Electrostatic interpretation of an electron density associated with the spherical exchange-correlation potential $V_{\mathrm{xc}}\mathrm{}\left(r\right)\mathrm{}$ in atoms: Application to Be

By means of an electrostatic analogy, an electron density is proposed that is related to the exchange-correlation potential $V_{\mathrm{xc}}\mathrm{}\left(r\right)\mathrm{}$ in atoms. More precisely, such an electron density is best characterized by the amount of electronic charge $Q_{\mathrm{xc}}\mathrm{}\left(r\right),$ say, enclosed within a sphere of radius r centered on the atomic nucleus. Then $Q_{\mathrm{xc}}\mathrm{}\left(r\right)\mathrm{}$ is related to the radial derivative of $V_{\mathrm{xc}}\mathrm{}\left(r\right)\mathrm{}$ by $Q_{\mathrm{xc}}\mathrm{}\left(r\right)=-r^2\partial V_{\mathrm{xc}}/\partial r.\mathrm{}$ $|Q_{\mathrm{xc}}\mathrm{}\left(r\right)\mathrm{}|$ tends to unity as $\overrightarrow{r}\infty$ and becomes zero in the limit $\overrightarrow{r}0.$ However, it increases at first as one comes away from the point at infinity, having the form at large r $|Q_{\mathrm{xc}}\mathrm{}\left(r\right)\mathrm{}\overrightarrow{|}1+2\alpha {/r}^3{+O(1/r}^4),$ where α is the dipole polarizability of the singly charged positive ion. This means that $|Q_{\mathrm{xc}}\mathrm{}\left(r\right)\mathrm{}|$ must have at least one maximum, its height $Q_{\mathrm{xc}}{(r}_m)$ and its position $r_m$ then being important parameters characterizing the shape of $Q_{\mathrm{xc}}\mathrm{}\left(r\right).$ The intersection(s) with the line $|Q_{\mathrm{xc}}|=1\mathrm{}$ are also plainly of importance in this same context. The exact form of $Q_{\mathrm{xc}}\mathrm{}\left(r\right)\mathrm{}$ involves both fully interacting one- and two-particle fermion density matrices, as well as the orbitals of the Slater-Kohn-Sham (SKS) reference system. However, the example of Be is worked out, where it is shown that, if the ground-state density $\rho \mathrm{}\left(r\right)={\rho }_{\mathrm{SKS}}\mathrm{}\left(r\right)\mathrm{}$ is known from either x-ray or electron diffraction experiments or from quantal computer simulation studies, then $Q_{\mathrm{xc}}\mathrm{}\left(r\right)\mathrm{}$ can be derived for this light atom.