Hohenberg-Kohn theory

In any practical application of the Hohenberg-Kohn theory, a specified density functional / s[p] restricted to ground-state densities defines an equivalent orbital functional A [ ,, ] that can be extended to all functions in the orbital Hilbert space. The OEL equations for occupied orbitals of the reference state of an A-electron ground state take the form, for i < N,... 

The Hohenberg-Kohn theory of /V-clcctron ground states is based on consideration of the spin-indexed density function. Much earlier in the development of quantum mechanics, Thomas-Fermi theory [402, 108] (TFT) was formulated as exactly such a density-dependent formalism, justified as a semiclassical statistical theory [231, 232], Since Hohenberg-Kohn theory establishes the existence of an exact universal functional Fs [p] for ground states, it apparently implies the existence of an exact ground-state Thomas-Fermi theory. The variational theory that might support such a conclusion is considered here. 

Hohenberg-Kohn theory proves this to be a functional of p, universal in the sense that it does not depend explicitly on v. 

For most applications, the external potential operator v reduces to a local potential v(r), and the Hohenberg-Kohn theory is valid for the ground state. The present derivation follows exactly the logic of standard Hartree-Fock theory. It is not restricted to ground states and remains valid for fractional occupation numbers. 

When Exc p is specified, the relevant ground-state density for Hohenberg-Kohn theory is p0, computed using the equivalent orbital functional Exc in the OEL equahons, (Q — e,-)local potential w(r) in the corresponding KS equahons is determined by the KSC by minimizing T for p = p0. Assuming the locality hypothesis, that w — v is the Frechet derivative of the model ground-state functional h p — Ts[p, this implies that w = vh + vxc + v is a sum of local potentials. If i>xc in the OEL equahons was equivalent to a local potential vxc(r), the KS and OEL equations would produce the same model wave function. 

As an application of the Hohenberg-Kohn theory to a noninteracting system, the KSC can be carried out using the Levy construction [6] with a Lagrange-mulhplier field w(r) for the density constraint,... 

It is illustrative to discuss the reformulation of the Hohenberg-Kohn theory originally carried out by Levy [54] (and later, also by Lieb [55-57]), where instead of the stronger v-representability condition, all that is asked for is compliance with the weaker TV-representability condition for the energy functionals. Our discussion is based on Eq. (18) plus the assumption that Av C M, where Av is the set of u-representable densities (namely, densities coming from ground-state wavefunctions for Hamiltonians // , with t/eV) and J f is the set of iV-representable densities. The latter is explicitly defined by... 

Figure 8.1 Schematic representation of the Hohenberg-Kohn theory. The short arrows denote [starting from 4xt(f)] the usual solution of the Schrodinger equation. The potential V4xt(f) determines all states including the...

Although the DFT method has been extensively applied to nonrelativistic calculations, the four-component DFT approaches have only recently appeared (see the book [335] and the review [499] and references therein). Relativistic versions of the Kohn-Sham equations have been developed based on the relativistic extension of the Hohenberg-Kohn theory [500]. 

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