Theory and Control of Thermal Photoinduced Electron Transfer

A Gaussian Wave Packet Propagation Approach to Vibrationally Resolved Optical Spectra at Non-Zero Temperatures. Ch. Sridhar Reddy and M. Durga Prasad...
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J. Phys. Chem. B 2001, 105, 6500-6506

Theory and Control of Thermal Photoinduced Electron Transfer Reactions in Polyatomic Molecules† Eli Pollak* and Yong He Chemical Physics Department, Weizmann Institute of Science, 76100, RehoVot, Israel ReceiVed: NoVember 21, 2000; In Final Form: February 27, 2001

The nature of the nascent vibrational distribution in the excited donor state in photoinduced electron transfer is shown to have a profound effect on the electron-transfer rate. In polyatomic molecules, excitation at wavelengths in the vicinity of the ground state to ground state excitation frequency may lead to significant cooling of the excited vibrational state distribution. This cooling is shown to lead to a slowing down of the electron-transfer rate. A theory for photoinduced electron transfer is developed to include the nonequilibrium nature of the excited donor vibrational distribution. The rate expression is shown to be the standard Golden rule thermal rate expression but at an effective temperature which depends on the ground electronic state temperature and the photoexcitation frequency. A simple numerical model is presented to demonstrate the cooling and control of the electron-transfer rate by variation of the excitation frequency.

I. Introduction The immense interest of the scientific community in electrontransfer processes is exemplified by two recent volumes of AdVances in Chemical Physics, entitled “Electron Transfer-From Isolated Molecules to Biomolecules”1. An excellent review by Bixon and Jortner2 may be found in the same volume. With this background, one may ask, what is there still to add from the theoretical side that has not yet been addressed? In recent years, we have stressed the fact that electronic excitation of a room-temperature polyatomic molecule for example from an S0 to an S1 state can lead to a significant change in the vibrational population of the molecule in the excited state.3,4 Within the Condon approximation, if the excitation wavelength is to the blue of the transition frequency from the ground vibrational state of the ground electronic state to the ground vibrational state of the excited electronic state (ω00), then the molecule is usually heated. Interestingly, at the ω00 transition frequency or somewhat to the red of it, one may expect under rather general conditions4 that the nascent distribution will be cooled. The cooling effect is predicted to be generic for polyatomic molecules4 and is caused by the lowering of vibrational frequencies in the excited electronic surface (in general, this lowering reflects the weakening of the chemical bonds due to the excitation of the electron). The cooling phenomenon has been used to explain3 the rather large differences between room-temperature fluorescence decay rates of stilbene in the gas and liquid phases. In the gas phase, at the ω00 frequency, the excited molecule is cooled, and the activated rate for isomerization is slow. In the liquid, the liquid rapidly reheats the molecular vibrations, and one observes a much faster decay rate. The purpose of the present paper, is to suggest that the same phenomenon can also cause significant slowing of electrontransfer processes. For this purpose, we consider the rate expression for electron transfer, within the golden rule limit, but include in it the effect of the nonequilibrium vibrational †

Part of the special issue “Bruce Berne Festschrift”. * Corresponding author.

distribution of the photoexcited donor molecule. The theory of nonequilibrium photoinduced electron transfer has been considered previously, for example, in refs 5 and 6; however, these authors considered a one-dimensional system coupled bilinearly to a dissipative heat bath, while here we concentrate on the multidimensional vibrational properties of the system. Anharmonic effects in photoinduced electron transfer have been recently studied by Evans;7 however, nonequilibrium effects due to the photoexcitation process were not discussed. A twodimensional system has been recently studied by Jean.8 Electron transfer in jets has been studied extensively,9,10,11 and the golden rule expression for the rate from a given energy is well-known and understood2. The new aspect presented here is that we consider the nascent distribution from a thermal distribution of ground electronic state donor molecules. The framework is thus of three electronic states and their vibrational manifolds. These are the ground electronic state of the donor molecule, the electronically excited state of the donor, and the acceptor state. The expression used for the rate constant reduces to the standard one if relaxation of the nascent excited donor state population is much faster than the electron-transfer rate. In the classical limit, the rate expression reduces to the Marcus expression but at a temperature which differs from the temperature of the ground electronic state donor molecule. This temperature depends on the photoexcitation frequency, photoexcitation pulse duration, and temperature of the ground electronic state donor molecule. In section II, we consider photoinduced electron-transfer rate theory for an isolated polyatomic molecule. Some examples, using harmonic models, are presented in section III. Relaxation of the nascent population of the donor molecule in the excited state is introduced in section IV. We end in section V with a discussion of the conditions needed and the experimental verification of the cooling phenomenon and its effect on electron-transfer reactions. II. Electron Transfer Rate Theory A. Two-State Case. To set the stage, we review briefly the golden rule expression for the electron-transfer rate for an

