GW calculations have become part of the standard toolbox in computational condensedmatter physics. Many details on foundations and putting into practice can be found in overviews and reviews, like [287, 334, 347, 408].

What does it mean to do a GWA calculation in practice? The formula for the GWA self-energy is as simple as its name, but GW calculations have a long history with continuous improvements. Modern GWA calculations are in the continuation of pioneering attempts to include correlations beyond Hartree–Fock using the concept of screening. Already in 1958 [409] correlation energies for the homogeneous electron gas were obtained from the study of the polarization of the gas due to an individual electron, and from the action of this polarization back on the electron, through the self-energy. These calculations, including several approximations, were limited to states close to the Fermi level. A GWA-like approach [410, 411] was applied to the electron gas in 1959, although these calculations didn't cover the range of densities rs ~ 2 - 5 which is typical for simple metals. Hedin's work [43] is fundamental in that it presented the GWA as the first term of a series in terms of the screened Coulomb interaction, and it contained an extensive description of the homogeneous electron gas on the GWA level. Many more studies on the HEG followed, including detailed investigation of the spectral functions [383, 384, 412], the importance of self-consistency [351, 352, 392], the electron gas in lower dimensions [369] (see also Ch. 11), and vertex corrections beyond the GWA (e.g., [413–415]), as discussed in Ch. 15.