THEORY OF VIBRATIONAL-ENERGY RELAXATION IN LIQUIDS - VIBRATIONAL-VIBRATIONAL ENERGY-TRANSFER

A theoretical treatment of the vibrational-vibrational (VV) contributi on to the vibrational energy relaxation time T1 of a solute normal mod e in a molecular solvent, which extends a previous treatment [S. A. Ad elman, R. H. Stote, and R. Muralidhar, J. Chem. Phys. 99, 1320 (1993), henceforth called Paper I] of the vibrational-translational-rotationa l (VTR) contribution to T1, is outlined and expressions for this VV co ntribution, valid for the infinitely dilute diatomic solution, are pre sented. The treatment is based on the formula T1 = beta-1(omega(l)), w here beta(omega) is the friction kernel of the relaxing solute mode an d where omega(l) is its liquid phase frequency. beta(omega) is evaluat ed as the cosine transform of the autocorrelation function [F(t)F]0nu 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. [F(t)F]0nu is expressed as a superp osition of the rigid solvent autocorrelation function [F(t)F]0nu and a correction which accounts for solvent vibrational motion. For diatomi c solvents one has [F(t)F]0nu = [F(t)f]0+N(S)M(D)(t) cos omega(D)t F(O MEGA(D)), where N(S) = number of solvent molecules, M(D)(t) is the vib rational force gradient autocorrelation function, omega(D) and OMEGA(D ) are solvent molecule liquid phase frequencies, and F(OMEGA) = 1/2hOM EGABAR-1 coth[hOMEGABAR/2k(B)T]. The Gaussian model is assumed for [F( t)F]0 and M(D)(t) yielding beta(omega) as a superposition of a Gaussia n centered at omega = 0 which mediates VTR processes and a Gaussian ce ntered at omega = omega(D) which mediates VV processes. Vector integra l expressions for M(D)(t), omega(D), and OMEGA(D) are presented which are similar to the expressions for omega(l) and [F(t)F]0 given in Pape r I. These expressions permit the evaluation of the VV contribution to T1 from the atomic masses bondlengths, vibrational frequencies, and s ite-site interaction potentials of the solute and solvent molecules an d from specified rigid solvent equilibrium site-site pair correlation functions of the liquid solution.