10.1021/jp004264j CCC: $20.00 © 2001 American Chemical Society Published on Web 05/02/2001

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J. Phys. Chem. B, Vol. 105, No. 28, 2001 6501

isolated donor acceptor system. We assume that the donor (acceptor) states are described by the vibrational Hamiltonian HD (HA). The two Hamiltonians have eigenfunctions |ψDj〉 (|ψAj〉) with associated energy eigenvalues EDj (EAj). The zero of the energy is taken to be the bottom of the potential well of the donor vibrational potential. The golden rule expression for the thermal rate at temperature T from the donor to the acceptor state is2

ΓArD(β) )

2π

∑j ∑k e-βE

|V|2

pZD(β)

|〈ψDj|ψAk〉|2δ(EDj - EAj)

Dj

(2.1)

where V is the electronic coupling term between the donor and acceptor states, β ) 1/kBT, and ZD(β) ) ∑je-βEDj is the partition function of the donor at the temperature T. Using the Fourier representation of the Dirac δ function, this expression may be conveniently rewritten as a time integral of a correlation function

ΓArD(β) )

1 |V|2 p ZD(β) 2

∫-∞∞ dtχAD(t,β)

(2.2)

PD(E; β) )

(2.3)

This correlation function has the same form as the one appearing in the golden rule expression for the electronic absorption spectrum in the Condon approximation.12 As also discussed below, a cumulant expansion up to second order for the correlation function may be justified. Thus, we may write down

χAD(t,β) = ZD(β)ei(t/h)XDt-(t /2p )XDtt 2

2

(2.4)

where brackets are used to denote the thermal average 〈Y〉 ≡ (TrYe-βHD)/(Tre-βHD)

XDt ≡ 〈∆H〉 ≡ 〈HA - HD〉

(2.5)

XDtt ) 〈∆H〉2 - 〈∆H〉2

(2.6)

Inserting this expansion into eq 2.2 gives the approximate expression for the thermal electron-transfer rate as6

x

|V|2 ΓArD(β) = p

2π -(X2Dt/2XDtt) e XDtt

(2.7)

∫08 dE PD(E;β)ΓArD(E)

(2.9)

FD(E) ) Trδ(E - HD)

(2.10)

we note that

ZD(β)ΓArD(β) )

∫0∞ dE e- βEFD(E)ΓArD(E)

(2.11)

The energy dependent transfer rate may therefore be written down formally as

ΓArD(E) )

ilt(ZD(β)ΓArD(β)) FD(E)

(2.12)

where ilt stands for the inverse Laplace transform operation. Formally, one may write down the energy dependent electrontransfer rate also as

ΓArD(E) )

2π 2Trδ(E - HD)δ(E - HA) |V| p Trδ(E - HD)

(2.13)

This expression is difficult to compute directly, unless one introduces a coarse graining procedure.2 We will use the inverse Laplace transform route as discussed further in section IV below. The experiment considered in this paper is one in which the donor molecule in the ground electronic state is inert while photoexcitation to an excited electronic state leads to an electrontransfer process. This is described in terms of a three-state model, in which Hg is the ground electronic state vibrational Hamiltonian of the donor molecule, HD is the excited electronic state vibrational Hamiltonian of the donor molecule, and HA is the vibrational Hamiltonian of the acceptor molecule. The energy distribution in the ground electronic state of the donor is thermal at temperature T. Under these conditions and assuming photoexcitation with frequency ω, it has been shown in ref 4 that the normalized nascent energy distribution in the excited electronic state of the donor is given in the Condon approximation by the expression

PD(E;ω,β) ) 1 (2π)2pZ(ω;β)

∫-∞∞ dt ∫-∞∞ dt′ e-i(t/p)(∆E

Dg-pω)

ei(t′/p)EχDg(t,t′;β) (2.14)

