Density functional theory Kohn-Sham construction

It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

Evidently, the LSD and GGA approximations are working, but not in the way the standard spin-density functional theory would lead us to expect. In Ref [36], a nearly-exact alternative theory, to which LSD and GGA are also approximations, is constructed, which yields an alternative physical interpretation in the absence of a strong external magnetic field. In this theory, Hf(r) and rti(r) are not the physical spin densities, but are only intermediate objects (like the Kohn-Sham orbitals or Fermi surface) used to construct two physical predictions the total electron density n(r) from... 

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]. 

The third and final approach to the electron correlation problem included briefly here is density functional theory (DFT), a review of which has been given by Kohn in his Nobel lecture [38], The Hohcnberg Kolin theorem [39] states that there is a one-to-one mapping between the potential V(r) in which the electrons in a molecule move, the associated electron density p(r), and the ground state wave function lP0. A consequence of this is that given the density p(r), the potential and wave function lf 0 are functionals of that density. An additional theorem provided by Kohn and Sham [40] states that it is possible to construct an auxiliary reference system of non-interacting... 

In this work we have given on overview of the mathematical foundations of stationary density functional theory. We discussed in great detail the question of differentiability of the functionals and showed that the Kohn-Sham theory can be put on a solid basis for all practical purposes, since the set of noninteracting E-V-densities is dense in the set of interacting E-V-densities. The question whether these two sets are in fact identical is still an open question. We further discussed two systematic approaches for the construction of the exchange-correlation functional and potential. What can we say about future developments within density functional theory There have been many extensions of density functional theory involving... 

The question of the locality of density-functional potentials of Kohn-Sham type, a central issue of the foundations of DFT, has been controversial for some time. Robert Nesbet has argued in several articles in the literature, in opposition to most of the DFT community, that the locality of DFT potentials has never been rigorously proven and he claims by means of a counter example that a local potential cannot exist for a system with more than two electrons. His conclusion is that a consistent density functional theory does not exist and that the only rigorous way to proceed is by constructing an orbital functional theory (OFT). This result has been challenged in the scientific literature by several authors criticisms that have been vigorously refuted by Nesbet. In the present volume four chapters appear on the subject. 

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