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The exact universal functional of integer charge leads to an extension tofractional charge asymptotically when it is applied to a system made ofasymptotically separated densities. The extended functional is asymptoticallylocal and is said to be i-local. Applying the functional to a system withnuclei distributed in two asymptotically separated locales requires an explicitsearch of the electronic charge at each locale. The result of the search leadsto the molecular size consistency principle. It is physically sensible toextend the concept of molecule to include fractional number of electrons(called fractional molecule) as a localizable observable, with its electronicenergy defined as a Legendre transform of the universal functional offractional charge. A one-to-one mapping between the density and the externalpotential of a fractional molecule exists. The well-known piecewise linearityof the functional with respect to the number of electrons is shown to hold onlyfor asymptotically isolated v-representable densities. On the other hand, thiscondition is necessary for an approximate i local universal functional to beaccurate for integer number of electrons. The KS kinetic functional for afractional molecule is well defined and has the same form as that for a systemof integer charge. It is shown to be i-local. A nondegenerate ensemblev-representable fractional density is simultaneously noninteractingwavefunction representable with the KS noninteracting assumption. A constrainedsearch over a set of those representing wavefunctions yields anexchange-correlation functional pertaining to fractional occupancies. It isshown to be an upper-bound to the formal KS exchange-correlation energy of thefractional molecule and includes strong correlation. The new functional yieldsthe correct result for a well-designed example of effective fractionaloccupancies in literature.

Density functional theory for molecular size consistency and fractional charge