Direct Determination of Pure-State Density Matrices. III. Purely Theoretical Densities Via an Electrostatic-Virial Theorem

A combined electrostatic virial theorem is introduced and used to derive a differential equation for the scale factor $\zeta$ in a diatomic molecule. This equation can either be used to compute $\frac{d\zeta }{\mathrm{dR}}$ or it can be integrated to yield $\zeta \left(\overline{\mathrm{R}}\right)={\left(\frac{T\left({\overline{R}}_0\right)}{T\left(\overline{R}\right)}\right)}^{\frac{1}{2}}\left(\zeta \left({\overline{R}}_0\right)-\frac{1}{2T\left({\overline{R}}_0\right)}\int {\overline{R}}^{{\overline{R}}_0}\frac{2F\left(R\right)-\frac{\mathrm{RdF}\left(R\right)\mathrm{}}{\mathrm{dR}}}{T{\mathrm{}\left(R\right)\mathrm{}}^{\frac{1}{2}}}\mathrm{dR}\right),$ where $R$ is the internuclear distance, $\overline{R}\equiv \zeta R$, $F=〈-\frac{\partial V_1}{\partial R}〉$, with $V_1$ the one-electron potential, $T\equiv 〈\Sigma i^{}-\frac{1}{2}{{\nabla }_i}^2〉$, and ${\overline{R}}_0$ is an integration limit. It is shown that if $\zeta \left({\overline{R}}_0\right)$ is a variational scale factor, then $\zeta \left(\overline{R}\right)$ is also a variational scale factor provided the electron density ${\rho }_1$ involves no other unoptomized variational parameters. Unlike the conventional variational expression for $\zeta$, which contains two-electron integrals, the above formula involves only the one-electron force and kinetic energy integrals. Using this $\zeta$, electron densities and energies are calculated for ${\mathrm{H}}_2^{+}$, ${\mathrm{H}}_2$, ${\mathrm{He}}_2$, and ${\mathrm{Li}}_2$ and compared with experimental and variationally calculated values. Qualitative agreement is obtained in general, and, in particular, our theoretical energy curve for ${\mathrm{He}}_2$ is in very good agreement with the best variational results for