Evaluation of $\mathit{GW}$ Approximations for the Self-Energy of a Hubbard Cluster

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.