J. Phys. Chem. 1994,98, 7395-7401
7395
Analytic Quantum Theory of Electron Transfer with a Reaction Mode Strongly Coupled to the Electron and Weakly Coupled to the Bath Katja Lindenberg' Department of Chemistry and Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0340
Emilio Cortest Physics Department, Indiana University-Purdue University at Indianapolis, Indianapolis, Indiana 46202- 3723 Robert M.Pearlstein* Physics Department, Indiana University-Purdue University at Indianapolis, Indianapolis, Indiana 46202-3723, and Chemistry Division, Argonne National Laboratory. Argonne, Illinois 60439-4831 Received: March 14. 1994'
We reconsider the old problem of donor-acceptor electron transfer with a strongly coupled reaction mode. In the past, this problem has been treated either heuristically or numerically, but not analytically. Here, we develop an effective analytic framework built around our own extension of standard small polaron theory. We assume a Hamiltonian in which the electron operators are strongly coupled to a single linear oscillator, the reaction mode, which is in turn weakly coupled to the heat bath. The Hamiltonian is transformed so that the previously bare electron is dressed in the quanta of the reaction mode but not in the quanta of the rest of the bath modes. The dressed electron, or "reacton", has no residual interaction with the transformed reaction mode; its remaining interactions with the bath can be treated perturbatively. Although our formalism describes the full kinetics, we present in detail here only the results of a Golden Rule calculation of the electron-transfer rate constant. We find that in the strict high-temperature limit the rate constant is of the Marcus form but with a reorganization energy that is simply the product of the reaction mode quantum energy and the dimensionless (strong) coupling constant squared, independent of the details of the phonon spectrum. This contrasts with earlier findings based on the standard polaron model that the reorganization energy is a weighted sum over the bath mode frequencies. We observe that our result may provide a basis for explaining the anomalously small values of the reorganization energy deduced for primary charge separation in photosynthetic reaction centers. Finally, we discuss the lowest order corrections to the high-temperature rate constant, noting the sensitivity of these to the nature and symmetry of the coupling between electron and reaction mode.
1. Introduction
The capture and conversion of solar energy in natural photosynthesis is a process of great scientific, and potentially great practical, interest. Among the steps not yet thoroughly understood are those involved in the photochemistry itself, the primary separation of positive and negative charges that proceeds with near-unit quantum yield. In trying to describe these initial steps, one usually assumes that the Marcus theory of electron transfer (in quantized form, if necessary) can be used to calculate the electron-transfer rate constant.' However, this leads to difficulties. A problem of long standing has been the temperature dependence of the rate constant, experimentally much weaker than predicted.* More recently, it has become apparent that a critical parameter in the Marcus rate-constant formula, the reaction reorganization energy, must be assigned an order-ofmagnitude smaller value in photosynthesis (as compared to electron-transfer reactions in solution) to obtain agreement with observed rate constants.3 There is no basis in existing theory to explain such a large difference in reorganization energies. This paper represents a f i t step to develop an analyticquantum theory of electron transfer that may eventually explain these and other anomalies. Here, we lay the general groundwork for our approach and outline the steps that we follow to arrive at our ~~
~~
t Permanent address: Departamento de Ffsica, Universidad Aut6nma
Metropolitana Iztapalapa, P.O.Box 55-534, 09340 Mexico D. F., Mexico t Argonne Fellow. Sabbatical address: Argonne National Laboratory. 0 Abstract published in Aduance ACS Absrrocrs, July 1, 1994.
