Hohenberg-Kohn theorems orbital functional theory

The Levy construction [222] can be used to prove Hohenberg-Kohn theorems for the ground state of any such theory. It should be noted that any explicit model of the Hohenberg-Kohn functional F[p] implies a corresponding orbital functional theory. The relevant density function p(r) is that constructed from an OFT ground state. This has the orbital decomposition , as postulated by Kohn and Sham [205]. Unlike the density p,, for an exact A-electron wave function T, which cannot be determined for most systems of interest, the OFT ground-state density function is constructed from explicit solutions of the orbital Euler-Lagrange equations, and the theory is self-contained. 

So the highest occupied Kohn-Sham orbital has a fractional occupation number to. The fact that is uniquely defined by follows directly from the Hohenberg-Kohn theorem applied to the non-interacting system. The proof of the Hohenberg-Kohn theorem for systems with noninteger number of electrons proceeds along the same lines as for systems with integer particle number (alternatively it can be obtained from the zero temperature limit of temperature dependent density functional theory [82]). We further split up v, as... 

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