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Physical review. B, Condensed matter

Canonical orbitals of the self-interaction-corrected local-spin-density approximation in doped systems

Shöji Kobayashi, M. Arai, Takeo Fujiwara

Published Nov 15, 1995• 1 citation
Publisher

Abstract

The property and role of canonical orbitals in the self-interaction-corrected local-spin-density approximation are investigated in hole- and electron-doped systems. One-electron properties and states in doped systems can be described by the canonical orbitals. It is also shown that the many-electron wave functions of (N\ifmmode\pm\else\textpm\fi{}1)-electron systems can be nicely approximated by a single Slater determinant constructed of the canonical orbitals of the N-electron system. The notion of the canonical orbitals is significant in the one-electron picture and closely related to many-electron wave functions.

DOI
10.1103/physrevb.52.13718

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The property and role of canonical orbitals in the self-interaction-corrected local-spin-density approximation are investigated in hole- and electron-doped systems. One-electron properties and states in doped systems can be described by the canonical orbitals. It is also shown that the many-electron wave functions of (N\ifmmode\pm\else\textpm\fi{}1)-electron systems can be nicely approximated by a single Slater determinant constructed of the canonical orbitals of the N-electron system. The notion of the canonical orbitals is significant in the one-electron picture and closely related to many-electron wave functions.
Using two different interpretations of the concept of the Fermi hole curvature, the author derives a class of correlation energy expressions for the inhomogeneous interacting electron gas. Their properties include freedom from spurious orbital self-interaction, invariance under unitary transformations among occupied orbitals, correct values in the homogeneous limit, correctly normalized correlation hole, inclusion of kinetic energy (KE) as well as potential energy of correlation, and non-vanishing values for fully spin-polarized systems (in contrast with some similar schemes developed for chemical applications). Minimization of the energy with respect to the orbitals leads to a Euler-Lagrange equation resembling the Hartree-Fock one-electron effective Schrodinger equation, with the addition of a term resembling the KE operator for an inhomogeneous effective mass. For current-carrying states there is a further term involving an effective dynamically induced vector potential. Despite these complications the effective one-electron Hamiltonian is Hermitian, so that the canonical orbitals are orthogonal, in contrast with those of the commonest self-interaction correction scheme.
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
A self-consistent scheme is presented to predict excitations energies of highly correlated many-electron systems within the self-interaction-corrected local-spin-density approximation. It is based on the minimization of the self-interaction-free total energy. This method yields a global picture of the excitation energies in the form of an effective wave-vector-resolved band structure for occupied and empty eigenstates which are automatically orthogonal to each other. The theory has been tested for the Hubbard-Peierls model in one and two dimensions. The electron removal energies and fundamental energy gaps are found to be substantially more accurate than in the standard local-density theory.
Svane and Gunnarsson's method of the self-interaction corrected local-spin-density approximation (SIC-LSDA) in the density-functional theory was applied to the one-dimensional Hubbard model and the two-dimensional pd Hubbard model in order to study the one particle energy spectrum. The calculated energy band in the SIC-LSDA gives a reasonable band gap in contrast to serious underestimation of it by the LSDA and slight overstimation by the Hartree-Fock approximation. Behavior of SIC localized orbitals is also discussed
Svane and Gunnarsson's method of the self-interaction corrected local-spin-density approximation (SIC-LSDA) in the density-functional theory was applied to the one-dimensional Hubbard model and the two-dimensional pd Hubbard model in order to study the one particle energy spectrum. The calculated energy band in the SIC-LSDA gives a reasonable band gap in contrast to serious underestimation of it by the LSDA and slight overestimation by the Hartree-Fock approximation. Behavior of SIC localized orbitals is also discussed.
We show that the self-interaction-corrected (SIC) local spin-density (LSD) approximation can describe localization in a system where the Coulomb interaction is large compared with the band-width. In contrast with the normal LSD approximation, the SIC approximation can be used for a band with orbital degeneracy, even if the band is not half filled. The method is applied to the one-dimensional Hubbard model for which there is an exact solution. The total energy, the band gap, the local moment, and the momentum distribution are substantially better described than in the LSD approximation. For some of these properties the SIC approach is shown to compare favorably with the Gutzwiller ansatz.