Green functions and self-consistency: insights from the spherium model

TL;DR

This study evaluates various Green function methods on the spherium model, revealing insights into their performance across correlation regimes and highlighting issues like self-screening and artificial discontinuities.

Contribution

It provides a comprehensive comparison of GW, self-consistent GW, and GF2 methods on the spherium model, illustrating their strengths and limitations in different correlation regimes.

Findings

Self-consistency often worsens results compared to perturbative approaches.

Self-screening can be mitigated with second-order screened exchange (SOSEX).

Partial self-consistency can cause artificial discontinuities in the weakly correlated regime.

Abstract

We report an exhaustive study of the performance of different variants of Green function methods for the spherium model in which two electrons are confined to the surface of a sphere and interact via a genuine long-range Coulomb operator. We show that the spherium model provides a unique paradigm to study electronic correlation effects from the weakly correlated regime to the strongly correlated regime, since the mathematics are simple while the physics is rich. We compare perturbative GW, partially self-consistent GW and second-order Green function (GF2) methods for the computation of ionization potentials, electron affinities, energy gaps, correlation energies as well as singlet and triplet neutral excitations by solving the Bethe-Salpeter equation (BSE). We discuss the problem of self-screening in GW and show that it can be partially solved with a second-order screened exchange correction (SOSEX). We find that, in general, self-consistency deteriorates the results with respect to those obtained within perturbative approaches with a Hartree-Fock starting point. Finally, we unveil an important problem of partial self-consistency in GW: in the weakly correlated regime, it can produce artificial discontinuities in the self-energy caused by satellite resonances with large weights.