The short-time intramolecular dynamics of solutes in liquids. I. An instantaneous-normal-mode theory for friction

Abstract

It is sometimes useful to be able to think of the energy relaxation of a solute dissolved in a liquid as being caused by some sort of solvent-inspired friction. This intuitive association can, in fact, be made literal and quantitative in classical mechanics by casting the dynamics into a solute-centered equation of motion, a generalized Langevin equation, in which the dissipative character of the solvent is embodied in a (generally time delayed) friction force. An exact prescription is available for finding this friction, but the process is formal and the connection with microscopic degrees of freedom is rather indirect. An alternate approach due to Zwanzig, which portrays the solvent as a harmonic bath, makes explicit use of a set of solvent coordinates, but these coordinates have no immediate relationship with any of the real solvent degrees of freedom. We show here that by taking a short-time perspective on solute relaxation we can derive a generalized Langevin equation, and hence a friction kernel, which is both exact (at least at short times) and has a completely transparent connection with solvent motion at the molecular level. We find, in particular, that under these conditions the instantaneous normal modes of the solution fill the role of the Zwanzig harmonic oscillators precisely, meaning that one can analyze friction in molecular terms by appealing to the explicitly microscopic definitions of the instantaneous modes. One of the implications of this perspective is that fluctuations of the solvent are automatically divided into configuration- to-configuration fluctuations and dynamics resulting from a given liquid configuration. It is the latter, instantaneous, friction that we shall want to decompose into molecular ingredients in subsequent papers. However, even here we note that it is the character of this instantaneous friction that leads to the fluctuating force on a solute having slightly, but measurably, non-Gaussian statistics. Our basic approach to liquid-state friction and a number of results are illustrated for the special case of the vibrational relaxation of a diatomic molecule in an atomic liquid.