The article by Aryasetiawan and Gunnarsson provides an overview of the Green function method, particularly focusing on the GW approximation, which is a widely used approach for calculating excited-state properties of materials. The authors begin by discussing the limitations of standard density functional theory (DFT) in accurately predicting excited-state properties, especially in systems with strong electron correlations. They introduce the Green function theory, which is more suitable for studying excited-state properties due to its ability to account for dynamic screening and correlation effects. The GW approximation is then described in detail, including its derivation from many-body perturbation theory and its physical interpretation. The self-energy in the GW approximation is non-local and energy-dependent, addressing the major deficiency of the Hartree-Fock approximation (HFA) in capturing screening effects. The authors also discuss the numerical methods used to calculate the self-energy, such as the plane-wave basis and localized basis, and the plasmon-pole approximation. The article further explores simplified versions of the GW approximation, such as the static Coulomb-hole and screened-exchange (COHSEX) approximation, and their applications to various materials, including alkali metals, semiconductors, transition metals, and surfaces. The authors also address the self-consistency issue and new developments beyond the GW approximation, highlighting both its successes and limitations. Overall, the GW approximation has become a standard method for including correlation effects beyond the HFA, providing accurate predictions of quasiparticle energies and excitation spectra in a wide range of materials.The article by Aryasetiawan and Gunnarsson provides an overview of the Green function method, particularly focusing on the GW approximation, which is a widely used approach for calculating excited-state properties of materials. The authors begin by discussing the limitations of standard density functional theory (DFT) in accurately predicting excited-state properties, especially in systems with strong electron correlations. They introduce the Green function theory, which is more suitable for studying excited-state properties due to its ability to account for dynamic screening and correlation effects. The GW approximation is then described in detail, including its derivation from many-body perturbation theory and its physical interpretation. The self-energy in the GW approximation is non-local and energy-dependent, addressing the major deficiency of the Hartree-Fock approximation (HFA) in capturing screening effects. The authors also discuss the numerical methods used to calculate the self-energy, such as the plane-wave basis and localized basis, and the plasmon-pole approximation. The article further explores simplified versions of the GW approximation, such as the static Coulomb-hole and screened-exchange (COHSEX) approximation, and their applications to various materials, including alkali metals, semiconductors, transition metals, and surfaces. The authors also address the self-consistency issue and new developments beyond the GW approximation, highlighting both its successes and limitations. Overall, the GW approximation has become a standard method for including correlation effects beyond the HFA, providing accurate predictions of quasiparticle energies and excitation spectra in a wide range of materials.