The use of a single surface of the PESs in Ref. [5] makes sense because the electron from the metal comes from levels in a neighborhood of width kT around the Fermi level EF [11].
For our purposes, it is possible to classify the electrode reactions into at least three classes.
To a first class belong electrode reactions which happen with chemical bonds broken or formed, as, for instance, in the famous reaction:
where M is the electrode. To this class of bond rupture atom transfer type belong also the electrode processes involving deposition of metal cations [17].
In a second class are reactions in which no chemical bonds are broken or formed, such as:
The ions in the above reactions are “tightly knit” (M.), that is, the interatomic distances in the oxidized and reduced forms are about the same (vide infra). In these reactions, the changes of atomic configurations consist primarily of a large reorientation of the solvent molecules about the species. Such a reorientation, which arises from the change in ionic charge, naturally occurs in reaction (3.28) as well, but is of particular importance in the cases (3.29) and (3.30) in determining the reaction kinetics. To this class also belong reactions in which bonds are considerably stretched [25] but not broken, such as the Co–N bond in:
To a third class might belong electrode reactions in which bonds between ions and electrode surface are formed to facilitate ET and then broken. Such a class of “bridged activated complexes” occurs in certain electron transfers in solution where an anion may form a bridging species between two cations [26]. One may consider a reaction in which the ET between cation and electrode happens through a bridging anion adsorbed on the electrode.
From the point of view of Marcus theory, it is then convenient to classify the ET reactions according to whether:
(a)Chemical bonds are broken or formed during ET.
(b)No bond rupture or formation occurs, but merely an ET.
(c)Bonds are formed to facilitate ET and then broken.
Fig. 2. Same plot as in Fig. 1 but for an electrode reaction. The finite spacing between levels, reflecting the finite size of the electrode, is enormously exaggerated. Only three of the numerous electronic energy levels of this system are indicated. The splitting differs from level to level, and the spacing decreases as the size of the metal increases. (From Ref. [11]).
The Marcus theory of ET at electrodes deals with (b) processes. In these reactions, the reactant approaches the electrode close enough to effect those electronic interactions between its orbitals and those of the electrode which are necessary to induce ET. After ET, the product recedes from the electrode.
3.9.The Theory
In the electrochemical case, the rate of the reaction is measured by the electric current passing through the electrode and that equals the total probability of reaching the saddle point connecting the valley of the reagents and that of the products multiplied by an appropriate frequency factor for passage through and by the concentration of active ions in the solution. “This probability is calculated by means of equilibrium statistical mechanics and a knowledge of the potential energy surface in the valley of the reactants and at the passes (saddle points). The assumption normally made for this purpose is that the reaction hardly disturbs the equilibrium between systems in these two regions.” [17]
In describing the PESs of Fig. 1, “let us first examine the hypothetical case of zero electronic interaction.” There are then two electronic states to consider, “each having its own many-dimensional PES. In the first of these states, the ion is in an oxidized form Aox and the electrode has the electronic charge distribution appropriate to the metal-solution potential difference. In the second state, the ion is in the reduced form Ared, the electrode has lost to it n electrons and again has a charge distribution appropriate to the same potential difference.
The valley of one surface is centered at quite different atomic coordinates from that of the other surface. The difference in ionic charge between reactants and products results in differences in:
(i)Average stable orientation of the solvent molecules outside the coordination shell
(ii)Ionic atmospheres about the ion and electrode
(iii)Interatomic distances inside the inner shells
If these surfaces are plotted versus N atomic coordinates of the entire system, they will intersect on some (N − 1)-dimensional surface. At the intersection surface, the atomic coordinates have values which represent a compromise configuration of solvent molecules, atmospheric ions, and coordination shell. The compromise is between the stable atomic configurations of the reactants and those of the products. Because of the assumed absence of electronic interactions between ion and electrode, there will be no electron transfer if a system moving on one diabatic PES reaches the intersection region. The system merely stays on the surface appropriate to its electronic configuration. During the reverse fluctuation, the system again passes through this intersection region, always staying on the initial PES, and reverts to the stable atomic configuration of the latter.
Let us consider next the case of a weak electronic interaction between ion and electrode. Such a weak interaction hardly affects the two PESs, but it removes the degeneracy at the intersection, as indicated in Fig. 1, so that we no longer have an intersection and the system is now described by adiabatic PESs. The quantum mechanical reason for this is briefly the following. If ψI denotes the electronic wave function for one of the two electronic states in the hypothetical zero-interaction case of Fig. 5*(a) of Chapter 1 and if ψI I denotes the other, ψI and ψI I are functions of the atomic coordinates. They describe not only the motion of electrons of the central ion but also those in the molecules outside this ion. They describe for instance the electronic polarization Pe of the solvent by a central ion in each of the two valence states. Because of the ionic charge difference in the two states, this electronic polarization is quite different. The weak interaction may be shown from