SEMICLASSICAL SURFACE-HOPPING APPROXIMATIONS FOR THE CALCULATION OF SOLVENT-INDUCED VIBRATIONAL-RELAXATION RATE CONSTANTS

Approximate schemes for the calculation of the rates of transitions be tween 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 vib rations and the relatively slow bath motions. (The bath is composed of the solvent degrees of freedom plus all the molecular degrees of free dom other than vibrations.) As a result, the dynamics of the system ar e described in terms of bath motions occurring on adiabatic vibrationa l-energy surfaces, which are coupled by a nonadiabatic interaction. Th e time-dependent vibrational transition probability is evaluated by pr opagating the canonical density of the system, with the molecule in th e initial adiabatic vibrational state, forward in time, and then proje cting it onto the final adiabatic vibrational state of interest. The t emporal evolution of the density is carried out with a semiclassical s urface-hopping propagator, in which the motion of the bath on an adiab atic vibrational surface is described in terms of the familiar (adiaba tic) semiclassical propagator, while transitions are accounted for in terms of instantaneous hops of the bath paths between the adiabatic vi brational surfaces involved, with an integration over all possible hop ping points. Energy is conserved in the hops, and the only component o f momentum that changes is the one along the nonadiabatic coupling vec tor. 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 c onstant 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 demo nstrate that local approximations for the adiabatic vibrational surfac es and the nonadiabatic coupling yield accurate results, with great re duction 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.