0022-3654/94/2098-7395So4.50/0
results for the electron-transfer rate. Detailed results for a variety of specificmodels and conditions will be presented ~ubsequently.~ Most theoretical treatments of electron transfer between a donor and an acceptor assume the existence of a 'reaction coordinate", a particular vibrational mode that facilitates the transfer. In some instances, model Hamiltonians are written in which the electron operators are directly coupled to the many vibrational modes of the heat bath.' In other calculations, the reaction mode is separated from the bath modes, with the electron operators coupled directly only to the reaction mode and the latter coupled to the bath modesas The problem with direct electron-bath coupling has been studied analytically for a long time, sometimes using the small polaron approach.61~In the hightemperature limit in appropriate parameter regimes this approach yields the familiar Marcus formula for the electron-transfer rate constant. The electron-transfer rate in this formula depends not only on the energy difference between the donor and acceptor sites but also on the so-called 'reorganization energy" that embodies the response of the medium to the electronic excitation. This reorganization energy depends on the phonon spectrum of the heat bath and on the coupling between the electron and the heat bath. In contrast, the problem with separated reaction mode, presumably a more realistic representationof the physical process, has up to now only been treated numerically* or heuristically.' To accommodate the effects of the strong electron-phonon coupling in the small polaron problem, the Hamiltonian is transformed so that the entity being transferred is no longer a bare electron but one "dressed" with phonons.' The motion of Q 1994 American Chemical Society
7396 The Journal of Physical Chemistry, Vol. 98, No. 30, I994
this heavier quasiparticle is perturbed by residual interactions with the heat bath. In other words, the original electron moves through the crystal carrying with it a strongly coupled cloud of phonons. We adapt these methods to the reaction coordinate problem. We assume that the electron is strongly coupled to the reaction mode, which is in turn weakly coupled to a heat bath. We then transform our Hamiltonian so that the entity being transferred is now an electron dressed with quanta of the reaction mode. The dynamics of the reaction mode are in turn affected by the heat bath, which thus indirectly affects the dynamics of the electron. The resulting reorganization energy will then depend on the frequency w of the reaction mode as well as on the spectrum oftheheat bathandthewaythat thereactioncoordinateiscoupled to the heat bath. In section 2 we set forth our model Hamiltonian and describe its canonical transformation. Section 3 outlines the Golden Rule calculationthat yields the transfer rate constant. In section 4 we discuss the equations of motion for the bath and reaction-mode operators and provide explicit analytic solutions of the latter in terms of the former. Section 5 contains our results for the transition rate, with recovery of the Marcus formula in the hightemperaturelimit,but with a reorganization energy A that depends explicitly on reaction-mode frequency w. A brief discussion of future work is presented in section 6.
Lindenberg et al. these two energies to be occasionally equal (in resonance), thus facilitating the electron tranefer. The simplest modulational interaction of this form is embodied in the interaction Hamiltonian
where the gj are (strong) dimensionless coupling constants. This coupling induces a modification of the electronic energies as a function of the displacement of the reaction coordinateoscillator. In the other model, in analogy to the quantum theory of solidstate excimers? one assumes that the reaction coordinate modulates the electronic transfer coupling parameter J. In this modelone thinksofthereactioncoordinateas thedistancebetween the monomers and the effective transfer rate as a function of this distance, so that the transfer may occur more easily when the monomers are closer together. Such an interaction is embodied in a Hamiltonian contribution of the form HEC= hw(gAltA2 + g*AztAi)(o + C). The interaction between the reaction coordinate and the heat bath phonons is in any case assumed to be bilinear in the respective displacements, I
2. The Hamiltonian and Its Canonical Transformation
We consider the electron-transfer problem between a donor and an acceptor. The donor and acceptor electron states are each coupled strongly to a reaction coordinate, which in turn is weakly coupled to a surrounding thermal bath. Our aim is to calculate the electron-transferrate and the way in which this rate depends on the characteristic frequency of the reaction coordinate. We propose a model Hamiltonian that is sufficiently simple to allow the explicit analytic determination of the dependence of the transfer rate on the frequency of the reaction coordinate. We begin with the Hamiltonian
The dimensionless coupling constants y. are small (weak coupling), and hermiticity of the Hamiltonian requires that y-" = y,*. Note that the full Hamiltonian conserves the number of electrons but that thelevelsofexcitationofthereactioncoordinate and of the phonons are unrestricted. The Hamiltonian H cannot be diagonalized exactly, and HEC cannot be treated as a perturbation because the electron-reaction coordinatecoupling is assumed to be strong. It is therefore useful to introduce a canonical transformation that incorporates the main contributionsof this coupling. This is accomplished via the standard small polaron transformation.' We define the operator
s = (g,*A,tA, + g,*A,tA,)Ct - (g,A,+A,+ g,A,tA,)C
Here HEis the isolated electronic Hamiltonian
