Range relaxation 2. Determination of a reaction coordinate over an energy surface of several dimensions

This paper introduces the pass function, the second of three “relaxation” functions to be used in a general functional formulation of parameters necessary for the a priori calculation of molecular structure and dynamics. The approach is intended to simplify the calculation of a reaction coordinate in a complex many-dimensional energy surface. In principle it involves the minimisation of a path integral across a transformed energy surface; and it converges on an analytical expression for the pass of minimum energy across the surface, and the optimal wavefunction of any given form at each point along that pass. In practice numerical evaluation of the integral by Gauss-Legendre quadrature necessitates calculation of the energy at a small number of nuclear configurations. A bridge may be formed with experiment by allowing possible use of empirical expressions or limiting molecular data as a guide to make the optimisation more rapid, though the method converges on a non-empirical result. The scheme is applied here to the hydrogen exchange reaction. For such a two dimensional surface the labour involved is estimated as roughly 5%-15% of that required for a conventional calculation of comparable scope. The economy increases with the number of dimensions considered. The approach can be applied in conjunction with the use of any type of molecular wavefunction.