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

Abstract

The exact universal functional of integer charge leads to an extension to fractional charge asymptotically 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. 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. The result of the search leads to the molecular size consistency principle. It is physically sensible to extend the concept of molecule to include fractional number of electrons (called fractional molecule) as a localizable observable, with its electronic energy defined as a Legendre transform of the universal functional of fractional charge. A one-to-one mapping between the density and the external potential of a fractional molecule exists. The well-known piecewise linearity of the functional with respect to the number of electrons is shown to hold only for asymptotically isolated v-representable densities. On the other hand, this condition is necessary for an approximate i local universal functional to be accurate for integer number of electrons. The KS kinetic functional for a fractional molecule is well defined and has the same form as that for a system of integer charge. It is shown to be i-local. A nondegenerate ensemble v-representable fractional density is simultaneously noninteracting wavefunction representable with the KS noninteracting assumption. A constrained search over a set of those representing wavefunctions yields an exchange-correlation functional pertaining to fractional occupancies. It is shown to be an upper-bound to the formal KS exchange-correlation energy of the fractional molecule and includes strong correlation. The new functional yields the correct result for a well-designed example of effective fractional occupancies in literature.