J. Electroanal. Chem., 204 (1986) 31-43

Elsevter Sequoia S.A., Lausanne - Prmted m The Netherlands

31

A THEORY OF ADIABATIC ELECTRON-TRANSFER REACTIONS l

WOLFGANG SCHMICKLER

Instrtut fiir Phpkahsche Chemre der Unrvemtiit Bonn, Wegelerstr 12. D-5300 Bonn (F. R G.)

(Received 16th December 1985)

ABSTRACT

A theory for adiabatic electron-transfer reactions at metal electrodes IS presented The model

Hamiltonian is similar to that of the Levich and Dogonadze theory for non-adiabatic reactions, but the

rate is not calculated from perturbation theory but by techmques fanuhar from the Anderson-News

model for adsorption. In the hmtt of comparatively weak mteractions, this model reduces to the Marcus

theory; for stronger Interactions there are significant deviations. A possible classification of electrochem-

ical electron-transfer reactions ts discussed.

(I) INTRODUCTION

For certain outer-sphere redox reactions occurring on metal electrodes, the rate constant seems to be essentially independent of the nature of the metal. Thus, Iwasita et al. [l] showed that the exchange current density of the [Ru(NH,),]*+/~+ couple is essentially constant on a series of metals comprising sd-, sp- and transition metals; this is true even when the metal surface is modified by metal adatoms deposited at underpotential [2]. This result is qualitatively in line with previous studies by Capon and Parsons [3], and Samuelson and Sharp [4], who showed that the rate of the system benzoquinone-benzoquinone anion radical is practically the same on several substrates. Whether there are any outer-sphere reactions whose rate

does vary with the nature of the metal is still an open question; early studies which did show such a dependence [5-71 were performed with redox couples which later proved not to be of the outer-sphere type.

These experimental observations cannot be explained within the Levich and Dogonadze theory [8] for electron-transfer reactions. According to this theory, the current density is proportional to the electronic overlap integral between the metal surface and the redox centre, and to the electronic density of states near the Fermi level of the metal. It is highly unlikely that the product of these two factors should be constant over a series of metals of different chemical character. A natural explanation is that reactions whose rate is independent of the metal proceed adiabatically. Classical theories for adiabatic electron-transfer reactions have been

l Dedicated to the memory of R.R. Dogonadze.

0022-0728/86/$03.50 8 1986 Elsevier Sequoia S.A.

32

developed by Marcus [9] and Hush [lo]. In these theories, electronic effects are not treated explicitly, but are included by way of a heuristic electronic transition probability, which is unity for adiabatic reactions; this is true also for modern versions of these theories [11,12]. A quantum-mechanical theory of adiabatic reac- tions has been presented by Dogonadze et al. [13]. It is based on a model in which there is a manifold of potential energy surfaces for the reaction. Even though the transition probability for each surface is small, the total transition probability can still be unity if a sufficient number of such surfaces is available.

In this work, we shall present a new quantum-mechanical model for adiabatic

outer-sphere electron-transfer reactions which is based on the Anderson-Newns model for adsorption from the gas phase (for a review, see ref. 14); recently, this has been extended to adsorption in electrochemical systems [15-171. Indeed, the model Hamiltonians employed for electron-transfer reactions and for adsorption are very similar, and we can therefore apply the techniques developed for the Anderson-Newns model to problems of electron transfer.

(II) THE MODEL HAMILTONIAN

We consider a redox centre situated close to the surface of a metal electrode. We restrict ourselves to the case where only one electron is exchanged, and denote by c the electronic energy of the electron on the redox couple. Let n be the number operator indicating whether or not the electron is present, and c+ and c the corresponding creation and annihilation operators. The electronic Hamiltonian for the isolated redox centre is then simply

H, = cc+c = cn (1)

For the electrons in the metal we use the model of quasi-free electrons, which are characterized by their momentum k and spin index u. Introducing the number operators nko, and the creation and annihilation operators cc0 and c~,. the corresponding terms in the Hamiltonian are

H,= c ko

‘ko =c Ehnho (2) ko

where the electronic energy cI is independent of the spin index. Electron exchange between the metal and the redox system is represented by the transfer Hamiltonian

H, =c ( V,c& c + h.c.) ko

(3)

where V, is the hopping integral characterizing the strength of the interaction between the redox system and the metal, and h.c. denotes the hermitian conjugate.

