  While the exact total energy of a separated open system varies linearly as a
function of average electron number between adjacent integers, the energy
predicted by semi-local density functional approximations curves upward and the
exact-exchange-only or Hartree-Fock energy downward. As a result, semi-local
density functionals fail for separated open systems of fluctuating electron
number, as in stretched molecular ions A$_2^{+}$ and in solid transition metal
oxides. We develop an exact-exchange theory and an exchange-hole sum rule that
explain these failures and we propose a way to correct them via a local hybrid
functional.
