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A computational approach to describe the energy relaxation of ahigh-frequency vibrational mode in a fluctuating heterogeneous environment isoutlined. Extending previous work [H. Fujisaki, Y. Zhang, and J.E. Straub, J.Chem. Phys. {\bf 124}, 144910 (2006)], second-order time-dependent perturbationtheory is employed which includes the fluctuations of the parameters in theHamiltonian within the vibrational adiabatic approximation. This means that thetime-dependent vibrational frequencies along an MD trajectory are obtained viaa partial geometry optimization of the solute with fixed solvent and asubsequent normal mode calculation. Adopting the amide I mode ofN-methylacetamide in heavy water as a test problem, it is shown that theinclusion of dynamic fluctuations may significantly change the vibrationalenergy relaxation. In particular, it is found that relaxation occurs in twophases, because for short times ($\lesssim$ 200 fs) the spectral densityappears continuous due to the frequency-time uncertainty relation, while atlonger times the discrete nature of the bath becomes apparent. Considering theexcellent agreement between theory and experiment, it is speculated if thisbehavior can explain the experimentally obtained biphasic relaxation the amideI mode of N-methylacetamide.

Dynamic treatment of vibrational energy relaxation in a heterogeneous and fluctuating environment