In considering the solvent and its interaction with the redox centre, we dis- tinguish between the librational and translational modes, which are slow compared to the time of electronic transitions, and the electronic modes of the interface, which are usually faster than the rate of electron exchange. Within the harmonic ap-

33

proximation, the slow solvent modes are conveniently described as a set of harmonic

oscillators:

K,, = E A%( PI + 4,‘) (4)

The p, and q, are dimensionless moments and coordinates. o, are the correspond- ing frequencies, and v labels the phonon modes. If inner-sphere modes are also reorganized during the electron exchange, they can be included in the summation.

For simplicity, we assume that all relevant modes are of a classical nature, i.e. nw < kT. The extension of the present work to the general case is quite straightfor- ward. The interaction between the phonons and the redox couple is assumed to be linear, and proportional to the charge:

(5)

where the g, are the coupling constants and z is the charge number of the redox system in the oxidized state. The fast electronic modes renormalize the electronic energy c of the redox system, and can thus be supposed to be included in eqn. (1).

The sum of eqns. (l)-(5) gives the usual model Hamiltonian H for electron- transfer reactions in second quantized form. In the Levich and Dogonadze theory, the transfer Hamiltonian H, is considered to be much smaller than the other terms, and the reaction rate is calculated through perturbation [8] or linear response theory [18]. We shall consider the case where this interaction is strong enough to establish electronic equilibrium between the metal and the redox couple, and higher-order terms in Ht must be retained. Formally, our model Hamiltonian is also very similar

to that used in the Anderson-Newns model as applied to electrosorption [16,17]. The main difference is that here only one spin state is considered for the electron on the redox centre. This is justified by the fact that for outer-sphere reactions the interaction with the metal is much weaker than for adsorption: no chemical bond is formed, and only one spin state is occupied.

(III) ADIABATIC ELECTRON TRANSFER

(II.]) General relations

If the electron-transfer reaction proceeds adiabatically, the electronic interaction must be fairly strong. We make the assumption that electron exchange between the redox system and the metal is faster than the characteristic times of the solvent modes, and also of those inner-sphere modes that are reorganized. Later, we shall see that this assumption need apply only near the transition region, but it is convenient to start with the stronger condition. The strength of the interaction can be characterized by the energy width, A, which the electronic state on the redox system acquires [14]:

A(w)=2n~]I/,]6(o-r,) (6) k

34

In principle, A is a function of the electronic energy w. However, for practical purposes we may consider A to be a constant, particularly since in our case it will be much smaller than in adsorption problems. In terms of this energy width, our assumption is that A > tzw,,,, where urnax is the fastest phonon mode that is reorganized in the electron-transfer reaction. If there is no reorganization of inner-sphere modes, as is almost the case for the [Ru(NH,),]‘+/~+ couple, an energy width of 10P3 eV suffices to establish electronic equilibrium. If inner-sphere modes are reorganized, an energy width of roughly 0.08 eV should be sufficient. These values may even be lowered if electronic equilibrium is required only near the transition region. Thus, relatively little interaction energy is needed to make a reaction adiabatic.

So far, there have been no attempts to calculate the electronic overlap between a redox couple and a metal electrode. However, for homogeneous electron-transfer reactions, calculations of the electronic overlap between donor and acceptor have been performed. For a series of Ru complexes, Meyer [19] has estimated that the interaction energy varies roughly between 0.015 and 0.07 eV. A much lower value of 0.001 eV was calculated by Newton [20] for the [Fe(H,0),]2+/3+ couple. Since the electronic density of a metal surface is higher than that of a redox couple, we expect the interaction to be stronger in the electrochemical situation. On the other hand, A should be much smaller than that for an adsorbed species. In this context, it should be noted that for alkali adsorbates the energy broadening has been estimated to be of the order of 0.5 eV [21].

