Theory of vibrational energy relaxation in liquids: Construction of the generalized Langevin equation for solute vibrational dynamics in monatomic solvents.

Algorithms which permit the explicit, albeit approximate, construction of a physically realistic generalized Langevin equation of motion for the energy relaxation dynamics of a specified solute normal mode coordinate y in a monatomic solvent are developed. These algorithms permit the construction, from equilibrium solute–solvent pair correlation functions, of the liquid state frequency ωl of the normal mode and of the Gaussian model approximation to the autocorrelation function 〈F(t)F〉0 of the fluctuating generalized force exerted by the solvent on the normal mode. From these quantities one may compute, from equilibrium solute–solvent pair correlation functions, the vibrational energy relaxation time T1 of the solute normal mode and also related quantities which permit one to assess the relative importance of direct [y coordinate→solvent] and indirect [y coordinate→solute translation–rotational coordinates→solvent] energy flow pathways in solute vibrational energy relaxation. The basis of the construction of T1 is the formula T1=β-1(ωl) where β(ω)=∫∞0 β(t)cos ω dt and where β(t)=[kBT]-1 〈F(t)F〉0 is the friction kernel of the solute normal mode. This formula is valid if T1>T2=vibrational phase relaxation time. The approximate formulas for T1 are worked out in detail for diatomic solutes. The approximations are tested for this diatomic solute case by comparing with molecular dynamics results. [ABSTRACT FROM AUTHOR]