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

TL;DR

This paper analytically derives the GW self-energy for two electrons on a sphere, enabling precise analysis of energy gap convergence and error extrapolation in a simplified model.

Contribution

It provides the first complete analytical derivation of the GW self-energy for a simple two-electron spherical system, facilitating convergence analysis.

Findings

The GW self-energy can be derived analytically for this model.

The energy gap convergence follows a cutoff energy to the power -3/2.

An asymptotic formula for truncation error is established.

Abstract

The GW approximation for the electronic self-energy is an important tool for the quantitative prediction of excited states in solids, but its mathematical exploration is hampered by the fact that it must, in general, be evaluated numerically even for very simple systems. In this paper I describe a nontrivial model consisting of two electrons on the surface of a sphere, interacting with the normal long-range Coulomb potential, and show that the GW self-energy, in the absence of self-consistency, can in fact be derived completely analytically in this case. The resulting expression is subsequently used to analyze the convergence of the energy gap between the highest occupied and the lowest unoccupied quasiparticle orbital with respect to the total number of states included in the spectral summations. The asymptotic formula for the truncation error obtained in this way, whose dominant contribution is proportional to the cutoff energy to the power -3/2, may be adapted to extrapolate energy gaps in other systems.