Derivation of the Marcus theory activation energy

We make the assumption that the red and blue potential energy surfaces in Figure 24.16 are both harmonic and have the same force constant. If we assume that the reactants are at Q = 0 and the products are at Q = Q⁰, we can write the two potential energy surfaces as

(1)

Figure 1. The potential energy surfaces for reactants (blue) and products (red) in Marcus theory.

We wish to know the height in energy of the barrier located at Q*.

(2)

We also know that

(3)

And

(4)

We can expand Eqn. 4,.

(5)

and substitute expressions for and

(6)

to obtain

(7)

And finally the activation energy,

(8)

Eqn. 8 is a remarkable for its simplicity and the relationship of the energy barrier to two fundamental parameters, the energy, e and reorganization energy, l.