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 ζ in a diatomic molecule. This equation can either be used to compute dζdR or it can be integrated to yield ${ζ}(R̄)={({T({R̄}₀)}{T(R̄)})}1/2({ζ}({R̄}₀){-}{1}{2T({R̄}₀)}{∫}{R̄}^{{R̄}₀}{2F(R){-}RdF(R)/dR}{T{(R)}1/2}dR),$ where $R$ is the internuclear distance, $R̄{≡}{ζ}R$, $F=〈{-}{{∂}{V}₁}{{∂}R}〉$, with ${V}₁$ the one-electron potential, $T{≡}〈{Σ}{i}^{}{-}1/2{{{∇}}ᵢ}²〉$, and ${R̄}₀$ is an integration limit. It is shown that if ${ζ}({R̄}₀)$ is a variational scale factor, then ${ζ}(R̄)$ is also a variational scale factor provided the electron density ${{ρ}}₁$ involves no other unoptomized variational parameters. Unlike the conventional variational expression for ${ζ}$, which contains two-electron integrals, the above formula involves only the one-electron force and kinetic energy integrals. Using this ${ζ}$, electron densities and energies are calculated for ${H}₂⁺$, ${H}₂$, ${He}₂$, and ${Li}₂$ and compared with experimental and variationally calculated values. Qualitative agreement is obtained in general, and, in particular, our theoretical energy curve for ${He}₂$ is in very good agreement with the best variational results for $1.5 a.u.<R<{∞}$. It is also shown how the electrostatic-virial theorem can be used as a condition in continuing density-matrix calculations from $R$ to $R+{Δ}R$.