(7) The operator A; (A,) creates (annihilates) an electron on sitej (we take site 1 to be the donor and site 2 to be the acceptot). These operators obey the usual Fermi commutation relations. E, is the energy of the electron on site j . J is the electron-transfer coupling parameter. We assume there is a singleelectronpresent. The isolated reaction coordinate is taken to be harmonic and therefore described by the Hamiltonian
Here 0 (C) creates (annihilates) a quantum of excitation of the reaction coordinate,and w is its frequency. The operators 0 and Cobey the Bose commutationrelations. The heat bath is assumed to be represented by a collection of phonons indexed by Y and of frequency w,: (4) Y
Here bJ and b, respectively are Bose operators that create and annihilatea phonon of frequency w,. All the A, C, and b operators commute with one another. Two different models might be considered for the coupling between the electron and the reaction coordinate. The usually chosen coupling modulates the energiesEl and E2 of the electron on the donor and acceptor sites.5 Such a modulation would cause
and introduce the transformed operators z related to the original operators Z by
z = esZes
(8)
One finds the transformed electron operators aj = esAf
= ApheO*fl
(9)
that describe an electron accompanied by a "distortion" of the reaction mode ( " d r d ' electron or "reacton*). The transformed reaction mode operator is
c = e%eS = C + ~ , * A , + A+, g , * ~ , t ~ ,
(10)
which simply represents a shift in the equilibrium position of the oscillator. The phonon operators are unaffected by the transformation. The transformation leaves all the commutation relations unchanged; furthermore, the a, c, and 6 operators still commute with one another. These relations lead to the products a,'a, = A,'A~
(that is, the electron number operator is unchanged by the transformation),
The Journal of Physical Chemistry, Vol. 98, No. 30, 1994 7397
Analytic Quantum Theory of Electron Transfer
where
In terms of the transformed operators the Hamiltonian (1) can then be written as
where
He = ( E , - hw)g,12)altal+ (E2- hw)g2l2)a2a2 (16)
Each term in the sum describes a transition from an initial state in which the dressed electron is on the donor (electronic state l), the reaction coordinate is in some state IC,), and the phonons are in a state Ibl), to a final state in which the electron is on the acceptor (electronic state 2), and the reaction coordinate and phonons are in states designated by a subscript 2. Since the changes in the states of the reaction coordinate and of the phonons are not of interest, these are summed over in (24). The time argument in the Hamiltonian HOE(?) indicates the usual interaction picture,
where HOis the Hamiltonian without the contribution Ha:
Ho
H - H, = He
+ H, + Hb + Hcb + Heb
(26)
(19) Y
The transformed problem thus describes a dressed electron, or reacton, that is transferred from donor to acceptor (or from acceptor to donor) more slowly than the original bare electron (as described by Jexp[f (g*ct -gc)]). The energies of the dressed electrons are lower than those of their bare counterparts. There is no other direct interaction between the dressed electron and the modified reaction mode. The interaction between the reaction coordinateand thephonon bath is thesameasin theuntransformed problem, and the canonicaltransformation has introduced a direct interaction Hcb that was not present (or assumed negligible) in the untransformed Hamiltonian. The Hamiltonian (1 5 ) is our point of departure for further analysis. In particular, weconcentrateon the transfer of a reacton from donor to acceptor.
Some matrix elements that appear in (24) can be evaluated trivially. The fact that there is only one electron in the system immediately leads to (lla2t(t) al(t)12) = 0 for all t. Also, it is straightforward to show that
where Ae is the (generally positive) donor-acceptor energy difference
A€ = €1 - €2
(28)
With these results and the explicit form of HOEin (24) we can carry out the sum over the "intermediate" (primed) phonon and reaction coordinate states (since each of these is a complete set of states) to write
3. Physical Observable: Rate Constant
Although the full k i n e t i c ~ ~ . ~can ~ J lbe obtained from our formalism, the simplest quantity to calculate is the transition rate constant for a reacton to transfer from the donor to the acceptor. The Golden Rule can be used if the coupling parameter J is small (especially since the canonical transformation further weakens the effective transfer parameter). To carry out this calculation, we define the eigenstates of the diagonal portion of the electronic Hamiltonian as 11) and 12), that is,
where ej are the energies of the dressed electrons,
The eigenstates of the modified reaction coordinate and bath Hamiltonians H, and Hb are written as In) and I{nY)), respectively. Here n denotes an occupation number for the reaction mode and (n,) denotes a set of phonon occupation numbers. The transition rate according to time dependent perturbation theory then is7
It is usually further assumed that the subsystem consisting of the phonon bath and the reaction coordinate are initially in thermal equilibrium. This allows us to rewrite (29) as
w(t)= ~2~iAc'/hL T~, ~ ~ ~ + H b + ~ ~ ) / k ~ , [ g c t ( 1 ) - g c ( t ) l e - ~ C t - g c 1 z (30)
where Tr denotes the trace over the heat bath and the reaction coordinate (sometimes explicitly denoted as Tr = Tr,Trb below), Z z c z b is the partition function, k is the Boltzmann constant, and Tis the temperature. In (30) and below we follow the usual convention whereby an operator without a time argument represents the operator at time r = 0. The further explicit calculation of W(t) requires the solution of the equation of motion for the c-operator. 4. Equations of Motion and F o d Solution
The Heisenberg equations of motion for the creation and annihilation operators in the interaction picture are
i
i.(t) = i;[HoAt)l
where
(31)
where the dot indicates a time derivative and the square brackets denote the commutator.
