Application of the Virial Theorem to Approximate Molecular and Metallic Eigenfunctions

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Fock showed that the virial theorem is automatically satisfied for any quantum-mechanical system whose potential is a homogenous function of the coordinates if a scale factor is introduced into the approximate charge distribution and varied so as to give the lowest energy. In this paper, we extend Fock's treatment to problems involving molecules or metals where it is customary to treat the internuclear separations as parameters. The introduction and variation of the scale factor requires very little additional work and results in considerable improvement in the physical properties. The Heitler-London-Sugiura charge distribution for molecular hydrogen has a mean potential energy, $V=\ensuremath{-}52.98$ ev; a mean kinetic energy, $T=22.79$ ev; and a total energy, $E=V+T=\ensuremath{-}30.20$ ev. When the scale factor is inserted into the eigenfunction and varied to form the Wang function: $E=\ensuremath{-}T=\frac{1}{2}V=\ensuremath{-}30.83$ ev and the virial theorem is satisfied. The total energy is improved by only 0.63 ev but the potential energy is improved by 8.67 ev and the kinetic energy by 8.04 ev. The eigenfunction for the diatomic hydrogen ion, $\ensuremath{\Psi}=N[\mathrm{exp}(\ensuremath{-}{r}_{a})+\mathrm{exp}(\ensuremath{-}{r}_{b})]$ where $N$ is the normalization factor and ${r}_{a}$ and ${r}_{b}$ are the distances of the electrons to nuclei $a$ and $b$, respectively, is another example where the scale factor results in striking improvement in the mean potential and kinetic energies. The James and Coolidge function for molecular hydrogen almost satisfies the virial theorem since $\frac{1}{2}V=\ensuremath{-}31.680$ ev and $\ensuremath{-}T=\ensuremath{-}31.588$ ev. When the scale factor, $s=1.0029$, is introduced, we obtain $E=\frac{1}{2}V=\ensuremath{-}T=\ensuremath{-}31.772$ ev. If the mean potential and kinetic energies, ${V}_{1}$ and ${T}_{1}$ respectively, are known for one nuclear separation, ${R}_{0}$, then on the introduction and variation of the scale factor, a lower energy is obtained for the internuclear separation, $R=\frac{\ensuremath{-}2{R}_{0}{T}_{1}}{{V}_{1}}$, for which $E=\ensuremath{-}T=\frac{1}{2}V=\frac{\ensuremath{-}\frac{1}{4}V_{1}^{2}}{{T}_{1}}$.