3168
J. Phys. Chem. B 1997, 101, 3168-3173
Temperature Dependence of the Transfer Coefficient of Simple Electrochemical Redox Reactions Due to Slow Solvent Dynamics Marc T. M. Koper Abteilung Elektrochemie, UniVersita¨ t Ulm, D-89069 Ulm, Germany ReceiVed: NoVember 14, 1996; In Final Form: February 11, 1997X
It is shown that for simple electrochemical redox reactions in which both outer- and inner-sphere modes are reorganized, the transfer coefficient R may decrease with increasing temperature if the solvent dynamics along the outer-sphere coordinate is sufficiently slow. A similar trend is observed for the activation energy. The effect is due to a change in the most effective reaction path with temperature and is intimately related to the phenomenon of saddle-point avoidance. It is also shown that the effect enhances the potential dependence of the transfer coefficient.
follow the classical Marcus-Hush expression:10,11
1. Introduction In both the theory and the experimental evaluation of electrochemical reaction rates, the transfer coefficient R plays a key role. Its physical meaning is equivalent to that of the Brønsted coefficient in chemical kinetics, as it describes the change in the reaction free activation energy ∆G# with a variation in the reaction free energy ∆G°; i.e., R ) d∆G#/d∆G°. In electrochemistry, changes in ∆G° are readily accomplished by changing the applied electrode potential; e.g., for an oxidation reaction, one has R ) -d∆G#/dη, where η ) |e0(φ - φ0)| is the (positive) overpotential. As R is a barometer for the symmetry of the free-energy barrier and a value of 0.5 would correspond to a perfectly symmetrical barrier, it also comes under the name of symmetry factor.1 The importance of R in electrochemical kinetics can hardly be overestimated. Its actual value and its potential and temperature dependence are indicators of the mechanism of the electrode reaction under study. Conway2 has reviewed the status of R or the Tafel slope RT/RF in electrode kinetics, and in particular, he has discussed the experimental evidence for and the possible origin of a temperature dependence of the transfer coefficient. For a more recent entry into the literature, the reader may consult the paper by Gileadi et al.3 and the references cited therein. Most of the experimental examples of a temperaturedependent R concern electrode reactions in which the reactant has to traverse the electric double layer and to absorb onto the metal surface.2 Models and computer simulations indeed predict that for proton and ion-transfer reactions on metal electrodes, R should be temperature dependent.4-6 This is usually rationalized in terms of a sizable entropic contribution to the activation free energy due to the temperature-dependent orientation of the solvent molecules at the interface or to the temperaturedependent properties of the electric double layer.2,5,6 For simple electrochemical redox reactions, which we define as reactions in which the reactant merely exchanges one electron without adsorbing onto the electrode during the reactive event, a temperature-dependent transfer coefficient is expected only at temperatures where nuclear tunneling contributes to the reaction rate.7-9 Near room temperature, R is generally believed to X
Abstract published in AdVance ACS Abstracts, April 1, 1997.
S1089-5647(96)03799-6 CCC: $14.00
η 1 R) 2 2λ
(1.1)
where λ is the total reorganization energy, including both the inner- and outer-shell contributions and |η| , λ. Several experimental studies have confirmed the temperature independence12,13 and the potential dependence14-16 for simple electrode reactions as predicted by eq 1.1. In this paper, we show the (theoretical) possibility of observing a temperature-dependent transfer coefficient under what we believe to be not too exotic conditions: a simple electrochemical redox reaction on a metal electrode involving a significant inner-sphere reorganization in a slow solvent. By a slow solvent we mean a solvent with a relatively high longitudinal dielectric relaxation time τL. The temperature dependence of R is not due to a change in the symmetry of the potential energy surface with temperature but rather to an anisotropic dynamics on this potential energy surface. Apart from the fact that there may be some fundamental interest in this observation from the viewpoint of electrochemical kinetics, we believe this result also has a more general significance in providing an alternative way of testing the predictions of the multidimensional stochastic Kramers theories of solution-phase reaction rates.17,18 We will briefly elaborate on this last point in the Conclusion section. 2. Model The electrode reaction we have in mind is of the following type:
M(z-1)+ - Ln f Mz+ - Ln + e-
(2.1)
where M is some metal ion coordinated to n ligands L, which may also be strongly bound solvent molecules or otherwise represent intramolecular modes which are reorganized during the electron transfer. The theory to be described in this section applies to the case for which the back reaction to eq 2.1 can be neglected or to the reverse case for which the oxidation reaction can be neglected with respect to the reduction reaction. The situation near equilibrium will be treated in a later section. In any case, this is no restriction as far as the experimental situation is concerned, as R may be easily determined at potentials sufficiently remote from the equilibrium potential. Ideally, the © 1997 American Chemical Society
Temperature Dependence of the Transfer Coefficient
J. Phys. Chem. B, Vol. 101, No. 16, 1997 3169 where L is the Smoluchowski operator
L)D
{
}
∂ d ∂ +β V(qout) ∂qout ∂qout dqout
(2.4)
and where P(qout,t) is the probability density distribution for those molecules at coordinate qout that have not yet transferred their electron at time t. D is the diffusion coefficient for solvent fluctuations and in our coordinate system is given by D ) (2βλoutτL)-1. Since in our model the reduced form is the reactant, we have for V(qout)
V(qout) ) λout(qout + (z - 1))2
(2.5)
For kin(qout), we assume the classical Marcus form:
kin(qout) ) νin exp[-β(λin - ηin(qout))2/4λin] Figure 1. Contour plot of the potential energy surface given by eq 2.2. λin/λout ) 0.5, η ) 0.1 eV. The two different reaction paths represent the high- and low-temperature reaction paths (see section 3).
where ηin(qout) is the effective driving force for the inner-sphere reaction at qout, given by
ηin(qout) ) η - λout + 2λout(qout + z - 1) reactant is tethered to the metal surface as in the experiments of refs 15 and 16, as this would minimize the temperature dependence of other important quantities such as the distance of closest approach of the reactant, as well as facilitate the correction for temperature-dependent double-layer properties. We will model the electrode reaction by an approach first applied to electron-transfer reactions by Sumi and Marcus.19 In their model, one singles out two effective “generalized” reaction coordinates of reaction 2.1: one corresponding to the outersphere solvent reorganization, which we will denote qout, and one corresponding to the inner-sphere vibrational reorganization, qin. The two-dimensional free-energy surface possesses two minima, corresponding to the oxidized and reduced forms of the reactant. In our case, the oxidized form has a free energy that is lower than that of the reduced form by the overpotential η. In the usual harmonic approximation,10 both the reactant and product surfaces are quadratic functions of qout and qin, with the respective force constants 2λout and 2λin, the λ’s representing the reorganization energies:
Vred(qout,qin) ) λout(qout + (z - 1))2 + λin(qin + (z - 1))2 Vox(qout,qin) ) λout(qout + z)2 + λin(qin + z)2 - η
(2.2)
Note that we have chosen our reaction coordinates to correspond to a fractional valency, following an earlier suggestion by Hush.11 This choice is not of any great importance; the variable X used in the Sumi-Marcus paper can be related to qout by X ) x2βλout(qout + z - 1), where β ) 1/kBT. A typical example of the two-dimensional free-energy surface is shown in Figure 1. It is now assumed in the Sumi-Marcus model that the motion along the outer-sphere coordinate is diffusive-like and slow, whereas the motion along the inner-sphere coordinate is regarded as relatively fast. As a result, one can define an inner-sphere rate constant at each qout, say kin(qout). This allows for the description of the reactive process as the diffusive motion on a one-dimensional potential energy surface V(qout) as described by the Smoluchowski equation, complemented by a sink term for each qout, given by kin(qout):