7398 The Journal of Physical Chemistry, Vol. 98, No. 30, 1994 Our Hamiltonian in (31) with the Bosewmmutation relations for the c and b, operators yield coupled equations of motion for these operators. Elsewhere? we derive these equationsand solve explicitly for the bath operators, thus arriving at the following exact equation of motion for the reaction coordinate oscillator:12 C(t)
= -iw'c(t)
+ iKct(t) - fit) w l d r K(f-T)[C(T) - C t ( 7 ) ] ( 3 2 )
Lindenberg et al. Note that all explicit operators in (42)are initial values. The combination of interest to us, g*ct(t) -gc(t), then has the Laplace transform obtained by combining (39) and its. hermitean conjugate:
+
1 g*Zt(s) - gZ(s) = --+[g*(s iw) - i(g* + g ) K L(s) i(g* g)sk(s)]ct- [g(s - iw) + i(g* + g)K i(g* + g ) s k ( s ) ] c+ [i&) - 2iReg,k(s)][(g* + g)s +
+
+
where the "dissipative kernel" is
(g* - g)iwll (44)
K ( t ) = 2 z w , l y , 1 2cos out
(33)
Y
the modified oscillator frequency of the reaction coordinate is
w'Ew-K, KrK(0)
(34)
and the "fluctuating operator" is
At) = F(t) + K(t)G
(35)
G=ct+c-2Regl
(37)
where
The asymmetry between sites 1 and 2 in (37)results from our explicit initial condition that places the electron on site 1 at time t = 0. Thesolutionof eq 32together with its hermiteanconjugate provides the input necessary to calculate the integrand W(t)of the transition rate w. Since (32)is linear and the integral term involves convolutions, we can solve it formally by taking its Laplace transform and that of its hermitean conjugate to obtain a set of algebraic equations for the transforms
L [ c ( t ) ]= E(s)
The full solution of the problem requires the Laplace inversion of (44). This inverse transform is determined by the singularities of (44),that is, by_thezeroes of the denominator L(s) and by the poles s = f i w , of F(s). The fact that the kernel K(t) decays in time in general leads to zeroes of L(s) that in turn lead50 decaying contributions to pct(t)- gc(t), while the poles of F(s) lead to oscillating contributions to g*ct(f) - gc(t). In the remainder of this paper we make one simplifying assumption that shortens the calculation considerably and yet exhibits many of the important results of this model, namely, that the coupling parameters gl and g2 and hence also the parameter differencedefined in (14)are purely imaginary. This in effect imposes particular phases and phase difference of the coupling of the donor and acceptor sites to the reaction coordinate.13 It is straightforward but cumbersome to allow the gi to be complex. With the gi purely imaginary (44)becomes
g*P(s) - gZ(s) = -g[Zf(s)+ E @ ) ]
The inverse Laplace transform of (45)then is
g'cf(t) - gc(t) = -g[cT(t) + c(t)]
c d t e"'&), with initial conditions
where L denotes the Laplace transform. This set can be solved ea'sily to yield
Here we have introduced the function
and its hermitean conjugate. Here
where L-1denotes the inverse Laplace transform and where
L(s) = s2
+ 2wsk(s) + w2 - 2wK
and k(s)is the Laplace transform of K ( t ) ,
Z(t) = L - y l / L ( s ) )
(49)
We have also introduced e(t)= cw,[rj,(r)e'"gb,t
+ y,*~,,*(t)e-'"~b,,]
(50)
Y
Insertion of (46)into (30)then gives Furthermore,
?&) = c - i&s) - ik(s)(ct + c - ZReg,)
W(t) = ~2~iAet/AL T~e ~ H s + H ~ + H ~ ) / k T e 2 ~ ( I ) e - g [ ~ * ( I ) ~ t + P ( r ) e ] e g ( c t + c )
z
and &) is the Laplace transform of (36),
(51)
5. Evaluation of Electronic Transition Rate Y
s-iw,
s+iw,
We now calculate the transition rate (23)with the integrand (30). using in the latter the inverse Laplace transform (46)of
Analytic Quantum Theory of Electron Transfer
The Journal of Physical Chemistry, Vol. 98, No. 30, 1994 7399
(45). In this section we discuss the results of this calculation as far as it can be carried out for an unspecified heat bath. The calculation of W(t)first of all requires the implementation ofthevarious traces whereby operator expressionsin theexponents are converted to c-number expressions. When this calculation is carried out, we arrive at the following result:
W(t) = jzeiAd/h
exp(-lA2[(N+ '/2)(lP(t)12+ 1) - ( N + 1 ) P W -
NP*(t)lI
rI
