The self-energy beyond GW: Local and nonlocal vertex corrections

Abstract

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.

Benchmark ofGWmethods for azabenzenes

“…84 Correcting these issues may require going beyond the GW approximation by introducing vertex corrections. Currently, such corrections are in the initial stage of exploration [85][86][87][88][89][90][91] and their implementation would come at the price of an even higher computational cost than scGW.…”

Analytic evaluation of the electronic self-energy in theGWapproximation for two electrons on a sphere

“…Even if a purely analytic treatment is then no longer possible, the computational cost will be much smaller than for real materials, allowing a highly accurate evaluation without the apparent artificialities and the parameter dependence of typical lattice models, which have repeatedly been chosen to study the influence of vertex corrections and self-consistency in the past. 10,11,43,46,56,57 Here this system was used to study the convergence of the self-energy with respect to the number of empty states included in the spectral summations. The results not only confirm previous empirical observations that transition energies converge faster than individual quasiparticle states due to a partial error cancellation, but the asymptotic expansion also demonstrates that the gap between the highest occupied and the lowest unoccupied state approaches its limiting value with an error that is, for practical purposes, proportional to the cutoff energy to the power −3/2.…”

“…Furthermore, the restricted Hilbert space does not allow us to address problems like the convergence behavior with respect to the number of empty states. 32 Peculiar symmetries, such as that between occupied and unoccupied states in the two-site model at half filling, 43 which are not obeyed by real solids, may also have an influence on the results. For completeness, it should be mentioned that the polaron model of individual electrons coupled to an external boson field can also be treated analytically, 48 but its usefulness as a test system for the GW approximation is even more limited, as there is no explicit renormalizable electron-electron interaction.…”

“…A related but even simpler two-site model with a pair of electrons can be treated analytically in the same way. 43 Lattice models with a wider range of parameters, for which the GW self-energy is accurately obtainable by numerical means, were also employed in several studies. 10,[44][45][46][47] The properties of Hubbard models deviate in many respects from those of real materials, however, and conclusions from such comparisons cannot always be directly transferred to the ab initio realm.…”

Two-site Hubbard molecule with a spinless electron-positron pair

“…Recently, transport properties 13 as well as the performance of the GW approximation have been studied in this context. 14 Among the lattice models, the Hubbard model is popular due to the simple form of its Hamiltonian. Though a general solution is not known, there are cases where the problem can be tackled analytically, for example, for small molecules.…”