Band narrowing in semiclassical nonadiabatic electron transfer

The two-state evolution of an electron which is coupled to a narrow band of optical phonons is considered in the nonadiabatic regime, where the interaction matrix element V is much smaller than the vibrational energy ħω. A separation of coordinates is achieved when the vibrational motion is described in terms of action-angle variables which allows for a consistent analytic treatment of the dynamics in the semiclassical limit. When V is much larger than the vibrational bandwidth it is found that the electron evolution is governed by a reduced matrix element Ṽ=VJ₀(4g√J/ħ) which is modulated by the Bessel function J₀, having an argument which depends on the electron-phonon coupling strength g and the vibrational action J, which is an adiabatic invariant. When V is much smaller than the vibrational bandwidth, the usual Huang-Rhys band narrowing factor is recovered.