It is convenient to collect the terms containing the occupation number n in eqns. (1) and (5) and to introduce the electronic energy i(q,) = c - ~Aw,g,q, which

depends on the phonon coordinates. The density of states of the elictronic state on the redox couple is then [14]:

‘(O)= &,A)1’+A’ (7)

and is thus a function of q,. The corresponding occupation probability is

(n(q,)) = &J_m f(W)P(W)dO cc,

where f(o) is the Fermi-Dirac distribution function. Of particular interest are those configurations for which the energy of the system has an extremum or a saddle point. Using the Hellman-Feynman theorem, these configurations are de- termined by the condition

1 ) aH 0 -= a4, (5)

which gives the following equations for the corresponding coordinates q,O and energies co :

q,o= -(Z-(n))& c0=c+2(z-(n))X (10) where h = xAa,gz is the energy of reorganization of the redox system.

35

The analysis is simplified if in eqn. (8) we replace the Fermi-Dirac distribution by a step function. This is permissible for A > kT, and, as is shown in the Appendix, it is also a good approximation if A is of the same order of magnitude as kT. The case A +C kT is considered below. The integral can then be performed explicitly to give the result

where the Fermi level has been taken as the energy zero. The corresponding expression for the potential energy of the system is

The integral diverges-a consequence of our assumption that the energy width A is a constant. However, the energy difference between two configurations is well defined, and that is all we shall require:

E[(n(q,,,))] -E[(n(qv,2))] = & In EIfqv~1)2f AZ 2(qys, I2 + A2

++l”.l)wLl))

-Gd~n(9P.2)) + fC~%(d,l - d.2) +cDWT~(9v.l - 9P.2) Y Y

The equation for the stationary solutions (n>O = (n(qY)) is

(131

(n). = i arccot C + 2( z - (n),)X

A 04)

Within a different context, viz. in studies of adsorption, this equation has been derived before by Kravtsov and Malshukov [22], and by us [17]. Inspection shows that this equation has either one or three solutions, two of which may coincide. It has three solutions if both the conditions

2X/7rA > 1 (15)

and

]e+X(2z-l)] 06)

where y2 = 7rA/2h, are fulfilled: otherwise, it has only one solution. If it has three solutions, numbered nr < n2 e n3, then n, and n3 correspond to local minima of the energy, and n, to a saddle point. n, can be identified with the oxidized state, n3 with the reduced state, and n, with the transition state.

The energy of reorganization X is typically of the order of 0.5-1.0 eV. As discussed above, in our case A -=E 0.5 eV; so condition (15) is certainly fulfilled. We shall discuss the physical meaning of the two conditions below.

36

The case where three solutions to eqn. (14) exist is obviously of particular interest. The energy difference between two stationary solutions is

E(n,) - E(n,) = & In [c+~(z-n,)X]*+A*

[C + 2(z - “,)h]‘+ A’

+(E+2h,)(n,-n,)-X(np-n:) (17)

The electronic energy e varies with the applied electrode potential. At equilibrium, the reduced state n3 and the oxidized state n, have the same energy; this occurs for c = X(1 - 2~). We introduce the effective overpotential 77 through c = TJ + X(1 - 22). Inspection shows that at equilibrium, for TJ = 0, the transition state is half occupied: n2( n = 0) = l/2, a result which was intuitively assumed in Hush’s theory [lo]. The energies of activation for the reduction and the oxidation reaction must then be equal. If all modes that are reorganized during the reaction are of a classical nature, this activation energy is given by the energy difference between the transition state and the initial state:

E,,. = (1 - 2n,)X/4 + $ In A’

X*(1 - 2~2,)~ + A’ for 77 = 0 (18)

The equilibrium situation is illustrated in Fig. 1, which shows the electronic density of state of the redox system for the reduced, the transition, and the oxidized state. Note that these electronic densities of states are entirely different from the densities

p (WI

Fig. 1. Demty of states p(w) of the redox couple for three nuclear configurations corresponding to the

reduced state (lower curve), the omdized state (upper curve) and the transitlon state, at vamshmg

overpotential.