If the donor and acceptor Hamiltonians are harmonic, this expression reduces in the classical limit to the famous Marcus expression13 for the rate (see also the next Section). B. Three-State Case. In deriving the standard electrontransfer rate, it was assumed that the vibrational energy distribution of the donor state is canonical at temperature T. If the donor state is prepared by photoexcitation, this assumption is no longer necessarily valid. A more detailed version of the golden rule rate expression is

ΓArD(β) )

Tre-βHD

and ΓArD(E) is the electron-transfer rate at donor energy E. Denoting the microcanonical density of states of the donor molecule at energy E as

where the donor-acceptor correlation function χAD is

χAD(t,β) ) Tre-iHAt/pe-(β-it/p)HD

Trδ(E - HD)e-βHD

(2.8)

where PD(E;β) is the probability that the thermal donor state is at energy E

where the two-point correlation function χDg(t,t′;β) is

χDg(t,t′;β) ) Tre-i(t+t′)HD/pe-(β-i(t/p))Hg

(2.15)

The “frequency-dependent partition function” Z(ω;β) appearing in eq 2.14 is

Z(ω;β) )

1 2π

∫-∞∞ dt e-it(∆E

Dg-pω)p

χDg(t,β)

(2.16)

The correlation function χDg(t,β) has the same form as the correlation function defined in eq 2.3, except that the acceptor Hamiltonian is replaced with the donor Hamiltonian, the donor Hamiltonian is replaced with the ground electronic state Hamiltonian, and t′ ) 0. The absorption probability from the

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Pollak and He

ground state to the excited donor state at photoexcitation frequency ω and ground-state inverse temperature β is

PDrg(ω,β) )

Z(ω;β) Zg(β)

(2.17)

where Zg(β) is the partition function of the ground-state Hamiltonian. Finally, ∆EDg is the energy difference between the bottom of the potential energy of the ground electronic state Hamiltonian Hg and the donor Hamiltonian HD. We will assume that the electron transfer process is an incoherent one such that the rate is an average of the energydependent transfer rate, where the averaging is over the nascent distribution resulting from the photoexcitation process. Thus, our three-state rate expression is written down formally as

ΓArD(ω,β) )

∫0∞ dE ΓArD(E)PD(E;ω,β)

(2.18)

We will simplify this expression considerably in the next subsection. C. Cooling and Control of Electron Transfer. The average energy in the electronically excited donor state is given exactly by the expression4

〈E〉(ω,β) ) pω - ∆EDg -

∂ ln(Z(ω;β)) ∂β

III. Harmonic Models A. Two-State Case. The simplest model which includes all the important physical features of the electron transfer process and for which one can readily undertake numerical computations is a harmonic one. First, we will review briefly the harmonic model for the two-state donor-acceptor system in which the donor is prepared in thermal equilibrium. The donor and acceptor harmonic Hamiltonians in mass weighted coordinates qj and momenta pj for a molecule with N degrees of freedom are taken as N

eff

HA )

(3.1)

(pj2 + ωj2(qj - qj0)2) + ∆GDA ∑ 2 j)1

(3.2)

1

One may make this model more general by allowing the frequencies ωj of the acceptor state to differ from the donor state; for the purpose of this paper, it suffices to allow the frequencies to be identical. In eqs 3.1 and 3.2, we assume that the equilibrium position of the jth normal mode in the acceptor state is shifted by qj0 relative to the donor state and that the energy difference between the bottom of the potentials of the donor and acceptor states is ∆GDA. For this harmonic model, the “reorganization energy” µDA is defined to be

(2.20)

µDA )

eff

The excited-state energy distribution is then well approximated as

PD(E;ω,β) =

FD(E)e-βeff(ω,β)E ZD(βeff(ω,β))

(2.21)

1

N

∑ωj2 qj02 2 j)1

(3.3)

With these preliminaries, one can show12 that the correlation function is

χAD(t,β) ) N

Inserting this approximation for the excited-state energy distribution into eq 2.18 using eq 2.12, one finds

ΓArD(ω,β) = ΓArD(βeff(ω,β))

