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 n1/2(→r), which may be solved by standard Kohn-Sham programs. The exact local effective (nonexternal) potential, veff(→r), is displayed explicitly in terms of wave-function expectation values, from which veff(→r)>~0 for all →r. A derivation for n as |→r|→∞ 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 ρxc(→r, →r^') about an electron at →r, as |→r|→∞.