Unified Theory on Rates for Electron Transfer Mediated by a Midway Molecule, Bridging between Superexchange and Sequential Processes

A typical example of electron transfer (ET) mediated by a midway molecule M is the initial ultrafast ET from the special pair to bacteriopheophytin in the reaction center of bacterial photosynthesis, where the donor D and the acceptor A are so far apart (∼17 Å) that ET is mediated by a bacteriochlorophyll monomer located in-between. An analytic formula for the rate constant k a,d of such an ET is presented with attention to its morphology to the resonance Raman scattering in second-order optical processes. When M is located in the same energy region as D and A, important roles are played by the dephasing-thermalization time of phonons τ m at M, relative to the lifetime of an electron lm at M. In the limit of τm ≫ lm, the superexchange ET occurs where M mediates the ET as a virtual intermediate state of quantum mechanics, while in the opposite limit of τm ≪ lm, the ordinary sequential ET occurs where ET to M from D is followed by ET to A from M after thermalization of phonons at M. The analytic formula correctly bridges the two limits. It describes intermediate cases as a single process, different from the expediency of assuming two channels by the superexchange and the ordinary sequential ET's, which cannot coexist. Occurring earlier than τ m in the course of ET are the superexchange ET and the subsequent hot sequential ET where ET to A from M occurs during reorganization of the medium around M after ET to M from D. Since they cannot be unambiguously separated, we can determine only the degree of ordinary sequentiality D OS of the ET, with D OS ≪ 1 for the superexchange ET and 1 − D OS ≪ 1 for the ordinary sequential ET. An analytic formula for D OS is also presented. D OS, in combination with k a,d, describes reasonably various aspects of the initial ET in bacterial photosynthesis, including its artificial modifications with respect to energy positions relative among D, M, and A.