Theory of vibrational energy relaxation in liquids: Vibrational–vibrational energy transfer

Abstract

A theoretical treatment of the vibrational–vibrational (VV) contribution to the vibrational energy relaxation time T1 of a solute normal mode in a molecular solvent, which extends a previous treatment [S. A. Adelman, R. H. Stote, and R. Muralidhar, J. Chem. Phys. 99, 1320 (1993), henceforth called Paper I] of the vibrational–translational–rotational (VTR) contribution to T1, is outlined and expressions for this VV contribution, valid for the infinitely dilute diatomic solution, are presented. The treatment is based on the formula T1=β−1(ωl), where β(ω) is the friction kernel of the relaxing solute mode and where ωl is its liquid phase frequency. β(ω) is evaluated as the cosine transform of the autocorrelation function 〈ℱ̃(t)ℱĩ〉0v of the fluctuating generalized force exerted by the vibrating solvent on the solute normal mode coordinate conditional that this coordinate is fixed at its equilibrium value. 〈ℱ̃(t)ℱ̃〉0v is expressed as a superposition of the rigid solvent autocorrelation function 〈ℱ̃(t)ℱ̃〉0 and a correction which accounts for solvent vibrational motion. For diatomic solvents one has 〈ℱ̃(t)ℱ̃〉0v= 〈ℱ̃(t)ℱ̃〉0+NSMD(t) cos ωDt F(ΩD), where NS=number of solvent molecules, MD(t) is the vibrational force gradient autocorrelation function, ωD and ΩD are solvent molecule liquid phase frequencies, and F(Ω)=1/2ℏΩ−1 coth[ℏΩ/2kBT]. The Gaussian model is assumed for 〈ℱ̃(t)ℱ̃〉0 and MD(t) yielding β(ω) as a superposition of a Gaussian centered at ω=0 which mediates VTR processes and a Gaussian centered at ω=ωD which mediates VV processes. Vector integral expressions for MD(t), ωD, and ΩD are presented which are similar to the expressions for ωl and 〈ℱ̃(t)ℱ̃〉0 given in Paper I. These expressions permit the evaluation of the VV contribution to T1 from the atomic masses, bondlengths, vibrational frequencies, and site–site interaction potentials of the solute and solvent molecules and from specified rigid solvent equilibrium site–site pair correlation functions of the liquid solution.