The GW approximation to the self-energy is a surprisingly simple formula, with a clear physical content as discussed in Ch. 11, and the potential for efficient calculations of electronic properties beyond independent-particle methods as described in Ch. 12. It is now time to look at results for realistic systems and ask: which properties can be described successfully by the GWA, and for which materials? And what can we learn from the results?

The selection presented in this chapter covers only a fraction of what could be cited. The GWA has become a very popular method – for valuable reasons, as the chapter shows: having a good approximation to the self-energy allows one to calculate accurate band structures, electron addition and removal spectra, densities, density matrices, total energies, and many related properties. GWA calculations lead to qualitative and often quantitative agreement with experimental findings. There exist many more applications, but here we have selected examples meant to illustrate content, strong and weak points of the GWA, in its various flavors such as perturbative or non-perturbative calculations. The examples are chosen to stress certain aspects, and they are not necessarily the first or the most recent calculations. A time span of several decades is covered. As a consequence, the computational scheme that has been used, for example the level of self-consistency from G0W0 to fully self-consistent GW, is not always the same. However, the quality of the results is always sufficient to support the message that the example is supposed to illustrate.

One can find, for instance, a huge number of GW bandgap calculations in the literature: for our purpose, it is sufficient to give just a few representative examples. Far fewer results exist for spectral functions and satellites, although the dynamically screened interaction W contains electron–hole and plasmon excitations.