∂ P(q ,t) ) [L(qout) - kin(qout)]P(qout,t) ∂t out
(2.7)
If the reaction is assumed to be electronically adiabatic, as we will do, the parameter νin is the average vibration frequency of the intramolecular vibrational modes. Small deviations from adiabaticity will not significantly change the results to be described below, as opposed to a completely nonadiabatic electron transfer. Recent calculations indicate that certain electrochemical electron-transfer reactions in which the reactant is not adsorbed onto the metal surface may quickly become nonadiabatic with increasing separation from the surface.20 On the other hand, such simulations have been performed on only a single couple. There seem to remain a sufficiently large number of redox couples that, based on the experimental data, are believed to be (nearly) electronically adiabatic.21,22 If the inner-sphere reaction rate is very low, it is easy to see that the reaction rate will tend toward the thermal equilibrium expectation value of kin(qout):
keq ) 〈kin(qout)〉eq )
∫kin(qout)Peq(qout) dqout
(2.8)
where Peq(qout) ∝ exp[-βV(qout)] is the appropriately normalized equilibrium distribution function. If the inner-sphere reaction rate is not very low, the distribution function P(qout,t) at any time t can differ from the equilibrium one. The rate at which the solvent polarization can approach the thermal distribution is determined by τL-1, and if this is not very much higher than keq, it will affect the total reaction rate. A quantity of experimental relevance is the fraction of reactant molecules that have not transferred their electron by the time t, which is termed the survival probability Q(t):
Q(t) )
∫P(qout,t) dqout
(2.9)
It was discussed at length by Sumi, Nadler, and Marcus19,23 that valuable information about the time dependence of this quantity is contained in two average survival times:
∫0∞Q(t) dt
(2.10)
∫0∞tQ(t) dt/∫0∞Q(t) dt
(2.11)
τa ) τb )
(2.3)
(2.6)
These quantities both give an estimation of the time scale of
3170 J. Phys. Chem. B, Vol. 101, No. 16, 1997
Koper
the Q(t) relaxation. They are equal only if the decay process can be described by a single exponential, but in general, they will be different. The quantity τa should monitor the shorttime dynamics of Q(t) and is equivalent to the mean first passage time; τb detects information of a time region later than τa. Since the short-time dynamics does not seem so relevant to the experimental situation in electrochemistry (any short-time measurement of the electric current will be dominated by the double-layer charging), in the following we will identify τb-1 with the reaction rate.24 However, in most of our calculations, τa and τb are not very different, since we will be mainly concerned with relatively high-barrier reactions. We calculate the quantities τa and τb by a numerical algorithm, making use of the generalized moment expansion, as described in detail by Nadler, Schulten, and Marcus.23,25 Since we will also use this algorithm for evaluating another quantity at the end of section 3, we briefly describe the basic equations. First, it is convenient to transform the Smoluchowski operator, eq 2.4, into a Hermitian operator. This operator L(s)(qout) is related to L(qout) in the following way:
L(s)(qout) ) peq-1(qout)L(qout)peq(qout)
(2.12)
peq(qout) ) xPeq(qout)
(2.13)
where
The idea of the generalized moment algorithm is to expand the Laplace transform of Q(t) in positive and negative powers of the Laplace variable s, with the generalized moments µn, n running from -∞ to ∞, as the expansion coefficients. With the operator L(s)(qout), these generalized moments can be written as the matrix elements of a Schro¨dinger operator:
µn ) 〈peq(qout)|[kin(qout) - L(s)(qout)]n|peq(qout)〉
(2.14)
Figure 2. Temperature dependence of R and Eact as calculated from eqs 3.3 and 3.4. λout ) 0.5 eV, λin ) 0.5 eV, η ) 0.1 eV, ωin ) 2πνin ) 100 ps-1, τL(T ) 300 K) ) 10 ps.
