Molecular Imaging. Markus Rudin

of PESs. The electron can be released from the electrode to the oxidized form A_ox of the ionic or neutral electroactive molecule only if a suitable previous thermal fluctuation of polarized solvent molecules, of ionic atmospheres and of inner shell vibrations have brought A_ox in an X^* configuration of energy equal to that of an X configuration of A_red [22].

       3.10.The Vibrational Motion within the Reactants

      After having extended the use of PESs to the description of ET reactions at electrodes, I shall now show how potential energy curves (PECs) can describe the vibrations of the electroactive species inside the inner shell which, in the first formulation of the theory, was approximated as a rigid sphere. The description is here very simple and follows a formulation to be found in Ref. [13].

      A simple ET reactions at an electrode can be written as

      while a simple ET between species A₁ and A₂ in solution is represented as

      with by now evident meaning of the symbols. In Fig. 1 of Ref. [13] curve R represents the PEC that describes how the potential energy of the reactant system, A₁(ox)+M(ne), varies as function of a single vibrational coordinate q varied in A₁(ox). The P curve likewise represents the products system A₁(red) + M, when a single vibrational coordinate is varied in A₁(red), all the other coordinates remaining fixed. Considering the simple but realistic case of harmonic vibrations, the figure represents the potential energies of R and P as two parabolas crossing at a point. Several facts are noted:

      (i)The minima of the two curves occur at different values of q, because the equilibrium bond length in A_ox is different (and usually shorter in the case of transition metal ions and a metal–ligand bond) from that in A_red.

      (ii)The relative height of the two minima ΔU^∘ depends on the electrostatic potential, the P curve being lowered vertically relative to the R curve by making the electrode more negative, that is, by decreasing e(ϕ_M − ϕ_S), where the ϕ’s represent the Galvani potentials [21, 24] of the metal and solution.

      (iii)For a given ΔU^∘, the barrier height ΔU^* is lower when the difference Δq^∘ of equilibrium bond lengths is smaller.

      (iv)When the P curve is lowered relative to the R, the height ΔU^* of the barrier between the minimum of the R curve and the intersection of the R and P curves, is reduced. This is not true in the “inverted region,” see Chapter 5.

Fig. 3. A plot of the potential energy U of the system consisting of reactants plus solvent (R), along some coordinate q, and of the system consisting of products plus solvent (P), holding all other coordinates fixed, for reaction [3] or [4]. (From Ref. [13]).

      The two curves have properties which are by now familiar. When the reactant is far from the electrode, the curves in Fig. 3 merely cross in the intersection region (dotted lines there, see also Fig. 5^*(a)). When the reactant is close to the electrode, the electronic interaction of the orbitals of the reactant and the electrode perturbs the dotted line curves in Fig. 3 in the intersection region.

      At the intersection, the unperturbed electronic R and P quantum states are degenerate, and the degeneracy is broken by the interaction. The new curves are the solid curves. The energy at the maximum of the lower solid curve is less than that at the intersection by an electronic interaction energy ε₁₂, the resonance energy. In this case we see that the fluctuation needed for ET to occur is that of the vibrational coordinate q which may induce the ET, of course, only if the reactant—reactant distance is sufficiently small for the electronic overlap to occur. Here the estimate of the barrier height in Fig. 3 is very simple. It will be assumed that the resonance energy ε₁₂ is small enough that the height of the crossing point is approximately that of the maximum of the lower solid curve in the figure. We let the potential energy of the R and P curves to be approximated by [34, p. 7]:

      Where and are the equilibrium bond lengths for A(ox) and A(red), respectively. ΔU⁰ is a linear function of ne(ϕ_M − ϕ_S). At the intersection of the dotted curves we have:

      and q = q^‡. From the earlier condition one gets:

      where

      Introducing q^‡ into Eq. (3.39) one obtains:

      where

      And we see that λ_i/4 is the barrier height when ΔU⁰ = 0.

      Figure 3 suffices to describe also the vibrational motion within the reactants in the ET reaction in solution:

      with completely analogous results.

      The rate constant k_r is given by the collision frequency Z (per unit area of electrode and unit time in the electrode case, per unit concentration and unit time in the homogeneous case) multiplied by the Boltzmann factor exp(−ΔU^*/RT) and by a Boltzmann factor exp(−w^r/RT), where w^r is the work term, if any, required to bring the reactants together to some separation distance R, and finally by κ: