Generalized Langevin theory for many-body problems in chemical dynamics: The method of partial clamping and formulation of the solute equations of motion in generalized coordinates

Abstract

A formulation of the analytical dynamics of an n-atom solute system present at infinite dilution in a monatomic solvent is presented. This treatment, which will aid in the understanding of liquid state spectroscopic, energy-transfer, and chemical reaction processes, is made by combining the methods of classical dynamics with recently developed generalized Langevin equation techniques [S. A. Adelman, Adv. Chem. Phys. 53, 611 (1983)]. The Hamiltonian of the solute system is formulated in generalized coordinates which are related to the underlying Cartesian coordinates by a point canonical transformation. The 3n generalized coordinates are partitioned into a set of p explicit coordinates whose dynamics are of direct interest and a set of q=3n−p implicit coordinates whose motion is of lesser interest. A generalized Langevin equation of motion for the explicit coordinates is formulated by computing the reaction force exerted by the solvent on the explicit coordinates in response to small displacements of these coordinates. This generalized Langevin equation will provide a realistic description of explicit coordinate dynamics if the liquid state motion of these coordinates is oscillatory on subpicosecond time scales. Solvent effects appearing in the generalized Langevin equation may be discussed in terms of the fluctuation spectrum which describes atomic motion in the solvation shells. A rigorous statistical mechanical method for assessing the influence of the implicit coordinate motion on this fluctuation spectrum is presented. The formulas for the liquid state quantities appearing in the generalized Langevin equation may be exactly evaluated for particular solute–solvent systems via molecular dynamics (MD) simulation of the motion of the implicit and solvent coordinates given that the solute is partially clamped, i.e., given that the explicit coordinates are fixed. The method of partial clamping provides an improvement of the method of full solute clamping developed earlier. This improvement should permit one to realistically treat many liquid state processes via generalized Langevin equation techniques for which the solute/solvent mass ratio is less than unity.