and
Eact ) (λ - η)2/4λ
where λ ) λout + λin. In the Sumi-Marcus model, we calculate them through
Introducing the auxiliary functions µn(qout)
µn(qout) ) [kin(qout) - L(s)(qout)]npeq(qout)
(2.15)
and making use of the fact that23
µ-1 ) τa
µ-2 ) τaτb
(2.16)
the average survival times can be calculated by the scalar products:
τa ) 〈peq(qout)|µ-1(qout)〉
(2.17)
τb ) 〈µ-1(qout)|µ-1(qout)〉
(2.18)
From the discretization procedure described by Nadler and Marcus,23 these expressions are easily evaluated numerically using a Gaussian elimination procedure for determining the vector µ-1(qout). 3. Numerical Results In this section, we will present some typical numerical results of the temperature dependence of the transfer coefficient and the activation energy of the reaction. First note that, in the thermal equilibrium model, these quantities are given by
η 1 R) 2 2λ
(3.1)
(3.2)
R ) -β-1 d ln(τb-1)/dη
(3.3)
Eact ) -d ln(τb-1)/dβ
(3.4)
and
In our calculations, we assumed that as we change the temperature, the shape of the potential energy surface remains unaltered. Although in reality there may be some temperature dependence of the reorganization energies, we neglect this effect in order not to distract too much from the point we want to make. We do allow for a temperature dependence of τL, as this effect seems of particular importance in a possible experimental verification of our result. Hence, we assume
τL ) τ°L exp(βEL)
(3.5)
where for slow solvents EL ≈ 0.10-0.15 eV.26 A typical result for the temperature dependence of R and Eact is shown in Figure 2 for EL of 0 and 0.10 eV and τL(300 K) ) 10 ps. We see that the temperature dependence can be quite pronounced, evidently the more so the smaller the temperature dependence of τL (the effect would ultimately disappear if EL would exceed the activation energy of eq 3.2). The origin of the temperature dependence is not difficult to understand: the lower the temperature, the lower the inner-sphere reaction rate and the easier a Boltzmann equilibrium along the outer-sphere coordinate is established. Figure 3 illustrates how the temper-
Temperature Dependence of the Transfer Coefficient
J. Phys. Chem. B, Vol. 101, No. 16, 1997 3171 To this end, we rewrite Pnoneq(qout) as
Pnoneq(qout) ) g(qout)h(qout)
(3.9)
where g ) peq ) xPeq. These functions g(qout) and h(qout) can be regarded as the coordinate representation of their ket vectors |g〉 and |h〉 and kin(qout) as that of an operator k. Substitution of eq 3.9 in eq 2.3 and making use of the transformation eq 2.12 give
ks|h〉 ) (k - L(s)|h〉 Figure 3. Temperature dependence of R for various ratios of λin/λout. λout ) 0.5 eV, η ) 0.1 eV, ωin ) 2πνin ) 100 ps-1, τL(T ) 300 K) ) 10 ps, EL ) 0 eV.
(3.10)
Since g is an eigenfunction of L(s) with eigenvalue zero, one has
L(s)|g〉 ) 0
(3.11)
and the rates keq and ks have the following short-hand notation
keq ) 〈g|k|g〉
ks ) 〈h|k|g〉 ) 〈g|k|h〉
(3.12)
where, of course, 〈g|g〉 ) 〈g|h〉 ) 〈h|g〉 ) 1. We rewrite eq 3.10 as
(k - L(s))|h〉 - ks|g〉 ) ks(|h〉 - |g〉) Figure 4. Dependence of R on the longitudinal relaxation time. λout ) 0.5 eV, λin ) 0.5 eV, η ) 0.1 eV, ωin ) 2πνin ) 100 ps-1, T ) 300 K.
ature dependence of R is affected by the ratio λin/λout; Figure 4 shows the variation of R at a particular temperature (300 K) with τL. We now show that the temperature dependence of R is due to a shift in the most effective reaction path with temperature, as illustrated by the two different reaction paths in Figure 1. First note that for the thermally equilibrated rate keq, R coincides with the saddle point of the integral in eq 2.8; that is,
R ) qsout + z R)
-(qsout
(reduction)
+ z - 1)
(oxidation) (3.6)
where qsout is that qout value for which the integrand in eq 2.8 is maximal. Incidentally, this is the reason why we chose qout rather than X as the reaction coordinate, as this choice leads to a physical meaning of R very similar to that originally suggested by Hush11 for the one-dimensional case. Hence, R is directly related to that value of qout for which the inner-sphere reaction occurs most effectively. One may inquire whether a similar interpretation of R exists when the probability distribution of qout is not thermally equilibrated. In the case of a single-exponential decay τa-1 ) τb-1 ) ks, Sumi27 has shown that the distribution function P(qout,t) can be separated as
P(qout,t) ) Pnoneq(qout) exp(-kst)
(3.7)
and that
ks )
∫Pnoneq(qout)kin(qout) dqout
(3.8)
where Pnoneq(qout) is an appropriately normalized nonequilibrium steady-state distribution function. Clearly our task is to calculate this Pnoneq(qout) and to evaluate the maximum of the integrand of eq 3.8.
