Representing Potential Energy Functions by Expansions in Orthogonal Polynomials. Generalized SPF Potentials

It long has been known that advantages attend employing, as a basic internuclear coordinate for determining a molecular potential energy surface, a variable S = 1 − R 0/R, where R 0 is a reference distance near to half of an equilibrium distance. For a diatomic molecule, starting from numerical or analytical representations of the energy, W(R) = W(S), it is shown how to generate the analytical series, W(S) = σ(S)∑ n b n P n (S), where P n (S) are orthogonal polynomials with weight function σ(S) over the range (−1,1) for S. By rearrangement, there result the series for W(R) in inverse powers of R. For neutral diatomics, the Jacobi polynomials, (S) with weight function (1 + S)(1 − S)6, seem particularly appropriate when the potential for large R is of special interest.