Free energy landscapes of electron transfer system in dipolar environment below and above the rotational freezing temperature

Electron transfer reaction in a polar solvent is modeled by a solute dipole surrounded by dipolar molecules with simple rotational dynamics posted on the three-dimensional distorted lattice sites. The interaction energy between the solute and solvent dipoles as a reaction coordinate is adopted and free energy landscapes are calculated by generating all possible states for a 26 dipolar system and by employing Wang-Landau sampling algorithm for a 92 dipolar system. For temperatures higher than the energy scale of dipole-dipole interactions, the free energy landscapes for the small reaction coordinate region have quadratic shape as predicted by Marcus Rev. Mod. Phys. 65, 599 (1993)] whereas for the large reaction coordinate region, the landscapes exhibit a nonquadratic shape. When the temperature drops, small notched structures appear on the free energy profiles because of the frustrated interactions among dipoles. The formation of notched structure is analyzed with statistical approach and it is shown that the amplitude of notched structure depend upon the segment size of the reaction coordinate and is characterized by the interaction energy among the dipoles. Using simulated free energy landscapes, the authors calculate the reaction rates as a function of the energy gap for various temperatures. At high temperature, the reactions rates follow a bell shaped (inverted parabolic) energy gap law in the small energy gap regions, while it becomes steeper than the parabolic shape in a large energy gap regions due to the nonquadratic shape of the free energy landscape. The peak position of parabola also changes as the function of temperature. At low temperature, the profile of the reaction rates is no longer smooth because of the many local minima of the free energy landscape.