The effect of quantum vibrations on electrochemical outer sphere redox reactions

The influence of an inner sphere vibration with a frequency in the region 10¹³ s^?1 ≤ ω ≤ 4 × 10¹⁴ s^?1 on simple electrochemical redox reactions is examined. The numerical calculations, based on first order perturbation theory and the Frank—Condon approximation, exhibit the following effects: the energy of activation decreases with temperature due to an increase in tunnel transitions of the system; the transfer coefficient is temperature dependent; the curvature of Tafel lines is greater at lower temperatures. All of these effects are of a quantum mechanical nature, and disappear both for very high and very low frequencies. These results are compared with classical and semi-classical approximations. At high overpotentials, the calculations predict limiting currents corresponding to activationless transfer. The magnitude of the limiting current is determined by the electronic matrix element and is independent of the model Hamiltonian for the solvation sphere.