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 (similar to 17 Angstrom) that ET is mediated b y 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 proces ses. When M is located in the same energy region as D and A, important role s are played by the dephasing-thermalization time of phonons tau (m) at M, relative to the lifetime of an electron l(m) at M. In the limit of tau (m) much greater than l(m), the superexchange ET occurs where M mediates the ET as a virtual intermediate state of quantum mechanics, while in the opposit e limit of tau (m) much less than l(m), the ordinary sequential ET occurs w here ET to M from D is followed by ET to A from M after thermalization of p honons at M. The analytic formula correctly bridges the two limits. It desc ribes 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 tau (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 o nly the degree of ordinary sequentiality D-OS of the ET, with D-OS much les s than 1 for the superexchange ET and 1 - D-OS much less than 1 for the ord inary 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 ini tial ET in bacterial photosynthesis, including its artificial modifications with respect to energy positions relative among D, M, and A.