Going Beyond the Point Nucleus Approximation to Satisfy the Hellmann–Feynman Theorem: Born–Oppenheimer H$^+_2$ in the Ground State
Résumé
Incomplete spaces are investigated for solving the Schrödinger equation under the Born–Oppenheimer approximation. It is shown that the Hellmann–Feynman theorem cannot be used for computing the electronic force exerted on a nucleus, when a variational wavefunction with floating centers is used, if multicenter polynomial components are added in order to describe the polarization effects through the chemical bond. This is because the minimum of the potential energy surface is not a stationary point in the direction of the float parameter. Such a failure can be fixed by considering a molecular model with finite size nuclei, as defined herein. The classical electronic force is computed for that model, as compared with the standard point charge approximation, and it is applied to the ${\text {H}_2}^+$ molecular ion. As a result, the former model is found more accurate by several orders of magnitude.