Hohenberg-Kohn theory energy density functionals

The local-scaling transformation version of density functional theory (LS-DFT), [1-12] is a constructive approach to DFT which, in contradistinction to the usual Hohenberg-Kohn-Sham version of this theory (HKS-DFT) [13-18], is not based on the IIohenberg-Kohn theorem [13]. Moreover, in the context of LS-DFT it is possible to generate explicit energy density functionals that satisfy the variational principle [8-12]. This is achieved through the use of local-scaling transformations. The latter are coordinate transformations that can be expressed as functions of the one-particle density [19]. 

In Kohn-Sham (KS) density functional theory (DFT), the occupied orbital functions of a model state are derived by minimizing the ground-state energy functionals of Hohenberg and Kohn. It has been assumed for some time that effective potentials in the orbital KS equations are always equivalent to local potential functions. When tested by accurate model calculations, this locality assumption is found to fail for more than two electrons. Here this failure is explored in detail. The sources of the locality hypothesis in current DFT thinking are examined, and it is shown how the theory can be extended to an orbital functional theory (OFT) that removes the inconsistencies and paradoxes. 

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... 

Press, Oxford Dreizler RM, Gross EKU (1990) Density junctional theory. Springer, Berlin Heidelberg New York Kryachko ES, Ludena EV (1990) Energy density functional theory of many-electron systems. Kluwer, Dordrecht March NH (1992) Electron density theory of atoms and molecules. Academic, London Hohenberg P, Kohn (1964) Phys Rev 136 B864 Kohn W, Sham LJ (1965) Phys Rev 140 All33... 

Density functional theory (DFT) uses the electron density p(r) as the basic source of information of an atomic or molecular system instead of the many-electron wave function T [1-7]. The theory is based on the Hohenberg-Kohn theorems, which establish the one-to-one correspondence between the ground state electron density of the system and the external potential v(r) (for an isolated system, this is the potential due to the nuclei) [6]. The electron density uniquely determines the number of electrons N of the system [6]. These theorems also provide a variational principle, stating that the exact ground state electron density minimizes the exact energy functional F[p(r)]. 

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