General classical variational treatment of the rates of bimolecular exchange and unimolecular reactions involving three bodies

Abstract

The classical variational theory of chemical reaction rates gives the rate as the equilibrium flux of systems through a trial surface in the phase space of the reaction system. The surface divides the phase space into reactant and product regions and is varied to obtain a least upper bound for the rate of product formation. For bimolecular reactions of the type A+BC→AB+C and the high pressure limit of unimolecular reactions of the type ABC→products, we have derived expressions which give the canonical and microcanonical rate coefficients for the most general dividing surface defined by internal configuration space coordinates. For bimolecular reactions, we have also derived an expression for the energy dependent mean reaction cross section for the most general dividing surface. Expressions for the rate coefficients and mean reaction cross section for any of the more restricted formulations of the dividing surface used in earlier work, and more flexible ones, can be obtained by the substitution of two terms in the appropriate general equation; namely, the partial derivatives of the function that defines the surface with respect to the internal coordinates that define the surface. For example, the flexibility of the surface can be improved systematically by introducing terms in a power series expansion of the surface. The application of a simplex algorithm to determine the coefficients (variational parameters) in an expansion of the surface that includes terms up to the second and third order may give a surface which is close to the best one possible. The minimization procedure corresponds to a search of the potential-energy function for a reaction coordinate that satisfies the variational condition. The variationally determined dividing surface identifies regions of configuration space in which the potential energy must be accurate in order to obtain accurate classical rate coefficients. Thecalculus of variations was applied to the general equations for the rate coefficients to obtain differential equations which give the best dividing surface defined by internal coordinates for the canonical and microcanonical formulations of transition state theory. The corresponding rate coefficients or mean reaction cross section are as close as possible to convergence with the results of classical trajectory calculations. The flux can thus be minimized completely by obtaining a numerical solution for the differential equations which define the best surface or nearly minimized for a series approximation to the best surface. Computational studies are required to determine the more tractable method. By using an appropriate generating function, the procedure described in this work can be extended to reactions involving more than three bodies. The variationally determined dividing surface, reaction coordinate, and reaction surface Hamiltonian could provide a basis for a semiclassical theory of reaction rates.