(3.13)
and observing that (i) the second lowest eigenvalue of L(s) is of the order of τL-1 and (ii) the approximation eq 3.7 holds only if ks , τL-1, implying that |h〉 and |g〉 cannot be very different, one may neglect the right-hand side of eq 3.13 to find the useful approximation
|h〉 ) ks(k - L(s))-1|g〉
(3.14)
In the notation of section 2, this is simply
h(qout) ) µ-1(qout)/τa
(3.15)
Therefore, we can easily calculate Pnoneq(qout) from the quantities used in the generalized moment algorithm:
Pnoneq(qout) ) xPeq(qout)µ-1(qout)/τa
(3.16)
Although in the above manipulations we have essentially followed Sumi27 (his eq 4.28 is our eq 3.14), our method for calculating Pnoneq(qout) is different from his and much easier to use if one has already implemented the generalized moment algorithm. In Figure 5a, we compare the maximum of the integrand of eq 3.8 with R as determined from τa and τb. We see that, although the trends are the same, there is no quantitative agreement between the saddle point of the integral and R, as in the equilibrium case. The reason for this is illustrated in Figure 5b. Since the inner-sphere oxidation reaction becomes faster with decreasing qout (see Figure 1), Pnoneq(qout) is depleted more strongly from equilibrium at those qout values and the integral is therefore no longer Gaussian. This makes a saddle-point approximation to the integral rather poor. Nevertheless, this calculation clearly shows that the temperature dependence of R is due to a change in the most effective reaction path, even though this path is no longer directly related to R, as in the equilibrium case. 4. Reversible Electron Transfer: Comparison with the Approximate Analytical Theory of Zhu and Rasaiah So far, our theory has been for electrode reactions sufficiently far from the equilibrium potential so that the back reaction
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Koper
Figure 6. Temperature dependence of R as calculated by the several theories: solid line, from τb; dotted line, from τa; broken line, from ZR. λout ) 0.5 eV, λin ) 0.5 eV, η ) 0.1 eV, ωin ) 2πνin ) 100 ps-1, τL ) 1 ps.
Figure 5. (a) Temperature dependence of R calculated from τa, τb, and the maximum of Pnoneq(qout)kin(qout). Where the curves corresponding to τR and τb start deviating from each other, the decay is no longer single exponential. λout ) 0.5 eV, λin ) 0.5 eV, η ) 0.1 eV, ωin ) 2πνin ) 100 ps-1, τL(T ) 300 K) ) 10 ps, EL ) 0 eV. (b) Logarithm of the integrands of eqs 2.8 and 3.8; parameters as in a for T ) 300 K.
Figure 7. Potential dependence of R as calculated from the various theories for two values of τL: broken line τL ) 1 ps, dotted line τL ) 10 ps. MH, Marcus-Hush eq 1.1; SM, Sumi-Marcus; ZR, ZhuRasaiah. λout ) 0.5 eV, λin ) 0.5 eV, T ) 300 K, ωin ) 2πνin ) 100 ps-1.
where ki,eq are the equilibrium rate constants
() () λin λ
k1,eq ) νin may be neglected. At equilibrium, for every electron being transferred to the electrode, reversibility requires that one will be transferred back to the reactant. Combining this symmetry of the detailed balancing with the Nernst law, one expects that, at equilibrium, Rforw ) Rback ) 0.5 for all temperatures. The crossover from equilibrium to the theory in the previous sections must involve the solution of two coupled Smoluchowski equations with sink and source terms.23,27-29 With P1(qout,t) and P2(qout,t) denoting the reactant and product probability density distributions, respectively, the diffusion reaction equations can be written as
k2,eq ) νin
λin λ
1/2
1/2
exp(-β(λ - η)2/4λ)
exp(-β(λ - η)2/4λ - βη)
(4.3)
with λ ) λout + λin and
(
Ri ) νin
)
πλin(1 + A) β(1 - A)A4
1/2
exp(yi(1 - A)/(1 + A)) × [1 - erf(xyi)]
i ) 1, 2 (4.4)
in which A ) λout/λ,
