Vibrational energy relaxation nonlinear interaction models

In this Section we apply the general formalism developed in Section 13.3 together with the interaction models discussed in Section 13.2 in order to derive explicit expressions for the vibrational energy relaxation rate. Our aim is to identify the molecular and solvent factors that determine the rate. We will start by analyzing the implications of a linear coupling model, than move on to study more realistic nonlinear interactions. 

Equation (13.39) implies that in the bilinear coupling, the vibrational energy relaxation rate for a quantum hannonic oscillator in a quantum harmonic bath is the same as that obtained from a fully classical calculation ( a classical harmonic oscillator in a classical harmonic bath ). In contrast, the semiclassical approximation (13.27) gives an error that diverges in the limit T 0. Again, this result is specific to the bilinear coupling model and fails in models where the rate is dominated by the nonlinear part of the impurity-host interaction. 

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