exp[-4(nu + 1/2)MZ~Z~,Zlr,W~J21 x
Y
ex~WQ:(t)
+ gQ,(t)[(N+
1)(P*(O - 1) + N(PD(t)- 1)11 (52)
Here Nis the Bose occupation number for the reaction coordinate,
MU
)+
Q(t) = 2 c cos w, ( t - iy , L(iw,) decaying contributions due to zeroes of L(s) (61) The first termpeakssharplyat t = t,=ihg/2, while theremaining contributions are expected to vary smoothly. It is therefore appropriate to carry out the integration indicated in (23) using a steepest descents approach. The procedure is standard and is described in detail in ref 7: one expands 4(t) about t,, ( t - tSl2 4(t) = 40,) + (2 - t,)b'(tS)+ 7 4 " ( t s ) +
*a*
(62)
The expansion is retained to second order. The integration path in (23) is deformed in complex space so as to go around t,, to obtain
(53)
Q,(t) = 2ghwfl[(n,
+ l)n,] I / Z w, lry12[z,(t)efoJe%hw,/2 + 2
z,* ( t )eJoJe-%hwJZ](54)
where nu is the Bose occupation number for the phonons, 6 = l/kT, and z,(t)
= L d . e-fwvrZ(7)
(55)
Our next step is to explore whether expression 52 can be written in a form that leads to a Marcus-type expression for the electron transition rate constant. Toward this purpose we write (52) in the form
To implement (63) requires the evaluation of 4 ( t ) and its first and second derivatives at tl. This in turn requires the corresponding evaluations of the various functions that enter in the expression for + ( t ) as well as their derivatives. To carry out this program we note that Laplace cansformation of (62) yields a series in inverse powers of s for +(s). On the other hand, all of the functions that we deal with in 4(t) have simple combinations of l/L(s) and the factors (s f iw,) as their Laplace transforms. It is a straightforward matter to expand each of them in inverse powers of s and then to recollect them into the transform of (62). The result is an explicit expression for each of the derivatives of interest. The procedure is tedious but straightforward. The result of this procedure in the high-temperature limit is an expression for the transition rate of the form
where
A€ 4(t) = i-r - 1A2[(N+ l/z)(lP(t)lz+ 1) - (N + l)P(t) h
where A(@) is in general temperature dependent and can be identified as the so-called "reorganization energy" that defines the effective activation barrier that the dressed electron or reacton must jump over to go from the donor to the acceptor. We find that to the lowest order in 6 this energy is given by the simple temperature-independent expression
A = M2hw The function
is sharply peaked around t = t, ih/3/2 [and therefore so is # ( t ) ] . To see this, consider its Laplace transform, which can easily be obtained from (54) because (55) is a convolution:
(65)
The form (65) is reminiscent of that obtained with a direct strong electron-heat bath coupling? to wit, A a E,& J2hwy,where u, is a direct electron-phonon coupling constant. Here, however, the reorganization energy to leading order is determined entirely by the coupling of the electron to the reaction coordinate and by the frequency of thereaction coordinate. The coupling of the reaction coordinate to the heat bath and the spectrum of the heat bath first enter in the corrections of order 62 in the exponent in (64) and in A@). These corrections, which can easily be found analytically, will be explored in detail in a subsequent paper.4 6. Discussion
where
M u 2ghofl[(n, + l)n,I
112
w,
2
lrJ2
(60)
The inverse transformation of Q(s) contains contributions from the poles at s = f i w , and also from the zeroes of L(s);the former lead tooscillating portions and thelatter todecaying contributions due to the dissipative nature of the phonon bath. Recalling that L(-io,) = L(iw,), we can write
In this paper we have introduced the idea of a reacton, an electron dressed in the quanta of a single,strongly-coupledreaction mode which is in turn weakly coupled to a heat bath, and we have laid the groundwork that shows that analytical means can be used to calculate the kinetics of reacton transfer. While many detailed results are yet to be completed and will be published subsequently? it is already possible to draw some conclusions of interest.
