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arXiv:1307.0139 · Physical Review Letters

N -Representability in Noncollinear Spin-Polarized Density-Functional Theory

David Gontier

Published Oct 7, 2013• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.111.153001

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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.
Pure-state NNN-representability in current-spin-density functional theory
This paper is concerned with the pure-state NNN-representability problem for systems under a magnetic field. Necessary and sufficient conditions are given for a spin-density 2×22 \times 22×2 matrix RRR to be representable by a Slater determinant. We also provide sufficient conditions on the paramagnetic current j\mathbb{j}j for the pair (R,j)(R, \mathbb{j})(R,j) to be Slater-representable in the case where the number of electrons NNN is greater than 12. The case N<12N < 12N<12 is left open.
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.
A particle-density distribution is said to be v-representable if it is the ground-state density for some potential v(r). By developing the idea of proper order, proposed earlier [J. Chen and M. J. Stott, Phys. Rev. A 44, 2816 (1991)], to any number of energy levels which are involved in possible ground-state configurations, we show that any reasonable density with an arbitrary number of spinless fermions is either pure-state or ensemble v-representable. This result supports the general applicability of the Kohn-Sham method [Phys. Rev. 140, A1133 (1965)] in density-functional theory.
Aiming at a unified treatment of correlation and inhomogeneity effects in superconductors, Oliveira, Gross, and Kohn proposed in 1988 a density functional theory for the superconducting state. This theory relies on the existence of a Kohn- Sham scheme, i.e., an auxiliary noninteracting system with the same electron and anomalous densities of the original superconducting system [Phys. Rev. Lett. 60, 2430 (1988)]. However, the question of noninteracting vvv-representability has never been properly addressed and the existence of the Kohn-Sham system has always been assumed without proof. Here, we show that indeed such a noninteracting system does not exist at zero temperature. In spite of this result, we also show that the theory is still able to yield good results, although in the limit of weakly correlated systems only.
The formulation of the density matrix functional theory (DMFT) in which the correlation component U of the electron–electron repulsion energy is expressed in terms of a model two-electron density cumulant matrix (a.k.a. the 2-cumulant) and the tensor of two-electron integrals g is critically analyzed. The dependence of on both the vector of occupation numbers n and g derived from the corresponding vector (1) of natural spinorbitals, and its concomitant invariance to the nature of the spin-independent interparticle interaction potential that enters the definition of g are emphasized. In the case of spin-polarized systems, (and thus U ) is found to be a function of not only n and g, but also of the vector and the matrix, where is the spin-flip operator. The presence of spin polarization imposes additional constraints upon, including a simple condition that, when satisfied, assures the underlying wavefunction being an eigenstate of both the and operators with the eigenvalues and, respectively. This feature allows targeting electronic states with definite multiplicities, which is virtually impossible in the case of the Kohn–Sham implementation of density functional theory. Among the four possible pairing schemes for the natural spinorbitals that give rise to approximations employing with only two independent indices, three are found to result in unphysical constraints even for spin-unpolarized systems, whereas the failure of the fourth one turns out to be precipitated by the presence of spin polarization. Consequently, any implementation of DMFT based upon “two-index” is shown to be generally unsuitable for spin-polarized systems (and incapable of yielding the spin-parallel components of U for the spin-unpolarized ones). A clear distinction is made between the genuine 1-matrix functionals that are defined for arbitrary N -representable 1-matrices and general energy expressions that depend on auxiliary quantities playing the role of fictitious 1-matrices subject to additional (often unphysical) constraints.
The problem of pure-state N-representability of the two-particle spin-dependent density function ρ(x1, x2) is considered for an N-electron system, and a procedure for finding an N-representable ρ(x1, x2) is advanced. The problem is formulated in the framework of a family of N × N matrices formed from integrals of auxiliary two-particle functions θn(x1, x2) converging at n → ∞ to ρ(x1, x2)/[N(N−1)]. The simple requirement of positive definiteness of these matrices is shown to play a decisive role in finding an N-representable ρ(x1, x2). The results obtained may open new possibilities for using ρ(x1, x2) in the density-functional theory. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 127–142, 1997