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Journal of Chemical Theory and Computation

Investigation of Self-Interaction Corrections for an Exactly Solvable Model System: Orbital Dependence and Electron Localization

Daniel Vieira, Klaus Capelle

Published Oct 6, 2010• 18 citations
Publisher

Abstract

A systematic investigation of two approximate self-interaction corrections (SICs), Perdew-Zunger SIC and Lundin-Eriksson SIC, and the local-density approximation (LDA) is performed for a model Hamiltonian whose exact many-body solution and exact LDA are known. Both SICs as well as LDA are applied in the calculation of ground-state energies, ground-state densities, energy gaps, and impurity densities of one-dimensional Hubbard chains differing in size, particle number, and interaction strength. The orbital-dependent potentials arising from either SIC are treated within the optimized-effective potential method, which we reformulate for the Hubbard model. The delocalization tendency of LDA is confronted with the localization tendency of SIC. A statistical analysis of the resulting data set sheds light on the role of SIC for weakly and strongly interacting particles and allows one to assess the performance of each methodology.

DOI
10.1021/ct100352r

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Physical review. B, Condensed matter•Nov 15, 1998•3 citations
A systematic investigation of two approximate self-interaction corrections (SICs), Perdew-Zunger SIC and Lundin-Eriksson SIC, and the local-density approximation (LDA) is performed for a model Hamiltonian whose exact many-body solution and exact LDA are known. Both SICs as well as LDA are applied in the calculation of ground-state energies, ground-state densities, energy gaps, and impurity densities of one-dimensional Hubbard chains differing in size, particle number, and interaction strength. The orbital-dependent potentials arising from either SIC are treated within the optimized-effective potential method, which we reformulate for the Hubbard model. The delocalization tendency of LDA is confronted with the localization tendency of SIC. A statistical analysis of the resulting data set sheds light on the role of SIC for weakly and strongly interacting particles and allows one to assess the performance of each methodology.
A systematic comparison of three approximate self-interaction corrections (SICs), Perdew-Zunger SIC, Lundin-Eriksson SIC and extended Fermi-Amaldi SIC, is performed for a model Hamiltonian whose exact many-body solution and exact local-density approximation (LDA) are known. For each of the three proposals we compare its implementation only for the potential, only for the energy, i.e., a post-LDA evaluation of the SIC energy), to none of them, i.e., a standard LDA calculation) and to both. Each of the resulting 10 permutations of methodologies is applied to 420 Hubbard chains differing in size, particle number and interaction strength. A statistical analysis of the resulting data set reveals trends and permits to assess the performance of each methodology. Overall, but not in each individual case, a post-LDA application of Perdew-Zunger SIC emerges as the recommended methodology.
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.
We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.
The density-functional theory for the half-filled extended Hubbard model in one and two dimensions has been studied within the self-interaction-corrected (SIC) local-spin-density (LSD) approximation. The SIC to the total energy is expressed in terms of the Wannier functions which have the most optimal degree of localization. The system at ground state can have spin-density wave (SDW) or charge-density wave (CDW) ordering depending on the relative strength of the on-site repulsion and the nearest-neighbor repulsion. Like the Hartree-Fock approximation, the SIC-LSD scheme predicts a first-order transition between CDW and SDW. However, in the SDW, the nearest-neighbor repulsion V near the transition point tends to increase the band gap for small U and decrease the band gap for large U, with a crossover around $U\ensuremath{\approx}4$ in one dimension, which corresponds to the tricritical point predicted by other theories. The ground-state energies are better described in SIC-LSD than in the Hartree-Fock approximation.