Exchange and correlation in open systems of fluctuating electron number

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 semilocal density-functional approximations is concave up and the exact-exchange-only or Hartree-Fock energy is concave down. As a result, semilocal density functionals fail for separated open systems of fluctuating electron number, as in stretched molecular ions $\mathrm{A}_{2}{}^{+}$ 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.