Extension of the Fermi–Amaldi functional to fractional electron number

The Fermi-Amaldi correction is a conceptually significant density-functional approximation with explicit dependence on the electron number N. This dependence raises the question of whether N should be treated as the integral of the total electron density for systems with noninteger (fractional) N. We answer this question negatively by observing that the Fermi-Amaldi functional is identical with the exact exchange functional for any number of fermions or bosons occupying the same spatial orbital and in systems where occupied orbitals have zero differential overlap. This conclusion is consistent with previous reports that the preservation of certain properties of N-dependent functionals requires treating N as a parameter. Our analysis also suggests that the Fermi-Amaldi functional is discontinuous within a given spin-channel of the total electron density and continuous on the boundaries between different spin-channels at N = 1. This implies that the jump discontinuity of the exact exchange-correlation potential observed at N = 1 is a correlation effect. The last conclusion is illustrated with numerical calculations.