1

(pj2 + ωj2 qj2) ∑ j)1 2

HD ) N

(2.19)

where Z(ω;β) is defined in eq 2.16. As shown in ref 4 (see especially Figure 10 of ref 4), the excited-state energy distribution may be well approximated as being thermal but with an effective (inverse) temperature βeff(ω,β), defined by demanding that the nascent average energy (as obtained from eq 2.19) is identical to the average energy of the donor state at the (inverse) temperature βeff(ω,β). That is, the effective temperature is defined by the relation

∫0∞dE EFD(E)e-β (ω,β)E 〈ED〉(ω,β) ≡ ∞ ∫0 dE FD(E)e-β (ω,β)E

process relative to what might have been expected at the groundstate (inverse) temperature β. Excitation of a polyatomic molecule from the ground electronic state to an excited state is usually accompanied by weakening of a number of vibrational bonds. As a result, one will typically find that Tr(Hg - HD)e- βHg > 0 so that the cooling phenomenon and the concomitant slowing down of the electrontransfer process should be generic.

(2.22)

This equation is the central result of this paper. It implies that the photoinduced electron-transfer rate constant may be estimated from the standard thermal electron-transfer rate expression but with an effective temperature which depends on the temperature of the ground electronic state and the excitation frequency. As shown in ref 4, the effective temperature has typically a parabolic dependence on the excitation frequency in the region where the absorption spectrum is maximal. This parabolic dependence has been recently verified experimentally in ref 14. If the excited state donor Hamiltonian is smaller than the groundstate Hamiltonian, in the sense that Tr(Hg - HD)e- βHg > 0, then one may expect cooling of the excited state for frequencies in the vicinity of the ω00 transition frequency. This cooling can lead to a significant slowing down of the electron-transfer

ZD(β)e

-i∆GDAt/p

e

∑(-ω q j

2

j0/p)((sin

2(ω t/2))/tanh(pβω /2) j j

- (i/2)sin(ωjt))

j)1

(3.4) One of the major differences between the electron transfer process and photoexcitation is that in the former, the position shifts are typically large. As can be seen from the harmonic expression for the correlation function (eq 3.4), this implies that the correlation function decays rapidly in time, justifying the short time expansion as in eq 2.4. The average energy and variance appearing in the approximate rate expression (eq 2.4) are found to be

XDt ) ∆GDA + µDA N

XDtt ) p

(3.5)

ωj3qj02

∑ j)1 2 tanh(pβω /2)

(3.6)

j

In the high-temperature limit, where the frequencies of all modes

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J. Phys. Chem. B, Vol. 105, No. 28, 2001 6503

are such that pβωj < 1, expansion of the variance (eq 3.6) leads to the famous Marcus expression13 for the electron-transfer rate

ΓArD(classical) =

x

2π 2 |V| p

β -β(∆GDA+µDA)2/(4µDA) e 4πµDA (3.7)

The reorganization energy appearing in this expression is that of the isolated molecule, whose molecular vibrations serve as the “bath” for the electron-transfer process. In the low-temperature limit, the variance becomes temperature independent, and so, the rate goes to a temperature independent constant. B. Three-State Case. In the three-state harmonic model, the excited donor Hamiltonian and the acceptor Hamiltonian are as given in eqs 3.1 and 3.2. Typically, position shifts between the ground-state Hamiltonian Hg and the excited-state donor Hamiltonian HD will be small, so for convenience, we will ignore them here. The frequencies of the ground-state Hamiltonian will differ from the excited state donor Hamiltonian; this difference is responsible for the possibility of controlling the excited-state electron-transfer process. As noted above, the frequency change is due to the electronic excitation of the donor molecule, which implies a weakening of a bond with a concomitant weakening of vibrational frequencies. Thus, the harmonic ground-state Hamiltonian considered here is N

Hg )

1

(pj2 + ωgj2 qj2) ∑ 2 j)1

(3.8)

To determine the effective temperature in the excited donor state, one needs to know the frequency and temperaturedependent partition function Z(ω;β). For the harmonic model in which Hg is given in eq 3.8 and HD is given in eq 3.1, one finds12 that the correlation function χDg(t,β)is

x(

agjaDj

N

χgD(t,β) )

∏ j)1

(bgj + bDj) - (agj + aDj) 2

)

2

(3.9)

ωgj sin(ωgjtc)

ωgj tan(ωgjtc)

aDj ) bDj )

ωgj sin(ωjt) ωgj tan(ωjt)

tc ) -ipβ - t

ΓArD(ω,β) ΓArD(β)

(3.15)

(3.10)

(3.11)

IV. Energy Relaxation Effects

and

bgj )

R(ω,β) ≡

of the photoinduced electron-transfer rate to the “standard” electron-transfer rate is plotted in Figure 3 as a function of photoexcitation frequency. As is evident from the figure, a noticeable slowing down of the transfer rate occurs in the region of maximum absorption, due to the cooling of the excited donor state population. The slowest rate is a factor of ∼20 slower than the rate at 300 K.

