Recent years have witnessed a tremendous improvement in the performance of supercomputing facilities as well as major advances in electronic structure techniques. Accordingly the scope of first-principles atomistic calculations has rapidly expanded in a discipline that could be dubbed as first-principles materials design. While ground-state properties of materials, such as structural configurations and energy barriers, and response properties which do not involve electronic excitations, such as lattice vibrations and response to static electric and magnetic fields, can now be predicted with very high accuracy even for large and complex systems, the progress on the study of electronic and optical excitations has been comparatively slower, arguably due to the complexity of such calculations, which typically scale less favorably as a function of system size than their ground-state counterparts.

Among the various methods to study electronic excitations the GW approximation to Hedins quasi-particle equations has had a number of successes and witnessed significant growth of interest within the computational electronic structure community. Besides its well-established predictive capability in the calculation of quasi particle band gaps, the GW method is especially appealing since (i) it performs consistently well across widely different materials, (ii) it does not require the use of adjustable parameters, and (iii) the underlying formalism is very general and non material-specific. These features make the GW method suitable for a number of applications, ranging from solids to molecules, liquids, surfaces, interfaces, and defects. The GW method is also of widespread use as a starting point for Bethe-Salpeter calculations of twoparticle neutral excitations. Current implementations find many diverse applications, also for correlated electrons systems, including among others the calculation of direct and inverse photoemission spectra, alignment of energy levels, quantum transport in nanoscale junctions, core-level spectroscopy and pump-probe spectroscopy. Despite its numerous successes the applicability of the GW method remains limited to systems of moderate complexity (less than 100 atoms), mostly due to the unfavorable scaling of GW calculations with the fourth power of the system size, and to the presence in the formalism of sums over a high number of unperturbed one-particle states. In addition, it is becoming clear that the use of the GW method at the perturbation theory level on top of standard density-functional calculations might not be sufficient to capture the physics of systems where electron correlations play an important role.

The majority of current GW implementations obtain the screened Coulomb interaction Wand the non-interacting Greens function G using a perturbative expansion over the Kohn-Sham eigenstates [1]. Several groups addressed the issue of eliminating the summation over the unoccupied states [2, 3, 4], or reducing the number of unoccupied states [5, 6]. Here it would be interesting to compare the various proposed methods and in particular to discuss the respective advantages from the computational viewpoint. While the GW method has traditionally been implemented in a basis of plane waves [1], some recent investigations have shown that it could be possible to improve the scaling of the calculation by exploiting localization in real space [7, 8]. In this case it would be interesting to understand how these new approaches compare to standard plane-waves implementations and what is the typical system size for which localized functions implementations become favorable over plane-waves. Similarly, implementations of the GW method within all-electron schemes and the projector-augmented wave method (PAW) have been proposed [9, 10]. Hence it would be interesting to see how these compare with more traditional pseudopotential approaches.

The use of GW at the perturbation theory level has been recently questioned, as in certain classes of materials including transition metal elements the energetics of localized electronic states cannot be corrected within perturbation theory. Within this framework several attempts have been made, in order either to go beyond perturbation theory [11, 12, 13], or to use better initial guesses for the electron wave functions and energies [14], or even to go beyond the GW approximation [15]. In this case it would be interesting to clarify the range of applicability of each one of these techniques, how they compare, and what are the computational overheads involved. The GW method is currently being used to investigate heterogeneous systems such as interfaces, surfaces, and nanoscale systems [16, 17, 18, 19, 20]. In this area it should be clarified what is the extra complexity associated with the inhomogeneity or reduced dimensionality and how to address these aspects in a systematic way. Although most GW calculations have been performed for bulk systems, they can also be used to evaluate photoelectron spectra and the quasi-particle excitation energies in molecular systems. In these cases it should be clarified how well the GW method performs and how the results depend on the approximations used.

Finally it would be interesting to discuss novel applications of the GW method to the calculation of total energies [21], electron-phonon interactions [22], and systems exhibiting magnetic ordering [23], as well as the use of the GW method for core-level spectroscopy [24, 25].