Whereas the determinantal wave functions constructed from the solutions of the Hartree-Fock equations satisfy the virial theorem, those constructed from self-consistent-field schemes with a local exchange approximation do not necessarily. A scaling procedure, which goes back to Fock, allows considerable improvement and good agreement with the Hartree-Fock results. The virial theorem criterion agrees with previous empirically based assertions that Slater's potential "overestimates" exchange effects. In addition, and also in an ab initio way, it can be added that Kohn and Sham's potential "underestimates" them, at least for light atoms. Numerical results and comparison of various methods are presented. A Slater-type local exchange potential is suggested, $V_{\mathrm{HFS}}^{}{}_{}{}^{\prime \prime }=-X6\mathrm{}{\left[\right(\frac{3}{8}\pi \left)\rho \right(\overrightarrow{\mathrm{r}}\left)\right]}^{\frac{1}{3}}$, where $X$ is determined by the virial theorem. Applications to solid-state physics are discussed.

• Received 14 March 1969

DOI:https://doi.org/10.1103/PhysRev.184.10