Condensed phase vibrational relaxation: calibration of approximate relaxation theories with analytical and numerically exact results

The dynamics of a small quantum system coupled to condensed phase bath is considered. Such dynamics is important in vibrational and spin relaxation of probe molecules in condensed phase media. Adapting and generalizing the condensed phase electron transfer analysis of Gayen et al. [J. Chem. Phys. 112 (2000) 4310], we show how to compute the reduced system density matrix exactly for a large class of Hamiltonians, namely those for which the system Hamiltonian and the system factor in the system-bath coupling term commute. For this class of problems, several approximate second order relaxation theory equations of motion for the reduced system density matrix also have special properties. In particular, the Markovian limit of these equations of motion forms a positive semigroup. Also, if the bath is a collection of harmonic oscillators, and the bath coupling operator is linear in these oscillator coordinates, then local second order relaxation theory is exact, even for strong system-bath coupling. The case of a degenerate two-level system coupled off-diagonally to the bath is among those that can be solved exactly. In order to treat the nondegenerate two-level analog, we show that the Hamiltonian describing population relaxation of such a system coupled linearly to a harmonic bath can be mapped to the canonical Spin-Boson Hamiltonian, albeit with nonstandard initial state conditions. Nevertheless, Path Integral methods can be utilized to compute numerically exact time-evolution of the equivalent Spin-Boson problem, from which the desired population relaxation dynamics can be extracted. Extensive comparisons to commonly utilized second order relaxation theory approximations are presented. (C) 2003 Published by Elsevier B.V.