7400 The Journal of Physical Chemistry, Vol. 98, No. 30,1994
Lindenberg et al.
is then of the form (64), with the reorganization energy identified as
A = $r)2
r
r
REACTION COORDINATE Figure 1. Schematic of the potential well configurationsunderlying our model. The donor and acceptor wells are both harmonic with the same force constant k. The donor well has a minimum at energy tl and coordinate rl, and the acceptor well has a minimum at energy tz and coordinate rz. The effective barrier that an electron must overcome to go from the donor to the acceptor well is Em.
In the high-temperature limit we obtain the usual form for the transition rate of the dressed electron in terms of a reorganization energy A that is a measure of the rearrangement of the medium that accompanies the electron-transfer process. The form of the reorganization energy to leading order in the inverse temperature is reminiscent of the form of the reorganization energy in the standard strong polaron model in which the electron is directly and strongly coupled to a heat bath but now involves only the coupling of the electron to the reaction mode. We find that A = i 2 h w , where g is the electron-reaction mode coupling parameter and w is the frequency of the reaction mode. Note that this result potentially explains the observation3 that the reorganization energy for primary charge separation in photosyntheticreaction centers is much smaller than in electron-transfer reactions in solution. This would be so if, for example, in the former many fewer modes are strongly coupled to the electron than in the latter. The phonon spectrum of the heat bath to which the reaction mode is in turn coupled enters in the 0(b2) corrections to this result. The effect of these corrections on the electron-transfer rate will in general depend on whether w does or does not fall within the phonon band, since the reaction mode acts as a filter between the electron and the heat bath. A number of points should be stressed, some straightforward and some more subtle. They are perhaps most easily understood in the context of Figure 1, where we sketch the potential well configurations that one should bear in mind when dealing with the familiar form (64) for the electron-transfer rate. Some of these points are not peculiar to our particular model but are indeed relevant regardless of the way in which the electronic energy states are ultimately coupled to the heat bath. The form (64) is easily understood in terms of the configuration shown in the figure. In the figure (as in our model) each of the potential wells is harmonic,
A straightforwardanalysisbased on this sketch leads to the relation
where Ar = rl - r2. An activated transition over the barrier with rate
The reorganization energy thus depends on the separation Ar between the minima of the donor and acceptor potential wells. The separation Ar is in turn determined by the coupling of each of the electron states to the reaction mode (or, in the standard polaron model, to the heat bath). The form (68) with (69) is precisely (64) with no corrections if one assumes that Ar is a constant. More accurately, we should recognize that Ar is a fluctuating quantity and that the average activation energy should more accurately be written as
Thedifferencebetween ( ( A r ) - 2 ) and ( ( A r ) Z ) - l will then introduce a temperaturedependent correction to the high-temperatureresult for the electronic transition rate. Another point to note is that the result (64) with (65) to the leading order in the inverse temperature could have been arrived at without even introducing a phonon bath, that is, by simply considering the electron states in the donor and acceptor to be strongly coupled to a reaction coordinate. This is certainly a simpler problem than the one we have considered and would not have required the extensive machinery that we have introduced above. The temperature in this simpler model enters even in the absence of an explicit heat bath to which the reaction mode is coupled because we have assumed that this mode is initially thermalized. There