One-body potential theory in terms of the phase of wave functions for the ground state of the Be atom

From the work of Slater, which was formally completed by Kohn and Sham, a one-body potential V(r) can be constructed which will generate the ground-state density n(r) of a spherically symmetrical atomic charge cloud. For the case of the Be atom, the 1s and 2s wave functions are written in terms of the density amplitude [n(r${\left)\right]}^{1/2}$ and a common phase angle θ. It is then shown that V(r) can be characterized solely by this phase angle, and this motivates the setting up of a variational principle only in terms of the phase θ. As an illustration of the method, the Hartree-Fock ground-state density ${\rho }_{\mathrm{HF}(\mathrm{r}}$) is employed to numerically calculate ${\theta }_{\mathrm{HF}(\mathrm{r}}$), which is then used to calculate $V_{\mathrm{HF}(\mathrm{r}}$). This provides the practical completion of Slater’s proposal for treating exchange for Be. The effect of electron correlation on V(r) is finally estimated using a correlated wave function for Be due to Bunge.