Diatomic Forces and Force Constants. II. Variation—Perturbation Method

Abstract

According to Rayleigh—Schrödinger perturbation theory the quadratic (k2) and cubic (k3) force constants of a diatomic molecule are completely determined by the unperturbed and first-order wavefunctions, ψ0 and ψ1. We approximate ψ1 using the Hylleraas variation technique which optimizes a trial function, ψ̃1, by minimizing an expression for k2. Calculations were carried out on H2 with the virial form of the Hellmann—Feynman theorem. Several approximate ψ0's, all of the scaled variety, were tested along with two ψ̃1's containing one and two variation parameters, respectively. Although it is not required by the theory the best results for both k2 and k3 were obtained with the more flexible trial function. Furthermore, with this ψ̃1 we found that improving ψ0 (in the sense of lower E0) had a salutary effect in all but one case. The major error in the better calculations arises not from the ψ1 terms but from evaluating 〈ψ0 | ∂2H/∂R2 | ψ0〉 and 〈ψ0 | ∂3H/∂R3 | ψ0〉. But the latter are readily available experimental quantities since they depend only on the total electronic energy and equilibrium internuclear distance. A semiempirical method for determining force constants is thus suggested. The results are excellent. For example, with the Weinbaum function as ψ0, k2=0.362 and k3=−1.43 (in atomic units) as compared to the experimental values of 0.368 and −1.30.