are at least two reasons (to be pursued in future work) for us to have introduced the more complex model, even though the results exhibited here are only the simplest ones: (1) The reaction coordinate might not be initially thermalized, that is, it might be in an initial specific configuration created together with theelectron at thedonor site. In this casean explicit heat bath is certainly necessary for a temperature to appear in the problem, and the reorganization energy in that case would depend on the way that the reaction coordinate couples to that heat bath; (2) the higher order inverse temperature corrections to the results exhibited here are interesting in their own right since the temperature in physical situations is not necessarily so high as to make these corrections negligible. These corrections of course require that the explicit coupling of the reaction coordinate to the heat bath be retained. We also stress that the behavior of the reorganization energy and more generally of the transfer rate constant are expected to be sensitive to the nature and symmetry of the coupling between the electron and the reaction mode. In this paper we have treated only diagonal coupling, that is, coupling that modulates the donor and acceptor electron levels separately. This is the case usually treated in electron-transfer theory. Even within this constraint we have dealt only with purely imaginary coupling coefficients. Diagonal coupling allows the electronic energy levels that are on the average separated by energy At to now and then become resonadt by this modulation. A more general diagonal coupling model with complex coupling coefficients may be important and can easily be accommodated. An alternative model to the one considered in this paper is a nondiagonalcouplingmechanism;that is, one in which thereaction mode modulates the electron-transfer rate J so that, for example, the electron finds it easier to jump from donor to acceptor when the two are physically closer together rather than farther apart.
Analytic Quantum Theory of Electron Transfer This sort of coupling is expected to lead to a different form for the reorganization energy, a prediction that will also be explored el~ewhere.~ Acknowledgment. K.L. gratefully acknowledges support from the US.Departmentof Energy Grant No. DEFG03-86ER13606 and from the National Science Foundation Grant No. NSF/ DMR 91-6731. R.M.P. is grateful to Dr. James R. Norris for an introduction to the subject of electron transfer and for his hospitality during the summers of 1992 and 1993, when R.M.P. was a Faculty Research Participant at the Argonne National Laboratory.
References and Notes (1) Marcus, R. A,; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. (2) Fleming, G. R.; Martin, J. L.; Breton, J. Nurure 1988, 333, 190. (3) Wang, Z.; Pearlstein, R.; Jia, Y.; Fleming, G. R.; Norris, J. R. Chem. Phys. 1993, 176, 421. DiMagno, T. J.; Rosenthal, S. J.; Xie, X.; Du, M.;
The Journal of Physical Chemistry, Vol. 98, No. 30, 1994 7401 Chan, C.-K.; Hanson, D. K.;Schiffer, M.; Norris, J. R.; Fleming, G. R. In The Photosynthetic Bucreriul Reaction Center I& Breton, J., Venniglio, A,, Eds.; NATO AS1 Series; Plenum: New York, 1992; pp 341-350. (4) Pearlstein, R.; Lindenberg,K.;Cortes, E. Manuscript in preparation. ( 5 ) Jean, J. M.; Friesner, R. A.; Fleming, 0. R. J . Chem. Phys. 1992, 96, 5827.
( 6 ) Warshel, A.; Chu, S.T.; Parson, W. W. Science 1989,246, 112. (7) Mahan, G. D. Many Particle Physics; Plenum: New York, 1981. (8) Jean, J. M.; Fleming, G.R.; Friesner, R. A. Ber. Bunsen-Ges. Phys. Chem. 1991.95, 253. (9) Wu, T.-M.; Brown, D. W.; Lindenberg, K.Phys. Reo. B 1993,47, 10122. (10) Skourtis, S. S.;da Silva, A. J. R.; Bialek, W.; Onuchic, J. N. J . Phys. Chem. 1992.96. 8034. (11) Gehlen, J. N.; Marchi, M.; Chandler, D. Science 1994, 263, 499. (12) Lindenberg, K.;West, B. J. Phys. Reo. 1984,30,568. Lindenberg,
K.;West, B. J. TheNonequilibriumStatistical Mechanicsof Openand Closed Systems; VCH: New York, 1990. (13) Brown, D. W.; Lindenberg, K.; West, B. J. J. Chem. Phys. 1987,87, 6700.