31

E I eV

W

Fig. 2. Potential energy curves for a one-dimensional model at equilibrium. h = 1 .O eV. (- -) A = 0.01

eV; (......) A=O.l eV; (---) A=2X/a; A=1 eV.

of states of the reduced and of the oxidized state which play an important role in Gerischer’s theory [23]. In our case, the energy broadening is caused by the

electronic interaction with the metal; in Gerischer’s work, the electronic levels are sharp but are fluctuating due to the interaction with the solvent.

Equations (11) and (13) specify the potential energy surfaces of the system

<n>

I

-0.5 0 05 1.0 919

15

Fig. 3. Occupation probabdity (n) as a function of the heavy particle coordmate in a one-dimensional model at equihbrium. h = 1 eV. (- ) A = 0.01 eV; (- - -) A = 0.1 eV; (. .) A = 2X/n.

38

completely. These multi-dimensional surfaces are, however, difficult to visualize, so it is instructive to consider the one-dimensional case in which the slow modes of the

solvent and of the inner sphere are represented by one effective coordinate q. Figure 2 shows the corresponding potential energy curve in the equilibrium situation for various values of A. For small A, two minima are observed, corresponding to the reduced and the oxidized state; they are separated by a potential energy barrier of height close to X/4, the value known from the theories of Marcus, Hush, and Levich and Dogonadze. With increasing energy width, the barrier is lowered; it disappears at A = 2X/r, and for even higher values of A only one minimum exists. Thus, Fig. 2 illustrates the transition from a redox system with two well-defined electronic states to a strongly adsorbed state with a partially occupied orbital, and explains the meaning of relation (15). Qualitatively, such transitions have been discussed before within the context of partial charge transfer to adsorbates [24,25].

The corresponding occupation numbers (rz) are plotted in Fig. 3. For small A, (n) is either close to zero or to unity, with a small transition region near the top of the energy barrier. With increasing A, the transition region is broadened, and the occupation number becomes more fractional.

By varying the overpotential, the potential energy surfaces are shifted and distorted. This is illustrated in Fig. 4 for a relatively small energy width A, so that at equilibrium there are two minima separated by an energy barrier. With increasing cathodic overpotential, the energy of the reduced state is lowered, and so is the energy of activation for the reduction. At high cathodic overpotentials, the barrier vanishes, and only the reduced state exists. This illustrates the meaning of relation (16) which we rewrite as

q -c 2hl[ arccos( y) - y,il-,z] I/n (16a) Well-defined reduced and oxidized states, separated by an energy barrier, exist only in the “normal” reaction energy region, i.e. for not too high overpotentials. We note that with increasing A the normal region is narrowed.

(III.2) Relations for small electronic overlap

For a typical adiabatic outer-sphere reaction, the energy width A will typically be much smaller than the energy of reorganization. So it is of interest to study the case A K X separately. We have to distinguish two subcases, viz. kT 5 A -C X and A -=z kT, respectively, for which different mathematical approximations are ap- propriate.

(III.2.a) kT 2 A << h

In this region, we recover many results of the conventional electron-transfer theory. As is demonstrated in the Appendix, for this range of A it is still permissible to replace the Fermi-Dirac distribution by the step function. So eqns. (ll)-(18) are still valid, but can be simplified. Relation (16), which guarantees the existence of three solutions, reduces to n < A(1 - 4y/?r), where the leading term gives the usual

39

- 0.2

-04

-0t

I eV

\ 1, 08 ,'I6 \ \

/ 919 \ \ I \ \ /I \ / 1-N

Fig. 4. Potential energy curves for a one-dimensional model. X = 1 eV, A = 0.05 V. (- )v=O;(- --) q=O.25 V; (,.....) q=O.75V.

