Electrochemical Electron Transfer: From Marcus Theory to Electrocatalysis - Fuel Cell Catalysis: A Surface Science Approach - page 43

2.5 SIMPLE BOND-BREAKING ELECTRON TRANSFER

The breaking of a chemical bond requires additional activation energy. In the simplest
case, the reaction proceeds according to
R X þ e ! R † þ X (2 : 11)

and neither the reactant nor the products are adsorbed. A quantitative theory can be
based on an extension of the Hamiltonian for simple electron transfer. Koper and
Voth [1998] proposed to add terms describing the interaction between R and X in
the molecule, and R and X 2 after the reaction, as a function of the separation r.
The corresponding expression in the Hamiltonian can be written in the form

H bb ¼ (1 n a )H i (r) þ n a H f (r) (2 : 12)

and thus consists of a term that acts before and one that acts after electron transfer.
Explicit forms for the potential energy in the terms H i and H f have been proposed
by Sav ´ ant [1993], who has developed a semiclassical version, along the lines of the
Marcus theory, and applied it successfully to several reactions. In his model, the poten-
tial curve for the reactants is a Morse curve, and that for the products is the repulsive
branch of a Morse curve:

d 2
dr 2 þ D(e 2r = l 2e r = l )

H i ¼ 1
2mh 2

(2 : 13)

d 2
dr 2 þ De 2r = l

H f ¼ 1
2mh 2

(2 : 14)

where m is the effective mass along the bond, 2D is the binding energy, and l is the
decay length of the Morse potential. The corresponding potential energy surfaces now
depend on the solvent coordinate q and the bond distance r; a typical example is shown
in Fig. 2.11. The initial state, the intact molecule, corresponds to the minimum
centered at q ¼ 0, r ¼ 0. In the final state, the bond has been broken, and the
energy surface shows a trough centered at q ¼ 1. The two regions are separated by
an energy barrier, whose height is determined by both the energy of reorganization
and the binding energy.
In the case where the bond coordinate can be treated as classical, and when the
electronic interaction D is much smaller than the solvent reorganization, the energy
of activation can be calculated explicitly in Sav´ant's [1993] model:

E act ¼ (l þ D e 0 h) 2
4(l þ D)

(2 : 15)

Thus, the energy of reorganization l from Marcus theory is replaced by the sum
l þ D, and the activation energy is significantly enhanced.