Stationary phase approximation of electron transfer rate constants for the spin-boson model with Debye spectral density

The electron transfer rate formalism based on stationary phase approximation for imaginary-time flux-flux correlation function is often used to investigate the dynamic processes of complex systems in condensed phase, but the applicability of the electron transfer rate formula from stationary phase approximation has never been clarified. In this paper, we evaluated the electron transfer rates for spin-boson model with Debye spectral density from nonadiabatic to adiabatic regime. The comparison between the electron transfer rates from stationary phase approximation and the accurate results from the hierarchical equations of motion were made. The results from several approximate electron transfer methods were also shown. We found that the electron transfer rate formalism from stationary phase approximation can reasonably predict the electron transfer rates from nonadiabatic to intermediate regime and is invalid in adiabatic regime. The merit of the electron transfer rate formalism from stationary phase approximation was discussed.