Local Approximation of Potential‐Energy Surfaces by Surfaces Permitting Separation of Variables

Abstract

In the immediate vicinity of a potential‐energy minimum or of a saddle point, it is shown that major topographical features of a "nonseparable" potential‐energy surface can be imitated by those of a surface permitting separation of variables. For each extremal path of descent or ascent to the cited critical point of the surface, there is an exact match of the tangent, the first curvature vector in configuration space, and the force constant along that path provided that the known curvature vector satisfies an equation containing the metric tensor of the selected coordinate system and known force constants. Because of the wide choice of coordinate systems available for selection, it is anticipated that this relation may be fulfilled for each extremal path, partly by choice of the coordinate system and partly by subsequent choice of the curvilinear coordinates of the critical point. There are several possible applications of this local approximation, including those to problems involving anharmonic coupling of normal modes and those involving n‐dimensional tunneling and other calculations in reaction‐rate theory. Use will be made of the formalism to extend the activated complex theory in chemical kinetics. As a preliminary test of the local‐approximation concept, literature data on n‐ and one‐dimensional tunneling rates are compared. They are found to be fairly similar when proper cognizance is taken of zero‐point energies.