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 Schr\"odinger-like differential equation for ${n}^{\frac{1}{2}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, which may be solved by standard Kohn-Sham programs. The exact local effective (nonexternal) potential, ${v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is displayed explicitly in terms of wave-function expectation values, from which ${v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})>~0$ for all $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$. A derivation for $n$ as $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|\ensuremath{\rightarrow}\ensuremath{\infty}$ 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 ${\ensuremath{\rho}}_{\mathrm{xc}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ about an electron at $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$, as $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|\ensuremath{\rightarrow}\ensuremath{\infty}$.