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.

Figure 1

Total energies of the F atom as functions of the average electron number $N$ in the PBE approximation (with and without correlation) and in Hartree-Fock theory. The dotted lines represent the exact-xc result based on the experimental ionization potential and electron affinity of the F atom [16].

Figure 2

Spin-restricted exchange-only dissociation curves for the ${\mathrm{C}}_2$ molecule in the unusual symmetric singlet electron configuration. This configuration is the lowest-energy state in the PBEx approximation near the equilibrium internuclear distance but not at larger $R$, and not at any $R$ in Hartree-Fock.