Time dependence of donor–acceptor electron transfer and back transfer in solid solution

Abstract

Electron transfer from an optically excited donor to randomly distributed acceptors followed by electron back transfer is treated theoretically for donors and acceptors in a rigid solution. The forward electron transfer process is described in terms of the excited state population probabilityPex(t) of the donor molecules, while the electron back transfer from the radical anion to the radical cation is characterized by Pct(t), the donor cation state population probability. Exact expressions for the ensemble averages 〈Pex(t)〉 and 〈Pct(t)〉 are derived. Numerical calulations are presented for the cation probabilities, the average cation–anion separation distance 〈R(t)〉, and the average cation existence time 〈τ(R)〉, using parameters which characterize the forward and back transfer distance dependent rates. Relationships among 〈Pex(t)〉, 〈Pct(t)〉 and the intermolecular interaction parameters provide detailed insights into the distance and time dependence of the flow of electron probability in an ensemble of donors and acceptors. The theoretical expressions can be used to calculate experimental observables. In particular, picosecond transient grating experiments are analyzed, and it is shown that by combining grating experiments (or other ground state recovery experiments) with fluorescence experiments it is possible to obtain the intermolecular interaction parameters for both forward and back transfer and a detailed description of the dynamics. The calculations presented here for rigid solutions are the precursor to the inclusion of diffusive motion of donors and acceptors to describe the dynamics of coupled electron transfer and back transfer in liquid solutions.