Effects of bath relaxation on dissipative two-state dynamics

A formal solution to the two-state Liouville equations is used to derive quantum equations of motion for dissipative two-state systems without making the assumption of a harmonic bath. The first-order equation of motion thus obtained is equivalent to the noninteracting blip approximation and can be systematically improved by introducing high-order cumulants. The second-order equation of motion incorporates effects of bath relaxation on two-state dynamics and leads to an effective nonadiabatic rate expression, which in the classical limit reduces to the well-known electron transfer rate formula. Numerical results with an Ohmic bath show saturation at large coupling constants due to the rate-limiting effect of relatively slow bath relaxation, and a comparison with classical calculations demonstrates larger rate constants at low temperature when quantum coherence is taken into account. (C) 2000 American Institute of Physics. [S0021-9606(00)50515-7].