Microscopic based density matrix treatments of electron-transfer reactions in condensed phases

Several non-phenomenological density matrix treatments of electron-transfer (ET) reactions in condensed phases are developed and examined. The methods consider the donor and acceptor system (the solute) under the influence of the surrounding fluctuating solvent. The main emphasis is placed on semiclassical methods, where the starting point is the Hamiltonian of the quantum mechanical electronic states of the solute. The diagonal elements of the Hamiltonian include the fluctuations of the solute electronic energies as a result of the interaction between the solute and the field from the classically moving solvent molecules. The fluctuating Hamiltonian is used to construct a Liouville equation, which is treated by three approaches. The first method is based on a direct numerical integration of the relevant Liouville equation. The second involves the use of a second-order Liouville equation, and the third involves the use of a Redfield type equation. The methods are examined by simulating electron transfer between two sodium-like atoms that are held at a 4 Angstrom separation in water. The simulations generate the fluctuations of the electronic energies of the states that are involved in the electron-transfer process. The fluctuating energies are then used in evaluating the rate constant of the reaction as a function of its assumed free energies. The results of the three approaches are similar to the corresponding results obtained from the Marcus equation. However, the Redfield equation converges much more quickly than the direct Liouville equation and its second-order version. The problems associated with the semiclassical treatments are briefly considered, emphasizing the approximation involved in treating the solvent motion classically. Some of these problems can be overcome by a previously developed density matrix approach(1) that uses classical simulations to evaluate the Franck-Condon factors of the solvent vibronic states. This vibronic density matrix treatment is briefly described and used in simulating an electron-transfer reaction in the reaction center from Rps. viridis.