∂ P (q ,t) ) [L1(qout) - k1(qout)]P1(qout,t) + ∂t 1 out k2(qout)P2(qout,t) ∂ P (q ,t) ) [L2(qout) - k2(qout)]P2(qout,t) + ∂t 2 out k1(qout)P1(qout,t) (4.1) where we suppressed the “in” subscripts on the inner-sphere rate constants. Instead of solving these equations numerically, we will make use of the approximate expressions for the reaction rate constants derived by Zhu and Rasaiah (ZR) for the case of a singleexponential time dependence of the survival probabilities Qi(t), i ) 1, 2. Their rate constants are given by
ki ) ki,eq/[1 + τL(R1/|x1c| + R2/|x2c - x0|)]
i ) 1, 2 (4.2)
y1 ) β(1 - A)x1c2/4 y2 ) β(1 - A)(x2c - x0)2/4 x0 ) (2λout)1/2 x1c ) (λ - η)/x0 x2c ) (λ - η - 2λout)/x0 One may verify that these expressions conform to the principle of chemical equilibrium, i.e., k1/k2 ) exp(βη), and indeed satisfy our expectation that R(η ) 0,T) ) 1/2. In Figures 6 and 7, we present some typical results obtained with eq 4.2 and compare them with the numerical results of the Sumi-Marcus (SM) theory outlined in section 2 and with the classical Marcus-Hush (MH) expression, eq 1.1. Figure 6
Temperature Dependence of the Transfer Coefficient shows that for η ) 0.1 eV, the agreement between ZR and SM is quite good, supporting the usefulness of the ZR theory. In Figure 7, we show that the potential dependence of the transfer coeffiicient, as predicted by the MH theory and as verified in quite a few recent experiments,14-16 is enhanced by a slow dynamics of the outer-sphere. It is seen that for τL ) 1 ps, the curve is still predicted to be approximately linear. If such a curve would be fitted by eq 1.1 in order to estimate the total reorganization energy λ, a too small value would be obtained (the error can amount to a factor of 2). We conclude that in electrochemical redox reactions in which both inner- and outersphere modes are reorganized, care must be exercised in estimating the total reorganization energy from such a “MarcusHush” plot. 5. Conclusion In this paper, we have shown how the transfer coefficient R of simple electrochemical redox reactions may decrease with temperature if the reaction involves a large inner-sphere reorganization in a slow (Debye) solvent. The same conditions also tend to enhance the potential dependence of the transfer coefficient. We find it difficult to estimate the feasibility of an experimental verification of the former effect, as the temperature dependence of the dielectric relaxation time may suppress it somewhat. We have not found experimental data in the literature that could be related to our theory; this is basically because researchers in the field have never measured or reported the temperature dependence of activation parameters in the systems to which our theory would apply (for possible candidates, see refs 30-32). We realize that additional practical problems may arise due to the temperature dependence of quantities that are neglected in our model, in particular, the role of the double layer. As mentioned, ideally our model applies to a redox couple tethered to the metal surface,15,16 since the overpotential in our model is the easieast to relate to the experimental overpotential in such a system. Such experiments have indeed provided the most convincing evidence to date of the long-debated overpotential dependence of the transfer coefficient, as predicted by classical electron-transfer theories. Nevertheless, in spite of these reservations, there is a qualitative aspect to our result that we believe goes beyond the field of electrochemical kinetics. First, it may provide an additional means for experimentally verifying the Kramers-Zusman33 or Sumi-Marcus theories of electron-transfer reactions. The usual method involves measuring the electron-transfer rate in many different solvents and subsequently disentangling the dynamic solvent effects from the energetic solvent effects, a procedure which is couched with many pitfalls and uncertainties.22 Our method involves measuring the temperature dependence in only one solvent. Second, the effect we reported on in this paper is intimately related to the phenomenon of friction anisotropy in multidimensional Kramers theory. It is, of course, well known that friction anisotropy may induce a reaction path that avoids the saddle point of the potential energy surface and that a variation in the friction anisotropy may cause a change in the most effective reaction path.34-38 That this may lead to non-
