Semiclassical surface-hopping approximations for the calculation of solvent-induced vibrational-relaxation rate constants

Abstract

Approximate schemes for the calculation of the rates of transitions between vibrational states of a molecule due to the interactions with a solvent are devised based on a rigorous, general semiclassical surface-hopping formalism developed earlier. The formal framework is based on an adiabatic separation of time scales between the fast molecular vibrations and the relatively slow bath motions. (The bath is composed of the solvent degrees of freedom plus all the molecular degrees of freedom other than vibrations.) As a result, the dynamics of the system are described in terms of bath motions occurring on adiabatic vibrational-energy surfaces, which are coupled by a nonadiabatic interaction. The time-dependent vibrational transition probability is evaluated by propagating the canonical density of the system, with the molecule in the initial adiabatic vibrational state, forward in time, and then projecting it onto the final adiabatic vibrational state of interest. The temporal evolution of the density is carried out with a semiclassical surface-hopping propagator, in which the motion of the bath on an adiabatic vibrational surface is described in terms of the familiar (adiabatic) semiclassical propagator, while transitions are accounted for in terms of instantaneous hops of the bath paths between the adiabatic vibrational surfaces involved, with an integration over all possible hopping points. Energy is conserved in the hops, and the only component of momentum that changes is the one along the nonadiabatic coupling vector. When the nonadiabatic interaction is taken into account to first order, the transition probability is predicted to become linear in the long-time limit. Various methods for extracting the relaxation rate constant in this limit are presented, and a simple model system with a one-dimensional bath is employed to compare their practical efficiency for finite time. In addition, this system is used to numerically demonstrate that local approximations for the adiabatic vibrational surfaces and the nonadiabatic coupling yield accurate results, with great reduction of the amount of computation time. Since a local approximation for the vibrational surfaces makes an N-dimensional problem separable into N effectively one-dimensional ones, this treatment is seen to be more generally applicable to realistic systems.