definition of the normal reaction energy region; but note that the second term is not negligible. In expression (17) for the energy, the electronic terms can be neglected so that the energy is determined solely by the phonon subsystem. So at equilibrium we have n, = 0, n, = l/2, n3 = 1, and the energy of activation is X/4. In the region of small and intermediate overpotentials, for n CC A, eqn. (3) gives

n, = 0, n, = 0.5 +7)/2x, n3 = 1 (19)

which is reminiscent of Hush’s theory. In this range of overpotential, the energy of activation for the oxidation reaction is

E(n,) -E(n,) = (h+TJ)?/4X (20)

40

which is again familiar. Note that the transfer coefficient is equal to n2, the

occupation probability at the saddle point.

(III.2.b) A << kT In this case, the width of the Lorenzian is smaller than that of the Fermi-Dirac

distribution, and the former can be approximated by a delta function. Equation (8) is then transformed into

(n(q,)) = (1 + exp(c(q,)/kT))-’

In particular, the stationary solutions are determined by

(21)

(22) Inspection shows that this equation has three solutions in a range which approxi- mately, i.e. within an error of the order of A/kT, is also given by eqn. (16b). At equilibrium, it has the usual solutions n, = 0, n, = 0.5, n3 2: 1. For n K h, the

solutions are approximately

n, = 0, nz=0.5+q/2X, n,=l (23)

So they are approximately the same as in the case kT ,< A < h. Since the first term in the energy expression (13) is again negligible, the energy of activation is also

given by eqn. (20). Thus, for A .=c X, eqns. (16b), (19) and (20) hold both for A +C kT and A >, kT even though the stationary solutions are derived from different equa- tions.

(III.3) The pre-exponential factor

So far, we have been concerned mostly with occupation probabilities and energies of activation. To complete the theory, we still have to specify the pre-exponential factor A. The latter depends on the dynamics of the solvent and the inner-sphere modes and hence on the shape of the potential energy surfaces. In our model, there is no explicit algebraic expression for the energy of the system, and it is therefore not possible to given a general formula for the pre-exponential factor A. However, in the important case where the relation A c X holds, the potential energy surface is practically parabolic up to the saddle-point region. If the assumption of transition state theory holds, that the number of systems crossing the saddle-point region can be calculated from the equilibrium distribution in phase space, then the pre-ex- ponential factor is [26,27]:

A = (1,24 (C&h” j,h]“2 Y

where X, = Aw,gz/2, and A is dominated by the highest frequency mode which contributes substantially to the energy of reorganization.

41

Recently, it has been suggested [11,12] that the motion of the solvent may be so strongly damped that the equilibrium hypothesis of transition state theory does not

hold. In this case, the pre-exponential factor is approximately

A = l/7, (25)

provided that the inner-sphere modes do not contribute significantly to the energy of reorganization. T, is the so-called longitudinal relaxation time of the solvent and is related to the usual Debye relaxation time T,, through T, = ~ne~/c~, where cl0 and es are the high frequency and the static dielectric constants of the solvent. Depending on the system under consideration, either eqn. (24) or eqn. (25) may apply; there seems to be experimental evidence for both situations (for a recent discussion, see, for example, ref. 28).

(III.4 The rate constant

We now sum up the results of our considerations. If the interaction between the redox couple and the metal is so small that 2X > ?TA, and if the applied overpotential is so small that the reaction is still in the normal reaction energy region, i.e. if relation (16) is fulfilled, then both an oxidized state with occupation probability n, = 0 and a reduced state with n3 = 1 exist. They are separated by an energy barrier with a saddle point of occupation probability 0 < n, < 1. Within our model, the exact values of n,, n, and n3 can be calculated from eqn. (14). The electro- chemical rate constant for the reduction reaction can then be written in the form

k,,=ACexp[E(n,)-E(n,)]/kT (26)

where the energy of activation is calculated from eqn. (17). C is a factor which converts bulk to surface concentrations, and is of the order of 10m9 cm. If both A +Z X and n < h are fulfilled, eqns. (19) and (20) apply, and the energy of activation is given by the familiar Marcus expression.

