Molecular Imaging. Markus Rudin

λ_o (λ outer) is what we have until now indicated with λ, that is:

      and λ_i (λ inner) is given by:

      where k_j and denote the force constants of the jth vibrational coordinate in a species involved in the reaction when that species is a reactant and when it is a product, respectively. The summation is over both reactants in the homogeneous case and over the one reactant in the electrode case (vide infra). When (ΔF⁰ + w^p − w^r)/λ is small, say 1/4, then:

      ΔF^* is then linear in ΔF⁰ with a slope 0.5. When w^r and w^p are small and when ΔF⁰ = 0 we have λ = 4(ΔF^*)₀ (the subscript indicates of ΔF⁰ = 0). Thus, the earlier condition for linearity in ΔF⁰ can be written as

      a condition often fulfilled in practice. More generally, the instantaneous slope of a plot of ΔF^* versus ΔF⁰ is, according to Eqs. (3.17) and (3.19), ¹/₂[1 + ΔF⁰/4(ΔF^*)₀] when the work terms are small.

       3.8.An Outline of the Theory of Electron Transfer Processes at Electrodes—Introduction

      The Marcus theory of the rates of electron transfer reactions at electrodes follows in the steps of the ET theory in solution because chemical and electrochemical redox reactions have many characteristics in common from the point of view of experimental results and theory.

I shall follow here mainly an abridged very clear qualitative description of the theory given by M. in Ref. [5]. The theory had been originally developed in two ONR Reports [19, 20] later published in Ref. [13]. The original theory of 1956 appears here already extended considering not only the effect of the nonequilibrium solvent polarization in inducing ET processes but also the effect of molecular vibrations and of ionic atmospheres.

      The electrochemical processes are characterized by various levels of complexity. In the simpler case, they consist of only one elementary reaction, the redox step. But the process is usually more complicated because the elementary step may be preceded or followed by various chemical reactions and equilibria involving the electrochemically active species. It is then essential to analyze the experimental data in sufficient detail so that the electric current is known as a function of the overpotential of the redox step, and not simply of the more usual overpotential of the overall reaction sequence. “When such an analysis has been performed for a complicated electrode reaction, attention can then be focused on the actual redox step itself,” that is, the one dealt with in the M. theory.

      The redox step can be visualized in the way usual in reaction rate theory: the atomic motions in the electrochemical systems happen on PESs which are functions of the coordinates of all atoms in the system, which means that to the usual atomic coordinates of solvent and solute we should add the coordinates of the atoms making up:

(i)The electrical double layer at the electrode/solution interface [17, 21–23]

      (ii)The electrode itself

      By a suitable fluctuation, that is, by a suitable concerted motion of the atoms, the system moves from a region of the many dimensional PES “where the electrochemically active species exists in one valence state to a region where it exists in the other valence state, with the electrode having undergone a corresponding change.”

      Figure 1 from Ref. [5] represents the simplest schematic potential energy diagram for the electrochemical process;

The diagram is very similar to those already seen in Chapter 1 and the electrode appears here as a “giant central ion” [21, p. 774] but with at least two important differences from those of the usual ions. First, the electrode has numerous energy levels and then connecting it to an external potential source it is possible to control its potential, charge, and energy levels. If the curves in Fig. 1 refer to a situation of electrodic equilibrium potential E_eq, changing the external potential making the electrode more positive the curve to the left is lowered and is instead shifted to higher energies setting the potential to a level more negative [17, 21, 22, 24, pp. 148, 149]. This gives the possibility of changing the activation energies for the cathodic and anodic currents at the electrode, as described in Refs. [17, 21, 22, 24].

      Fig. 1. Cross section of two intersecting electronic energy surfaces in N-dimensional atomic configuration space. Electrochemical process: A_ox + ne⁻ (metal) → A_red + (metal). The intersecting dashed lines indicate zero overlap of the electronic orbitals of A and metal (From Ref. [5]).

The qualitative description of an electrodic ET using Fig. 1 from Ref. [5] is very simplified because the potential energy diagram represents only a mean of the electronic energy levels of the electrode taking part in the ET process. For a metal piece of finite size, a more realistic representation is given in Fig. 2 from Ref. [11] — another Marcus’ logo. Each surface there is “a many-electron energy level of the entire reacting system and is a function of the nuclear coordinates.” The different R and P surfaces differ in the distribution of the electrons among the one-electron quantum states in the metal piece. As one sees there are