Here, the a’s and b’s are

agj )

the medium frequencies range from 800 to 1220 cm-1 with the same spacing, and the high frequencies range from 2000 to 2700 cm-1 with an equal spacing of 50 cm-1. These three groups mimic the typical frequency distribution of a polyatomic molecule. In the excited donor state, the frequencies of the lowfrequency group are reduced by a constant factor of 0.95, the medium-frequency group by 0.98, and the high-frequency group by 0.99. The energy gap between the ground and excited donor states is unimportant, since it simply sets the scale of frequencies for the photoexcitation laser. Thus, for the excitation frequency, we use the difference ω - ∆EDg. The position shifts qj0 in the acceptor Hamiltonian are chosen such that the shift energy for a given mode 1/2ωj2 qj02 ) xpωj. The parameter x is chosen to be 0 for the medium-frequency and high-frequency groups of modes and 2 for the low-frequency group of modes.2 This choice is made to ensure that we are not in the tunneling limit, where the electron-transfer rate beomes temperature independent. The intramolecular reorganization energy is thus 7410 cm-1. This implies an activation energy of approximately a quarter of an electronvolt for a symmetric transfer process. The absorption spectrum for this model (see eq 2.17) is plotted in Figure 1. Since most of the absorption occurs in the region -200 e ω - ω00 e 200 cm-1, we will consider in the next figures only this frequency range. In panel a of Figure 2, we plot the average energy in the excited donor state as a function of the photoexcitation frequency at a ground-state temperature of 300 K. The effective temperature in the excited donor state is plotted as a function of photoexcitation frequency in panel b of the same figure. As already noted in ref 4, for the model considered, we find extensive cooling in the excited donor state, with the maximal cooling occurring somewhat to the red of the ω00 transition frequency. The ratio

(3.12)

(3.13) (3.14)

Z(ω;β) is then obtained by numerical Fourier transformation, as in eq 2.16. C. Numerical Example. As an example for the control of the photoactivated electron-transfer process, we will choose a system with 45 degrees of freedom, divided into three groupss low, medium, and high frequencies. For Hg, the low frequencies range from 50 to 470 cm-1 with an equal spacing of 30 cm-1,

A condensed phase will have a profound influence on the electron transfer process. As pioneered by Marcus, the reorganization energy will now include a large component due to the reorganization of the medium around the acceptor state. There is, however, a second aspect which is of interest for us. The medium remains in thermal equilibrium and is not affected by the photoexcitation process. In the absence of reaction, the medium will thermalize the nascent energy distribution in the photoexcited donor state. If this thermalization process is faster than the electron transfer, then the electron transfer rate will become independent of the photoexcitation frequency and will be dependent only on the temperature of the condensed phase. Conversely, if the electron-transfer process is fast compared to the thermalization process, then the dependence on the photoexcitation frequency will be similar to that for the isolated molecule. The more interesting case is when the two processes occur on the same time scale.

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Figure 1. Absorption spectrum (normalized to unity at the peak) from the ground state to the excited donor state for the model considered in section III.C. The ground-state temperature is T ) 300 K.

Figure 2. Average energy (a) and effective temperature (b) in the electronically excited donor state plotted as a function of excitation frequency for the model system considered in section III.C. The horizontal dotted line in panel a denotes the thermal average energy at 300 K, and in panel b, it denotes the ground state temperature of 300 K.

As discussed in ref 15, the time dependence of the population in the excited donor state may be approximated as being thermal but with a time dependent effective temperature. We assume that the temperature of the medium is T. If an averaged friction coefficient 〈γ〉 is denoted which characterizes the intermolecular energy transfer rate between the donor molecule and the medium, the time dependent temperature of the donor state becomes

Teff(t;ω,β) ) T + (Teff(0;ω,β) - T)e-〈γ〉t

FD(E)e-βeff(t;ω;β)E ZD(βeff(t;ω,β))

where βeff(t;ω,β) ) 1/(kBTeff(t;ω,β)).

