E+ and E, - E_~E~-E_,abetterE~~~is~~=~++2~(E+~.Thedom- inant o~illation of Pm(f) is then described by a Hamiltonian &b.&( 6;) that neglects weaker osci~ations of magnitude2h(g;)f(EB-~~).Sinceh(~~)f(EB-~~)ch(E+)/(EB-E+),~~(cB;)isamoreaccuratedescrip- tion of the TS dynamics than @&( E, ). In general the optimum choice for Et,, is 4 c E,, < (s; where 4, % are obtained iteratively.

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The two-state reduction for electron and hole transfer in bridge-mediated electron-transfer reactions