Interpolation of Multidimensional Potential Surfaces by Polynomial Roots

Abstract

Quantum mechanical calculations produce potential energy surfaces in the form of values of the energy E(q) at many geometries. The components of the vector q = q(q₁, q₂,...q_d) are parameters which define the molecular geometry, and d is the total number of internal degrees of freedom defining the nuclear configuration space. This pointwise representation of the surface is rarely of much use either in visualization of the surface or in calculations of the reaction dynamics. A continuous representation can be derived from the given points by fitting a function to them. While the form of the function is at the disposal of the researcher, the effort required to obtain a given accuracy of fit can depend strongly on the functional form chosen. Several classes of functions have been used in interpolations, e.g., cubic splines,¹ polynomials,² Padé approximants,³ and ad hoc functions suggested by the physics of the problem.^4,5 None of these, except perhaps the last, are particularly suited to the representation of potential surfaces in that these surfaces are not of the same form as the fitting function. For example, polynomials are not bounded as their arguments go to plus or minus infinity, have a fixed number of extrema and inflection points, and cannot be multi-valued as can potential surfaces if more than one state of the molecule in question is considered.