DIRECT DETERMINATION OF PURE-STATE DENSITY MATRICES .3. 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(R̄0)T(R̄)12ζ(R̄0)-12T(R̄0) R̄R̄02F(R)-RdF(R)dRT(R)12dR, where R is the internuclear distance, R̄ζR, F=VR, with V1 the one-electron potential, Ti-12i2, and R̄0 is an integration limit. It is shown that if ζ(R̄0) is a variational scale factor, then ζ(R̄) is also a variational scale factor provided the electron density ρ1 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 H2+, H2, He2, and Li2 and compared with experimental and variationally calculated values. Qualitative agreement is obtained in general, and, in particular, our theoretical energy curve for He2 is in very good agreement with the best variational results for 1.5a.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+ΔR. © 1969 The American Physical Society.