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Physical review. B, Condensed matter

Density-functional theory for spin-density waves and antiferromagnetic systems

Klaus Capelle, L. Nunes de Oliveira

Published Jun 1, 2000• 28 citations
Publisher

Abstract

An extension of density-functional theory, designed to treat spin-density waves and antiferromagnetic systems, is presented. The nonlocal nature of the antiferromagnetic correlations and possible noncollinearity in spin space are incorporated via an additional fundamental variable, the staggered density, which supplements the spin densities of conventional density-functional theory. Inclusion of this variable is justified by both physical and methodological considerations. We prove the corresponding Hohenberg-Kohn theorem, derive the pertinent Kohn-Sham equations, and present several approximate functionals depending explicitly on the staggered density. As a first test the formalism is applied to two simple model systems, a one-dimensional electron gas with a short-range interaction, and the three-dimensional electron gas with Coulomb interactions. These calculations serve to test the developed formalism, but also already allow us to draw a number of conclusions regarding the stability and nature of possible spin-density wave states in homogeneous electron systems.

DOI
10.1103/physrevb.61.15228

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An extension of density-functional theory, designed to treat spin-density waves and antiferromagnetic systems, is presented. The nonlocal nature of the antiferromagnetic correlations and possible noncollinearity in spin space are incorporated via an additional fundamental variable, the staggered density, which supplements the spin densities of conventional density-functional theory. Inclusion of this variable is justified by both physical and methodological considerations. We prove the corresponding Hohenberg-Kohn theorem, derive the pertinent Kohn-Sham equations, and present several approximate functionals depending explicitly on the staggered density. As a first test the formalism is applied to two simple model systems, a one-dimensional electron gas with a short-range interaction, and the three-dimensional electron gas with Coulomb interactions. These calculations serve to test the developed formalism, but also already allow us to draw a number of conclusions regarding the stability and nature of possible spin-density wave states in homogeneous electron systems.
We propose a novel density-functional approach to antiferromagnetism and spin-density waves. Nonlocal antiferromagnetic correlations and possible noncollinear spin configurations are incorporated via a staggered density, which enters the formalism on the same footing as the particle density. We establish the pertinent generalization of the Hohenberg-Kohn theorem, derive the associated Kohn-Sham equations, outline the construction of approximations for the corresponding exchange and correlation functional, discuss the relation of the present method to other density-functional approaches, and report a first application to a simple model system.
A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.
We analyze the ground-state energy and correlation energy of the Heisenberg model as a function of spin, both in the ferromagnetic and the antiferromagnetic case, and in one, two and three dimensions. First, we present a comparative analysis of known expressions for the ground-state energy E0(S){E}_{0}(S)E0​(S) of homogeneous Heisenberg models. In the one-dimensional antiferromagnetic case we propose an improved expression for E0(S),{E}_{0}(S),E0​(S), which takes into account Bethe-ansatz data for S=1/2.S=1/2.S=1/2. Next, we consider inhomogeneous Heisenberg models (e.g., exposed to spatially varying external fields). We prove a Hohenberg-Kohn-like theorem stating that in this case the ground-state energy is a functional of the spin distribution and that this distribution encapsulates the entire physics of the system, regardless of the external fields. Building on this theorem, we then propose a local-density-type approximation that allows us to utilize the results obtained for homogeneous systems in inhomogeneous situations too. We conjecture a scaling law for the dependence of the correlation functional on dimensionality, which is well satisfied by existing numerical data. Finally, we investigate the importance of the spin-correlation energy by comparing results obtained with the proposed correlation functional to ones from an uncorrelated mean-field calculation, taking as our example a linear spin-density wave state.
The formalism developed in the first paper of the series [arXiv:0901.1060] is applied to two thermodynamic systems: (i) of three global observables (the energy, the total electron number and the spin number), (ii) of one global observable (the internal electron energy) and two local (position-dependent) observables (the total electron density and the spin density). The two-component potential of the many-electron system of interest is constructed of a scalar external potential and a collinear magnetic field (coupled only with the spin operator). Various equilibrium characteristics of two systems are defined and investigated. Conditions for the equivalence between two systems (the same equilibrium density matrix demanded) are derived and thoroughly discussed. The applicability of the Hohenberg-Kohn theorem is extended to the thermodynamic spin-density functional theory. Obtained results provide a rigorous mathematical foundation for future derivation of the zero-temperature limit of this theory and determination of its chemical reactivity indices.
Many-electron theory is treated here with much attention focused on the synthesis of the density functional and density matrix theory. This synthesis shows that the exchange–correlation potential, normally obtained as the functional derivative with respect to the electron density of an assumed approximate exchange–correlation energy, can be written exactly in terms of the “near-diagonal” behavior of the fully interacting first-order density matrix and the diagonal of the second-order density matrix, thereby bypassing the functional derivative. Also involved, since the exchange–correlation potential is an “artifact” of reducing the many-electron Schrödinger equation to one-body Slater–Kohn–Sham equations, are the one-body orbitals derived from these latter equations, not just the density thereby obtained. This situation has then motivated a discussion of what constraints are placed on the low-order density matrices of atoms and molecules, described by an assumed spin-independent Hamiltonian, by the requirement that these matrices are constructed from an N-electron wave function which is an eigenfunction of the spin operators S2 and Sz. Some briefer discussion is added on the optimally convergent determinantal expansion of such an N-electron wave function, the relation between Brueckner and Löwdin natural orbitals being commented on as a by-product. In this latter context, it is noted that the exchange-only potential of density functional theory can be represented as a sum of two line integrals, each of which is path-dependent, the sum being, of course, path-independent. But in addition to knowledge of Slater–Kohn–Sham orbitals appearing as in the Harbola–Sahni exchange-only potential (which is one of the path-dependent terms), there is a kinetic correction. This involves Löwdin natural orbitals and their occupation numbers, but only now in the weak coupling limit of the electron–electron interaction, as emphasized in the kinetic “correction” to the Harbola–Sahni formula exhibited by Levy and the writer. Some directions for further work are briefly set out. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 163–176, 1999
We use the exact strong-interaction limit of the Hohenberg-Kohn energy density functional to approximate the exchange-correlation energy of the restricted Kohn-Sham scheme. Our approximation corresponds to a highly nonlocal density functional whose functional derivative can be easily constructed, thus transforming exactly, in a physically transparent way, an important part of the electron-electron interaction into an effective local one-body potential. We test our approach on quasi-one-dimensional systems, showing that it captures essential features of strong correlation that restricted Kohn-Sham calculations using the currently available approximations cannot describe.