Multiphonon vibrational relaxation in liquids: should it lead to an exponential-gap law?

The profound differences between solids and liquids notwithstanding, high-frequency vibrational energy relaxation in liquids seems to be well described by assuming that the excess energy is being transferred into discrete overtones of some fundamental intermolecular vibrations-precisely the way it is in crystalline solids. In a solid-state context, this kind of analysis can be used to justify the observation that relaxation rates fall off exponentially with the energy being transferred. Liquids, however, have a substantial degree of disorder, causing their relevant intermolecular spectra to have correspondingly diffuse band edges and large bandwidths. It is therefore not at all obvious what should become of this exponential-gap-law phenomenology. We show in this paper how near exponential-gap-law behavior can still be derived for vibrational energy relaxation in liquids. To do so, we take advantage of the simple dynamics that the high-frequency relaxation has when it is launched from an individual instantaneous configuration. Interestingly, the physically relevant region turns out not to be true asymptotic limit of our formalism, but for realistic liquid parameters the behavior in the physical regime differs only slightly from an exact exponential-gap law and is strikingly independent of the details of the intermolecular spectra.