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International Journal of Modern Physics B

ON THE N-REPRESENTABILITY AND UNIVERSALITY OF F[ρ] IN THE HOHENBERG-KOHN-SHAM VERSION OF DENSITY FUNCTIONAL THEORY

Eduardo V. Ludeña, Francesc Illas, Alejandro Ramírez-Solís

Published Oct 20, 2008• 12 citations
Publisher

Abstract

We discuss two basic problems in the Hohenberg-Kohn-Sham version of density functional theory, HKS-DFT: the first, the N-representability of the functional F[ρ] and, the second, the universality of F[ρ]. In relation to the first, we show that F[ρ] must satisfy N-representability conditions that follow from those on the 2-matrix D2(r1, r2; r′, r′2). In the case of the second, we provide arguments based on the equivalence between ab initio DFT and HKS-DFT to show that the functional F[ρ] is not universal.

DOI
10.1142/s0217979208050395

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We discuss two basic problems in the Hohenberg-Kohn-Sham version of density functional theory, HKS-DFT: the first, the N-representability of the functional F[ρ] and, the second, the universality of F[ρ]. In relation to the first, we show that F[ρ] must satisfy N-representability conditions that follow from those on the 2-matrix D2(r1, r2; r′, r′2). In the case of the second, we provide arguments based on the equivalence between ab initio DFT and HKS-DFT to show that the functional F[ρ] is not universal.
Abstract Density functional theory (DFT) is usually formulated in terms of the electron density as a function of position n(r). Here we discuss an alternative formulation of DFT in terms of the orbital occupation numbers {n α } associated with a local-orbital orthonormal basis set {ϕ α }. First, we discuss how the building blocks of DFT, namely the Hohenberg–Kohn theorems, the Levy–Lieb approach and the Kohn–Sham method, can be adapted for a description in terms of {n α }. In particular, the total energy is now a function of {n α }, E[{n α }], and a Kohn–Sham-like Hamiltonian is derived introducing the effects of the electron–electron interactions via effective potentials, { V α eff = ∂ E e e [ { n β } ] / ∂ n α } . In a second step we consider the Hartree and exchange energies and discuss how to describe them, in the spirit of a DFT approach, in terms of the orbital occupation numbers. In this contribution special attention is paid to the description of the (intra-atomic) correlation energy and corresponding correlation potentials {V corr,α }. For this purpose, a model system is analyzed in detail, whereby an atomic Hamiltonian interacts with the environment via a simplified model; the use of this model allows us to obtain the correlation energy and potentials (in terms of {n α }) for different cases corresponding to low, intermediate and high electron correlations.
The N-representability problem for noncollinear spin-polarized densities was left open in the pioneering work of von Barth and Hedin [J. Phys. C 5, 1629 (1972)] setting up the Kohn-Sham density-functional theory for magnetic compounds. In this Letter, we demonstrate that, contrarily to the nonpolarized case, the sets of pure and mixed state N-representable densities are different in general. We provide a simple characterization of the latter by means of easily checkable necessary and sufficient conditions on the components ρ(αβ)(r) of the spin-polarized density.
All downfolding procedures have to start from an approximate treatment of the full Hilbert space. For strongly correlated solids, density functional theory (DFT) within the Kohn–Sham scheme provides a suitable starting point. In this chapter, we describe density functional theory and discuss the different merits and problems of Kohn–Sham DFT. Since it is not the main topic of the book, we will focus on the conceptual ideas and formulation and on how DFT is used in practice as a starting point for many-body treatments. DFT is such a well developed theory that there exist a wealth of literature on the topic. There are a number of lucid reviews such as that by Jones and Gunnarsson [1] and more recent reviews by Jones [2] and Becke [3]; the last two contain a fascinating historical account of DFT. A collection of articles with contributions from pioneers in the field was given by Dreizler and da Providencia [4] and many others, which are too numerous to be listed here. A Nobel Lecture [5] is another source of inspiration.
Abstract Reducing the many-fermion problem to a set of single-particle (s.p.) equations, the Kohn–Sham (KS) theory has provided a practical tool to implement ab initio calculations of ground-state energies and densities in many-electron systems. There have been attempts to extend the KS theory so that it could describe other physical quantities, or it could be applied to other many-fermion systems. By generalizing and reformulating the KS theory in terms of the 1-body density matrix, we expose the minimal composition of the theory that enables the reduction of the many-fermion problem to the s.p. equations. Based on the reformulation, several basic issues are reconsidered. The v- and N-representabilities for the KS theory are distinguished from those for the Hohenberg-Kohn theorem. Criteria for the extendability of the KS theory are addressed.