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Physical Review Letters

Evaluation of GW Approximations for the Self-Energy of a Hubbard Cluster

Claudio Verdozzi, Rex W. Godby, S. Holloway

Published Mar 20, 1995• 51 citations
Publisher

Abstract

We evaluate several approximations for the self-energy operator and dielectric function of systems of interacting electrons using a two-dimensional Hubbard cluster for which the self-energy, dielectric function, and one-particle Green's function may be calculated exactly. The results show the $\mathrm{GW}$ approximation (in the form in which it is commonly used in ab initio calculations for real materials) to be relatively successful in establishing the main features of the spectrum, even when the electron-electron interaction is not weak. It is also clear that improving the $G\mathrm{and}W$ used in this approximation without including vertex corrections in the self-energy does not lead to major improvements.

DOI
10.1103/physrevlett.74.2327

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We evaluate several approximations for the self-energy operator and dielectric function of systems of interacting electrons using a two-dimensional Hubbard cluster for which the self-energy, dielectric function, and one-particle Green's function may be calculated exactly. The results show the GW\mathrm{GW}GW approximation (in the form in which it is commonly used in ab initio calculations for real materials) to be relatively successful in establishing the main features of the spectrum, even when the electron-electron interaction is not weak. It is also clear that improving the GandWG\mathrm{and}WGandW used in this approximation without including vertex corrections in the self-energy does not lead to major improvements.
An important method in many-particle physics is related to the Hubbard model which can be used to investigate the behavior of fermions, particularly to study exchange and correlation of electron- electron interaction. For a special cluster, the molecule consisting of two atoms, the electrons interact via one hopping integral, and for a small number of N = 1; 2; 3 electrons, the model has the advantage that it can be solved exactly. Therefore, some important physical quantities like the Green's function, the spectral function and the self-energy are calculated and discussed. Exchange and correlation are fully treated. The self-energies obtained by two approximate methods, namely the Hartree-Fock and the GW approximation, are compared with the exact self-energy. The comparison allows a discussion of the validity of the approximations versus the ratio of Hubbard parameter and hopping integral.
Many observables such as the density, total energy, or electric current, can be expressed explicitly in terms of the one‐body Green's function, which describes electron addition or removal to or from a system. An efficient way to determine such a Green's function is to introduce a self‐energy, which is a nonlocal and dynamic effective potential that influences the propagation of particles in an interacting system. The state‐of‐the art approximation for the self‐energy is the GW approximation, where the system to (or from) which the electron is added (or removed) is described as a polarizable, screening, medium. This is expressed by the name of the approximation: ‘GW’ stands for the one‐body Green's function G and for W , the dynamically screened Coulomb interaction. The GW approximation is very popular for the calculation of band structures in solids, and increasingly used also to describe nanostructures, clusters, and molecules. As compared to static mean‐field approximations for the effective potential, the dynamical screening of the Coulomb interaction in GW leads to a renormalization of energies, to broadening and/or to the observation of additional excitations. An analysis of the approximations that lead to the GW self‐energy, and of the underlying picture, explains the successes and the limitations of the approach. This article is categorized under: Electronic Structure Theory > Density Functional Theory Electronic Structure Theory > Ab Initio Electronic Structure Methods Theoretical and Physical Chemistry > Spectroscopy Structure and Mechanism > Computational Materials Science
Calculations of ground-state and excited-state properties of materials have been one of the major goals of condensed matter physics. Ground-state properties of solids have been extensively investigated for several decades within the standard density functional theory. Excited-state properties, on the other hand, were relatively unexplored in ab initio calculations until a decade ago. The most suitable approach up to now for studying excited-state properties of extended systems is the Green function method. To calculate the Green function one requires the self-energy operator which is non-local and energy dependent. In this article we describe the GW approximation which has turned out to be a fruitful approximation to the self-energy. The Green function theory, numerical methods for carrying out the self-energy calculations, simplified schemes, and applications to various systems are described. Self-consistency issue and new developments beyond the GW approximation are also discussed as well as the success and shortcomings of the GW approximation.
In this thesis, the GW approximation (GWA, Green's function G times screened interaction W) and the Hubbard I approximation (HIA) are combined in a non-self-consistent one-shot calculation to determine the electronic structure of a one-dimensional strongly-correlated model. The scheme was chosen to incorporate both screening effects through the GWA and strong on-site correlations through the atomic HIA. The resulting self-energies of both methods are summed together and another term is subtracted to correct double-counting. The examined double-counting terms were the local self-energy of the GWA result, the GWA self-energy of the atomic model used in the HIA and lastly the impurity self-energy within the GWA. The self-energy of the atomic model in the HIA was solved analytically. It was found that all three approaches yield non-causal features, which increase with the on-site Coulomb repulsion, in the resulting spectral function. The local GWA self-energy was observed to perform best in terms of causality and computational effort. Changes in the resulting quasi-particle structure in the spectral function showed that screening effects and local correlations were included successfully.
It is commonly accepted that the GW approximation for the electron self-energy is successful for the description of the band structure of weakly to moderately correlated systems, whereas it will fail for strongly correlated materials. In the present work, we discuss two important aspects of this approximation: first, the "self-screening error," which is due to an incorrect treatment of induced exchange, and second, the atomic limit, in which, instead, correlation is directly responsible for the observed problem. Using the example of the removal of a particle from a box, we show that the self-screening error stems from the use of test charge-test charge screening and that it can be corrected by a two-point vertex contribution to the self-energy derived from time-dependent density functional theory (TDDFT). We explain why the addition of a particle, instead, requires the use of a different approximate vertex. This illustrates why the general vertex function, valid both for valence and conduction states, must be a three-point function. Moreover, we show that also the bad performance of GW in the atomic limit is due to the neglect of the vertex in the self-energy; in that case, the TDDFT-derived vertex correction is not sufficient in order to remove the error even qualitatively. We discuss the effects of the self-screening error as well as the atomic limit using GW for the exactly solvable two-site Hubbard model.
We have studied electron correlations in the doped two-dimensional (2D) Hubbard model by using the coupled-cluster method (CCM) to investigate whether or not the method can be applied to correct the independent particle approximations actually used in ab initio band calculations. The double excitation version of the CCM, implemented using the approximate coupled pair (ACP) method, accounts for most of the correlation energies of the 2D Hubbard model in the weak $(U/t\ensuremath{\simeq}1)$ and the intermediate U/tU/tU/t regions $(U/t\ensuremath{\simeq}4).$ The error in this case is always less than 11%.1 The ACP approximation gets less accurate for large U/tU/tU/t $(U/t\ensuremath{\simeq}8)$ and near half-filling. Further incorporation of electron correlation effects is necessary in this region. The accuracy depends neither on the system size, nor the gap between the lowest unoccupied level and the highest occupied level in the finite size effect. Hence, the CCM may be favorably applied to ab initio band calculations on metals as well as semiconductors and insulators.