ΓArD(t;ω,β) )

∫0∞ dE ΓArD(E)PD(E,t;ω,β) ) ΓArD(βeff(t;ω,β)) (4.3)

Finally, the survival probability for the population in the excited donor state becomes

(4.1)

where Teff(t;ω,β) is as determined in eq 2.20. The timedependent normalized energy population in the donor state then takes the form

PD(E,t;ω,β) =

The time-dependent electron transfer “rate constant” is then given by

(4.2)

SD(t) ) e

-

∫ dt′Γ t

0

ArD(t′;ω,β)

(4.4)

The simple theory presented here is valid provided that the electron-transfer process is activatedsthat is, provided that the activation energy is larger than the thermal energy kBT. If not, the theory becomes more involved, as described in ref 15. A second condition is that the friction is not too strong so that an energy diffusion equation provides a valid description of the relaxation dynamics. If the friction is too strong, as perhaps

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Figure 3. Ratio of the photoinduced frequency-dependent electron transfer rate to the thermal rate (see eq 3.15) as a function of excitation frequency for the model system considered in section III.C. The ground-state temperature is 300 K.

may be the case in a dipolar liquid, one must resort to a more sophisticated approach. The theory presented above for the photoinduced electrontransfer dynamics in the condensed phase is based on the assumption that the medium may be described as a continuum. The resulting energy diffusion equation would not be valid in the gas phase, where a binary collision theory is the more appropriate framework for describing the thermalization of the nascent distribution due to collisions with the gaseous surrounding. The Gaussian binary collision theory presented in ref 16 can be readily adapted to the electron-transfer case. The energy exchange and reaction process is described in terms of a master equation with a Gaussian energy-transfer kernel and a sink term. The sink is the energy dependent electron-transfer rate, as given in eqs 2.12 and 2.13. To apply the theory, one must know this energy-dependent transfer rate. The formalism of ref 16 may then be straightforwardly applied to the electron-transfer dynamics to obtain the time-dependent survival probability. The remaining missing ingredientsthe energy-dependent transfer ratesmay be obtained by coarse graining eq 2.13, but here we suggest a simpler route, by using the inverse Laplace transform expressionseq 2.12. Although the inverse Laplace transform is notoriously difficult,17 for the problem at hand, it should not pose any serious problem. We know that ΓArD(E) is a monotonic function of the energy, similar in nature to the RRKM unimolecular decay rate. For this type of function, the iterative short time Laplace inversion method (STILT), developed in ref 18 is ideally suited. The input needed for application of STILT is the thermal rate ΓArD(β) at all temperatures and its first and second derivatives with respect to the inverse temperature β. For this purpose, one may use the Gaussian approximation expression for the thermal rate as in eq 2.7. This procedure will be studied in some detail elsewhere.19 V. Discussion We have shown that vibrational cooling of the excited donor state may have a profound effect on the thermal electron transfer

rate. The photoexcitation frequency controls to a large extent the nascent vibrational distribution in the excited state so that one may use the excitation wavelength to control the electron transfer rate. In the presence of a medium, this control depends on the ratio of the electron transfer rate to the vibrational energy transfer rate between the locally excited donor state and the medium. If the energy transfer rate is much faster than the electron-transfer rate, then thermalization occurs before reaction and the initial wavelength dependence will be negligible. If the electron-transfer rate is much faster than the energy relaxation rate, one will be able to control the photoinduced electrontransfer rate by varying the photoexcitation frequency. In the theory presented thus far, we have assumed an infinitely narrow laser pulse. If one uses a pulse with a finite time duration, then the effective rate will be given by an average of the frequency dependent rate, averaged over the frequency distribution of the pulse. This means that for a short femtosecond pulse whose frequency bandwidth covers a few hundred cm-1, the extensive frequency averaging will reduce the sensitivity of the rate to the excitation wavelength, and the control will be diminished. This then implies that the control mechanism discussed in this paper is limited to “slow” electron transfers that is, the mean transfer time should not be smaller than ∼500 fs. This time scale is though sufficiently fast to compete with energy relaxation in a dense gas or a liquid so that it may be possible, even in the liquid phase, to find electron-transfer processes which may be controlled. The simple rate expression found in this paper, depends on the observation that the nascent vibrational energy distribution is well approximated as a thermal one with an effective temperature. This approximation is valid under a few conditions. One is that the molecule has “many” degrees of freedom. In small polyatomics, the excitation is mode specific, and one must resort to a more detailed theory which would take into account also the intramolecular vibrational redistribution rate. A second condition is that the electronic transition from the ground electronic state to the excited donor state is truly a Condon transition. If it is symmetry-forbidden, then the absorption

