On series expansions of orientation-dependent properties of pairs of non-spherical molecules

Abstract

The accuracies of expansions of pair-wise molecular properties in angle-dependent functions are re-examined. The usual expansions in orthogonal polynomials such as products of spherical harmonics are those which, for a given truncation of the infinite series, minimize the mean square error. It is argued therefore, that the magnitude of the mean square deviations is a good criterion for determining the accuracy of this kind of expansion. Numerical illustrations are given for expansions of a site-site intermolecular interaction and for two functions of it: the Boltzmann factor and the product of the Boltzmann factor and the potential. As an alternative to the spherical harmonics, expansions based on products of Chebyshev polynomials are also considered. It is shown that coefficients in these expansions can be calculated either by minimizing the mean square error (with a different weight factor than for the spherical harmonics) or by utilizing them as interpolating polynomials. An alternative criterion for accuracy is also explored, which is to determine the absolute magnitude of the maximum difference between a given approximation and the exact function. The various Chebyshev expansions and the spherical harmonic expansion are examined in the light of both of the criteria suggested. It is concluded that the Chebyshev interpolating polynomials possess some attractive features which make them useful in particular applications.