Effect of friction on electron transfer in biomolecules

In biological and chemical electron transfer, a nuclear reaction coordinate is coupled to other nuclear and/or ‘‘solvent’’ coordinates. This coupling, or friction, if strong enough, may substantially slow down motion along the reaction coordinate, and thus vitiate the assumption of electron transfer being nonadiabatic with respect to the nuclei. Here, a simple, fully quantum mechanical model for electron transfer using a one mode treatment which incorporates this coupling is studied. Path integral methods are used to study the dependence of the reaction rate on friction, and the limits of the moderate and the high friction are analyzed in detail. The first limit will prevail if the reaction coordinate is, e.g., an underdamped nuclear vibration, whereas the second limit will prevail if it corresponds to a slow or diffusive degree of freedom. In the high-friction limit, the reaction rate is explicitly shown to vary between the nonadiabatic and adiabatic expressions as the tunneling matrix element and/or the friction are varied. Starting from a path integral expression for the time evolution of the reduced density matrix for the electron and reaction coordinate, a Fokker–Planck equation is obtained which reduces in the high-friction limit to a Smoluchowski equation similar to one solved by Zusman.