J. Phys. Chem. B, Vol. 101, No. 16, 1997 3173 Arrhenius behavior and a temperature dependence of the activation energy has already been pointed out by several authors.19,27,38 In this paper, it was shown that this interesting effect has perhaps an even closer connection to another experimentally measurable quantity: the “good-old” Brønsted or transfer coefficient. Acknowledgment. I gratefully acknowledge financial support from the European Commission through a Marie Curie Fellowship in the framework of the Training and Mobility of Researchers (TMR) Programme and thank Professor Wolfgang Schmickler for discussions and his comments on the manuscript. References and Notes (1) Schmickler, W. Interfacial Electrochemistry; Oxford University: New York, 1996. (2) Conway, B. E. In Modern Aspects of Electrochemistry; Conway, B. E., White, R. E., Bockris, J. O’M., Eds.; Plenum: New York, 1985; Vol. 16, p 103. (3) Kirowa-Eisner, E.; Rosenblum, M.; Schwarz, M.; Gileadi, E. J. Electroanal. Chem. 1996, 410, 189. (4) Schmickler, W. J. Electroanal. Chem. 1990, 284, 269. (5) Pecina, O.; Schmickler, W.; Spohr, E. J. Electroanal. Chem. 1996, 405, 239. (6) Pecina, O. Ph.D. Thesis, Ulm, 1996. Pecina, O.; Schmickler, W. Preprint. (7) Schmickler, W. Electrochim. Acta 1976, 21, 161. (8) Ulstrup, J. Electrochim. Acta 1984, 29, 1377. (9) Koper, M. T. M.; Mohr, J.-H.; Schmickler, W. Chem. Phys., submitted. (10) Marcus, R. A. J. Chem. Phys. 1965, 43, 679. (11) Hush, N. S. Trans. Faraday Soc. 1961, 57, 557. (12) Weaver, M. J. J. Phys. Chem. 1979, 83, 1748. (13) Curtiss, L. A.; Halley, J. W.; Hautmann, J.; Hung, N. C.; Nagy, Z.; Ree, Y. J.; Yonco, R. M. J. Electrochem. Soc. 1991, 138, 2033. (14) Save´ant, J.-M.; Tessier, D. Faraday Discuss. Chem. Soc. 1982, 74, 57. (15) Miller, C.; Gra¨tzel, M. J. Phys. Chem. 1991, 95, 5225. (16) Chidsey, C. E. D. Science 1991, 251, 919. (17) Kramers, H. A. Physica 1940, 7, 284. (18) Ha¨nggi, P.; Talkner, P.; Borkovec, M. ReV. Mod. Phys. 1990, 62, 251. (19) Sumi, H.; Marcus, R. A. J. Chem. Phys. 1986, 84, 4894. (20) Smith, B. B.; Halley, J. W. J. Chem. Phys. 1994, 101, 10915. (21) Miller, R. J. D.; McLendon, G. L.; Nozik, A. J.; Schmickler, W.; Willig, F. Surface Electron Transfer Processes, VCH: New York, 1995. (22) Weaver, M. J. Chem. ReV. 1992, 92, 463. (23) Nadler, W.; Marcus, R. A. J. Chem. Phys. 1987, 86, 3906. (24) Marcus, R. A.; Sumi, H. J. Electroanal. Chem. 1986, 204, 59. (25) Nadler, W.; Schulten, K. J. Chem. Phys. 1986, 84, 4015. (26) Fawcett, W. R.; Opallo, M. Angew. Chem. 1994, 106, 2239. (27) Sumi, H. J. Phys. Chem. 1991, 95, 3334. (28) Zhu, J.; Rasaiah, J. C. J. Chem. Phys. 1991, 95, 3325; Rasaiah, J. C.; Zhu, J. J. Chem. Phys. 1993, 98, 1213. (29) Berezhkovskii, A. M.; Zitserman, V. Yu. Chem. Phys. 1991, 157, 141. (30) Weaver, M. J.; Nielson, R. M. J. Electroanal. Chem. 1989, 260, 15. (31) Fawcett, W. R.; Opallo, M. J. Electroanal. Chem. 1992, 331, 815. (32) Mu, X. H.; Schultz, F. A. J. Electroanal. Chem. 1993, 353, 349. (33) Zusman, L. D. Chem. Phys. 1980, 49, 295. (34) Northrup, S. H.; McCammon, J. A. J. Chem. Phys. 1983, 78, 987. (35) Agmon, N.; Kosloff, R. J. Phys. Chem. 1987, 91, 1988. (36) Berezhkovskii, A. M.; Berezhkovskii, L. M.; Zitserman, V. Yu. Chem. Phys. 1989, 130, 55. (37) Klosek-Dygas, M. M.; Hoffman, B. M.; Matkowsky, B. J.; Nitzan, A.; Ratner, M. A.; Schuss, Z. J. Chem. Phys. 1989, 90, 1141. (38) Carmeli, B.; Mujica, V.; Nitzan, A. Ber. Bunsenges. Phys. Chem. 1991, 95, 319.
