Exact differential equation for the density and ionization energy of a many-particle system

The ground-state density $n$ of a many-electron system obeys a Schrödinger-like differential equation for $n^{\frac{1}{2}}\left(\overrightarrow{\mathrm{r}}\right)$, which may be solved by standard Kohn-Sham programs. The exact local effective (nonexternal) potential, $v_{\mathrm{eff}}\left(\overrightarrow{\mathrm{r}}\right)$, is displayed explicitly in terms of wave-function expectation values, from which $v_{\mathrm{eff}}\left(\overrightarrow{\mathrm{r}}\right)>~0$ for all $\overrightarrow{\mathrm{r}}$. A derivation for $n$ as $\left|\overrightarrow{\mathrm{r}}\right|\to \infty$ implies that this new effective potential tends asymptotically to zero, as does the exact Kohn-Sham potential, with the highest occupied eigenvalue as the exact ionization energy. A new exact expression is also presented for the exchange-correlation hole density ${\rho }_{\mathrm{xc}}(\overrightarrow{\mathrm{r}}, {\overrightarrow{\mathrm{r}}}^{\prime })$ about an electron at $\overrightarrow{\mathrm{r}}$, as $\left|\overrightarrow{\mathrm{r}}\right|\to \infty$.