An integral equation approximation for the dynamics of reversible electron-transfer reactions

Abstract

The solution to an integral equation [J. Zhu and J. C. Rasaiah, J. Chem. Phys. 96, 1435 (1992)] for the survival probabilities in the Sumi–Marcus model of reversible electron-transfer (ET) reactions, in which ligand vibrations and fluctuations in the solvent polarization play important roles, is obtained numerically using a simple computer program suitable for use on a PC. The solutions depend on the time correlation function Δ(t) of the reacting intermediates along the reaction coordinate which is shown to be equal to the time correlation function of the Born free energy of solvation of these intermediates even in discrete molecular solvents provided its response is linear. This enables Δ(t) to be determined accurately from time-delayed fluorescence Stokes shift experiments or from dynamical theories of ion solvation; it is usually an exponential (Debye solvent) function of time or a sum of such exponentials (non-Debye solvent). The solutions to the integral equation, which can be obtained numerically for any given Δ(t), are found to predict the electron-transfer dynamics successfully over a wide range of model parameters. They can also be approximated by single or multiexponential interpolation formulas in which the thermally equilibrated rate constants are modified by a factor which reflects the relative importance of ligand (or inner-sphere solvent) vibration and outer-sphere solvation dynamics. The use of an effective longitudinal relaxation time in calculations of ET rates in solution is shown to be a poor assumption in some solvents. The theory is compared with an experiment in the inversion region, and its extension to include high-frequency vibrational modes that lead to an increased ET rate in other experiments is discussed.