Density functional theory for fractional charge: Locality, size consistency, and exchange-correlation.

We show that the exact universal density functional of integer electronic charge leads to an extension to fractional charge in an asymptotic sense when it is applied to a system made of asymptotically separated densities. The extended functional is asymptotically local and is said to be i-local. The concept of i-locality is also applicable to nuclear external potentials, and a natural association exists between the localities of a density and a set of nuclei. Applying the functional to a system with nuclei distributed in two asymptotically separated locales requires an explicit search of the electronic charge at each locale with the constraint of the global charge. The determined number of electrons at each locale can be fractional. The molecular size consistency principle is realized as the result of the search. It is physically sensible to extend the molecule concept to include a fractional number of electrons (called fractional molecule henceforth) as a localized observable. The physical validity of fractional molecules is equivalent to the asymptotic separability of molecules, a basic assumption in molecular research. A one-to-one mapping between a fractional molecule's density and external potential is shown to exist with a nondegenerate condition. The global one-to-one mapping required by the Hohenberg-Kohn first theorem is realized through the aforementioned global search for molecular charges. Furthermore, the well-known piecewise linearity of the universal functional with respect to the number of electrons is necessary for an approximate i-local universal functional to be broadly accurate for any integer number of electrons. The Kohn-Sham (KS) noninteracting kinetic energy functional for a fractional molecule is well-defined and has the same form as that for a system of an integer number of electrons. It is shown to be i-local. A nondegenerate, noninteracting ensemble v-representable fractional density is simultaneously noninteracting wavefunction representable. A constrained search over those representing wavefunctions yields the definition of an exchange-correlation functional pertaining to fractional occupancies of KS orbitals. The functional is shown to be an upper bound to the formal KS exchange-correlation energy of a fractional molecule and includes a strong correlation. It yields the correct result for a well-designed example of effective fractional occupancies in the literature.