Extensions of the density-functional theory (DFT) to systems with a fractional number of electrons: Consequences for the meaning of the DFT energy levels

The relation between the ionization potential I(N) and the energy ${\mathrm{\varepsilon }}_{\mathit{N}}$(N), relative to the value of the external potential at infinity, of the highest occupied orbital level in the density-functional theory (DFT) for finite systems is examined on the basis of the extensions of the energy functional to the case where the number of electrons of the system is not an integer. The difference between I(N) and -${\mathrm{\varepsilon }}_{\mathit{N}}$(N) is analyzed, as in other works, as a relaxation of the system following the removal of an electron. A problem closely related to the value of this difference is that of the constant value of the exchange and correlation contribution to the effective single-electron potential entering in the DFT, far away from the system. To illustrate this, we propose a sum rule that can be considered as a constraint for the exact energy functional.