Effects of vibrational energy relaxation and reverse reaction on electron transfer kinetics and fluorescence line shapes in solution

Abstract

The existing theoretical formulations of electron transfer reactions (ETR) neglect the effects of vibrational energy relaxation (VER) and do not include higher vibrational states in both the reactant and the product surfaces. Both of these aspects can be important for photo-induced electron transfer reactions, particularly for those which are in the Marcus inverted regime. In this article, a theoretical formulation is presented which describes the two aspects. The formalism requires an extension of the hybrid model introduced earlier by Barbara et al. [Science 256, 975 (1992)]. We model a general electron transfer as a two-surface reaction where overlap between the vibrational levels of the two surfaces create multiple, broad reaction windows. The strength and the accessibility of each window is determined by many factors. We find that when VER and reverse transfer are present, the time dependence of the survival probability of the reactant differs significantly (from the case when they are assumed to be absent) for a large range of values of the solvent reorganization energy (λX), quantum mode reorganization energy (λq), electronic coupling constant (Vel) and vibrational energy relaxation rate (kVER). Several interesting results, such as a transient rise in the population of the zeroth vibrational level of the reactant surface, a Kramers (or Grote–Hynes) type recrossing due to back reaction and a pronounced role of the initial Gaussian component of the solvation time correlation function in the dynamics of electron transfer reaction, are observed. Significant dependence of the electron transfer rate on the ultrafast Gaussian component of solvation dynamics is predicted for a range of values of Vel, although dependence on average solvation time can be weak. Another result is that, although VER alters relaxation dynamics in both the product and the reactant surfaces noticeably, the average rate of electron transfer is found to be weakly dependent on kVER for a range of values of Vel; this independence breaks down only at very small values of Vel. In addition, the hybrid model is employed to study the time resolved fluorescence line shape for the electron transfer reactions. It is found that VER can have a significant influence on the fluorescence spectrum. The possibility of vibrational state resolved spectra is investigated.