Adiabatic versus non-adiabatic approach to the inner sphere vibrational effects on electrochemical reduction rates

This work is a theoretical comparative study on the adiabatic and non-adiabatic mechanisms of electrochemical reductions. In our previous paper [A. Ignaczak, W. Schmickler, Electrochim. Acta 52 (2007) 5621] several aspects of the reaction kinetics were discussed for the adiabatic electron transfer from the electrode to the reactant, taking aquocomplexes as a testing case. In the present work the same reactions are studied in the non-adiabatic approach. First order perturbation theory is applied to analyze the rate dependence on quantum effects associated with inner sphere nuclear vibrations using the same model Hamiltonian. It describes independently the solvent reorganization and the intramolecular modification of the reactant. In case of complexes the ligand shell in the oxidized and reduced states is characterized by two different frequencies, ω1 and ω2, respectively, so the ratio θ = ω2/ω1 describes asymmetry of the potential energy surface. An influence of different θ values combined with high and low ligand frequencies on the rate enhancement due to the nuclear tunnelling is tested.The quantum correction, calculated as kq/ks (kq, the non-adiabatic rate; ks, the adiabatic rate), is found to depend strongly on θ and ω1 parameters as well as on the overpotential applied. The rate enhancement is especially sensitive to the initial state frequency ω1 for systems characterized by larger θ values. For overpotential η = 0 V, when ω1 = 250 cm−1 the ratio kq/ks is equal to 1.35 (θ = 2/3), 0.65 (θ = 1) and 0.38 (θ = 3/2), while for ω1 = 1000 cm−1 the corresponding values are 4.05, 3.94 and 4.77. These values increase significantly when the overpotential is applied, up to 14 for the largest values of ω1 and θ. The transfer coefficient is also presented for the two reaction mechanisms as a function of all parameters tested. An effect of the reactant–electrode interaction and the friction parameter on the rate constants is also analyzed for all cases.