Dynamical theory of electron-transfer reactions: Bridge-assisted transfer

We present an extension of a recently developed model for studying electron-transfer reactions. The model is based on the use of time-dependent statistical density operators for describing the time evolution and dynamics of the electron-transfer system. The system consists of the donor, the acceptor, and a selection of solvent/bridge molecules; this encounter complex is surrounded by the outer solvent, which is taken as a dielectric continuum. The evolution of the electron-transfer system is followed in real time by a nonlinear time-dependent Hartree-Fock method. For the initial state of the electron-transfer system the statistical density operator is chosen by a variational scheme. The electronic subsystem is dynamically coupled to the surrounding solvent. The actual calculations (Hartree-Fock) for the electron-transfer system have been performed with benzene anion radical as the donor, benzene as the acceptor, and water as the bridge/solvent molecule, with intermediate states corresponding to electron-type (as opposed to hole-type) superexchange permitted. We find probabilities for electron transfer that decrease strongly with distance and depend strongly on orientation. The variational selection of the initial state distorts the two rings away from coincident geometry, resulting in increased importance of bridge-assisted transfer. For some geometries, the second empty level on the bridging water contributes to bridge assistance. The dynamical approach clarifies some aspects of electron transfer, including the detailed role of the electronic structure and the role of solvent relaxations, that are more difficult to describe in the usual rate theory or golden rule rate formulations. Nevertheless, the dynamic approach suffers from some drawbacks, notably numerical problems in cases of slow transfer.