(IV) DISCUSSION

In the light of our investigations, we can divide electrochemical electron-transfer reactions at metal electrodes into four classes, which we list in the order of increasing strength of the interaction between the metal and the redox system.

(1) Near the saddle-point regton the rate of electron exchange is slower than the time

constants of the heavy-particle modes that are reorganized during the reaction (A < Fro,,,). In this case, the electron-transfer reaction is non-adiabatic. The transfer Hamiltonian can be treated as a perturbation, and the concepts of the Levich and Dogonadze theory apply. The reaction rate depends on the nature of the electrode surface. Examples are reactions at metals covered by a thin passive film.

(2) Near the saddle-pornt region, the rate of electron exchange is faster than the time constants of the heavy-particle modes that are reorganized during the reactton, but the interactron is sufftciently weak so that A < X. The reaction is adiabatic, but terms of

42

the order of A/x can be neglected. The concepts of the Marcus theory apply. When inner-sphere modes with a high frequency (o > kT/h) are reorganized, the rate

equation may, however, have to be modified for tunnelling effects [29,30]. The reaction rate is independent of the metal. The reaction of the [Ru(NH,),]~+/~’ couple seems to be an example, and probably also the benzoquinone’/- couple.

(3) The reaction is adiabatic, and kT -=K A < 2X/77. The energy barrier is lowered significantly by the electronic interaction; its height can be calculated from eqns. (14) and (17). The reaction rate depends on the nature of the metal. At present, there seem to be no examples for such reactions: they would be very fast.

(4) The reaction is adiabatic and A > 2X/m. In this case, we cannot distinguish between reduced and oxidized states; there is only one state with a partially filled valence orbital. The redox system is specifically adsorbed. This corresponds to an inner-sphere reaction.

It is interesting to note that our theory for adiabatic reactions is not equivalent to that of Dogonadze et al. [13]. The latter is based on a model in which the reaction occurs via a manifold of reaction surfaces, each corresponding to a different electronic orbital on the metal. Both the resulting rate expression and the criterion of applicability are different. These differences are due to the fact that the theory of

Dogonadze et al. is based on first-order perturbation theory, and electron exchanges with different metal states are viewed as occurring independently of one another. In our model, transitions to different energy levels are not independent; the metal and the redox couple share electrons.

ACKNOWLEDGEMENT

Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

APPENDIX

In the following, we shall demonstrate that replacing the Fermi-Dirac distribu- tion by a step function in eqn. (10) is a good approximation even if A is of the order

of kT. If this approximation is not made, the equation for the stationary states is

A

{w-[ v + X(1 - 2(n),]}2 + A2 dw (Al)

For 9 e X, and A - kT, we have the usual solutions n, = 0 and n3 = 1. Obviously, only the solution n 2, corresponding to a partially filled orbital, can be affected if we replace f(w) by 8(-w). Inspection shows that for q= 0 we have n, = l/2, and for small q we have the estimate n, = (1 + ~)/2h, which is the same as that in eqn. (19). As a further check, we have calculated n2 from eqn. (Al) for the critical value of A = kT, and compared it with the approximate values derived from eqns. (14) and (19) (see Table 1): the agreement is good. Since for A = kT the electronic terms

43

TABLE 1

Occupation probabihty at the saddle point in various approximations. X = 1 eV, A = kT = 0.026 eV

V/V nz (eqn. Al) nz (eqn. 14) rrz (eqn. 19)

0.0 0.5 0.5 0.50

0.1 0.554 0.552 0.55

0.2 0.609 0.604 0.60

0.3 0.664 0.657 0.65

0.4 0.720 0.710 0.70

0.5 0.776 0.764 0.75

0.6 0.836 0.821 0.80

0.7 0.905 0.884 0.85

in the energy expression are small, a good approximation for n, also yields good values of the energy of activation.

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