Electronic Bases of Molecular Vibrations. I. General Theory for Diatomic Molecules

The force constant of a diatomic molecule in a given electronic state is the second derivative with respect to the internuclear distance R of the molecular electronic energy, evaluated at the equilibrium distance Re. In principle it is determined by the electronic-charge distribution ψ*ψ, where ψ is the appropriate solution of the electronic-wave equation. Expressions for the force constant in terms of the electronic-charge distribution are derived which differ depending upon (1) what coordinate representation is used to describe the wavefunction, (2) whether the virial theorem or the Hellmann—Feynman theorem is employed to define the first energy derivative. Correspondingly, if the wavefunction is expressed in confocal elliptic coordinates, two different expressions for the force constant are obtained. These expressions give identical results for exact wavefunctions or for approximate wavefunctions which are constructed to satisfy the virial theorem for all R. If this condition is not satisfied, the two expressions will give, in general, different results. It is shown that if the Hellmann—Feynman theorem is used to define the first energy derivative, then the force constant assumes a particularly simple form which is suggestive of the force constant of a classical harmonic oscillator. The utility of the force constant expressions as probes to explore molecular electronic-charge distributions is discussed and related problems are suggested.