Revisiting N-continuous density-functional theory: Chemical reactivity and "atoms" in "molecules"

We construct an internally-consistent density-functional theory valid for noninteger electron numbers N by precise definition of a density functional that is continuous in N. In this theory, charge transfer between the atoms of a heteronuclear diatomic molecule, which have been separated adiabatically to infinity, is avoided because the hardness for fractional occupation of a single HOMO spin-orbital is negative. This N-continuous density functional makes possible a variational theory of "atoms" in "molecules" that exactly decomposes the molecular electron density into a sum of contributions from its parts. The parts are treated as though isolated. That theory, in turn, gives a deep foundation to the chemical reactivity theory provided that the hardness of entities with vanishing spin density is positive, as argued to be the case here. This transition from negative to positive hardness closely parallels the transition from the Heitler-London to the Hund-Mulliken picture of molecular bonding.