Vibrational energy relaxation in the condensed phases: Quantum vs classical bath for multiphonon processes

We develop a theory for relating quantum and classical time correlation functions in the context of vibrational energy relaxation. The treatment is based on the assumption that both the quantum and the classical systems are characterized by effective harmonic Hamiltonians with identical normal modes; and the solute-solvent interaction is taken to be linear in the solute vibrational coordinate, but nonlinear in the bath coordinates. We propose an approximate “quantum correction” which allows the determination of the quantum energy relaxation rates from the classical force-force time correlation functions in the limit of large solute’s vibrational frequency. We test the accuracy of this approximate correction against exact numerical results for two forms of the solute-solvent interaction (exponential and power law), and find it to be accurate for a wide range of solute vibrational frequencies and for different solvent thermodynamic states. A simple form of the “quantum correction” is proposed for the models based on Lennard-Jones interactions. In all cases it is found that the vibrational relaxation time in a fully quantum system is better approximated by a fully classical theory (classical oscillator in classical bath) than by a mixed quantum-classical theory (quantum oscillator in classical bath).