The wave functions ψI ± ψI I describe the state better than ψI or ψI I alone, one of them will have an energy lower than either ψI or ψI I , lower by the resonance energy in the usual manner familiar to chemists, as indicated in the figure.
The behavior of the system on passing through the avoided crossing region of the PES will depend on the magnitude of this resonance energy. If it is large enough, about 1 kcal mole−1,
If we consider next the case of an extremely small resonance energy, that is, of an extremely small electronic interaction between ion and electrode, we see that “On passing through the intersection region, the system would tend to stay on the same surface that it stays on in the case of zero interaction. That is, if the system is started on the lower surface in the left of Fig. 1, the atomic motion would tend to carry it to the upper surface on the right and conversely during the return fluctuation. We see that when the interaction is extremely weak, the system ‘jumps’ from the lower surface to the upper one at the intersection. Actually, in the intersection region the concept of PESs and of the Born–Oppenheimer approximation on which it is based breaks down to this extent when the interaction is so extremely weak. During a passage through this region there is, however, a certain probability that the system will not ‘jump’ and hence a certain probability that the system will end up on the lower surface on the right in Fig. 1, and so have effected an electron transfer.” [5]
M. calls electron transfer mechanisms involving weak interactions and extremely weak interactions the quantum mechanically adiabatic and nonadiabatic mechanisms respectively, according to customary terminology. At the time of writing of Ref. [5], the few experimental data on preexponential factors tended to favor the adiabatic mechanism.
I shall now report the results of the Marcus treatment of ET processes at electrodes. The demonstration of how they were obtained, in Refs. [19] and [20], will be outlined in Chapter 4.
In order to understand these results, it is necessary to anticipate that M. uses the method of electrostatic images to determine the ion-electrode potential and so the electron released by, say, the reactant to the electrode is formally released by the reactant to its electrostatic image. The use of images permits the condition of zero electric field at a metal electrode to be satisfied. If the ion lies in solution at a distance R/2 from the electrode surface, its image lies inside the electrode also at a distance R/2 from the surface. Consequently, the distance between the ion and its image will be R but the electrostatic image is just a fictitious ion and so the energy λ for the process will be one half of the value for the homogeneous case where two real ions exchange an electron.
The electrostatic method of images is described, for example, in Refs. [27–30].
The current density i as a function of the activation overpotential ηa is:
where
In these equations, wr and wp are the works required to bring the reactant and the product to the electrode, respectively, m and λ are defined by Eqs. (3.34) and (3.35), n is the number of electrons transferred, e is the unit electronic charge, F is the Faraday, a is the ionic radius, R is twice the distance between the electrode and the center of the ion, Dop and Ds have the usual meaning, A′ is of the order of 1 × 104 to 5 × 104 cm sec−2. The activation overpotential is:
where E is the electric potential, E0′ is the value the latter would have under equilibrium condition if the reactant and product concentrations were equal (thereby E0′ is the formal standard potential in the prevailing medium, that is, at given salt concentration [11, 17]).
If the activation overpotential is small relative to the free energy barrier prevailing at zero overpotential, on expanding m2 in Eq. (3.33) about its value at ηa = 0 one obtains a linear dependence of ΔF* on ηa:
One sees from this equation that for a salt concentration suffi-ciently large for wp and wr to be small, the transfer coefficient, ∂ ln k/∂(neηa), the slope of the Tafel plot [17, 21, 23], should be equal to 0.5.
The reason why, at sufficiently high salt concentration, the electric work necessary to bring the ion to the electrode or away from it is close to zero, is that in the presence of a sufficiently high salt concentration the transference number of the electroactive ion decreases practically to zero and the electrical resistance of the solution and thus the potential gradient throughout is made small. In such conditions, the electroactive ion plays only a negligible part in the electrolyte conduction and its motion is affected by the diffusive force alone, see, for example, Refs. [17, 31–33].
As in the case of homogeneous ET, the Franck–Condon principle plays here an essential role as it was first recognized by J. Randles [4]. Even now a nonradiative ET becomes possible with little or no change of electronic configurations