A density-matrix model of photosynthetic electron transfer with microscopically estimated vibrational relaxation times

Abstract

Abstract The dispersed-polaron (spin-boson) model is reviewed briefly and then used to develop a density-matrix model for studies of electron transfer in condensed phases. The frequencies and Franck–Condon factors for solvent vibrational modes that are coupled to electron transfer are obtained from molecular dynamics (MD) simulations by the dispersed-polaron treatment. Microscopic rate constants for vibrational relaxations, dephasing and coherence transfer between the solvent modes are obtained by fitting the time dependence of the solvent coordinates in the density-matrix treatment to the corresponding time dependence obtained from molecular-dynamics simulations with a classical linear-response approximation. This is done by adjusting a single parameter, the time constant for thermal equilibration of the two lowest levels of a solvent mode (T10). The model thus focuses on the coupling between solvent modes, rather than on the more widely studied coupling of solute modes by the thermal bath. The resulting density-matrix model is used to examine vibronic coupling in the initial electron-transfer step in photosynthetic bacterial reaction centers. Values of T10 in the range of 1–2 ps are consistent with molecular-dynamics simulations of the time-dependent energy gap between the reactant and product states (P* and P+B−), and also with the damping of coherent vibrational motions that are seen experimentally after excitation of reaction centers with a short pulse of light. In both the density-matrix model and the MD simulations, the autocorrelation function of the energy gap also has a decay component with a time constant of about 50 fs, which we ascribe to the group dephasing of oscillatory motions at many different frequencies. This component is insensitive to vibrational relaxations and is largely irrelevant to the electron-transfer dynamics. Using values of T10 in the range of 1–2 ps, a model with five vibrational modes reproduces the main features of electron transfer from P* to B, including stepwise formation of the product during the period when the system retains vibrational coherences. Although the rate does not depend strongly on whether P* is prepared coherently or incoherently, speeding up vibrational relaxations decreases the rate. At least part of the adverse effect of rapid relaxations can be viewed as a manifestation of the quantum Zeno paradox, which arises when off-diagonal elements of the density matrix decay very rapidly.