6506 J. Phys. Chem. B, Vol. 105, No. 28, 2001 probability in the vicinity of the ω00 frequency is small, and it is no longer clear that the distribution may be really described as a thermal one. This does not mean that cooling does not occur; we have shown in a recent study20 that cooling may also occur in benzene where the S0 to S1transition is symmetry disallowed. If the temperature is sufficiently low so that the rate from the donor to the acceptor becomes temperature-independent, then the excitation step will have a negligible effect on the rate unless one goes considerably to the blue or the red of the ω00 transition where the nascent distribution will be heated. This means that the cooling mechanism will be important mainly when the electron transfer is “classical”sthat is, the rate is given by the Marcus expression. We have ignored in this paper important issues such as intramolecular vibrational redistribution (IVR) effects and anharmonicities. We have assumed that the excited state distribution may be approximated as thermalsthat is, all states are populated. This is not necessarily so, especially when considering a very large polyatomic molecule, in which the electronic excitation is localized in a given region of the molecule. At this point, we do not know whether anharmonicities will reduce the cooling effect or increase it. These questions are left as topics for future research. Acknowledgment. This paper is dedicated by E.P. to Prof. B. J. Berne. Bruce has been a wonderful teacher, colleague, and friend! We thank Professors P. Ha¨nggi and A. Nitzan and

Pollak and He Dr. L. Plimak for stimulating discussions on the theory of electron transfer processes. This work was supported by a grant of the Minerva Foundation by the Meitner-Humboldt award of the Alexander von Humboldt Foundation, the Jubila¨umsfonds der Oesterreichischen Nationalbank, and the Volkswagen Foundation. References and Notes (1) Jortner, J.; Bixon, M. AdV. Chem. Phys. 1999, 106, 107. (2) Bixon, M.; Jortner, J. AdV. Chem. Phys. 1999, 106, 1. (3) Gershinsky, G.; Pollak, E. J. Chem. Phys. 1997, 107, 812. (4) Wadi, H.; Pollak, E. J. Chem. Phys. 1999, 110, 11890. (5) Coalson, R. D.; Evans, D. G.; Nitzan, A. J. Chem. Phys. 1994, 101, 436. (6) Cho, M.; Silbey, R. J. J. Chem. Phys. 1995, 103, 595. (7) Evans, D. G. J. Chem. Phys. 2000, 113, 3282. (8) Jean, J. M. J. Phys. Chem. A 1998, 102, 7549. (9) Tramer, A.; Brenner, V.; Millie´, P.; Piuzzi, F. J. Phys. Chem. A 1998, 102, 2808. (10) Levy, D. H. AdV. Chem. Phys. 1999, 106, 203. (11) Wegewijs, B.; Verhoeven, J. W. AdV. Chem. Phys. 1999, 106, 221. (12) Yan, Y. J.; Mukamel, S. J. Chem. Phys. 1986, 85, 5908. (13) Marcus, R. A. J. Chem. Phys. 1956, 24, 966, 979. (14) Warmuth, Ch.; Milota, F.; Kauffmann, H. F.; Wadi, H.; Pollak, E. J. Chem. Phys. 2000, 112, 3938. (15) Pollak, E.; Talkner, P.; Berezhkovskii, A. M. J. Chem. Phys. 1997, 107, 3542. (16) Talkner, P.; Pollak, E.; Berezhkovskii, A. M. Chem. Phys. 1998, 235, 131. (17) Rabani, E.; Krilov, G.; Berne, B. J. J. Chem. Phys. 2000, 112, 2605. (18) Plimak, L.; Pollak, E. J. Chem. Phys. 2000, 113, 4533. (19) Plimak, L.; Pollak, E. J. Chem. Phys., in press. (20) He, Y.; Pollak, E. J. Phys. Chem., in press.