Diatomic Forces and Force Constants. I. Errors in the Hellmann—Feynman Method

Abstract

Diatomic forces and force constants can be calculated from the Hellmann—Feynman (H—F) theorem and its derivatives but this theorem omits terms which do not vanish, in general, unless the wavefunction is exact. Using common approximate valence bond and molecular orbital wavefunctions for H2, of both the scaled and unscaled type, we have evaluated the terms which are omitted in this manner. In each case the total theoretical force and the force constants were determined as well. We note that the H—F theorem, when used with approximate wavefunctions, can assume a variety of inequivalent forms depending upon the set of electronic coordinates held constant during differentiation. Two such forms are studied in detail, namely the virial and electrostatic theorems. Various equivalent expressions for the quadratic and cubic force constants (k2 and k3) are developed by taking the first two derivatives of these theorems. Some of our expressions are new and one, in particular, has immediate computational advantages. A new approximation for k3 is tested and the implications of our study for the perturbation treatment of force constants is noted.