The SAOP exchange-correlation potential, which statistically averages the potential with the correct Coulombic asymptotics and the potential arising from the step structure of the atomic and molecular electron shells, produces a good-quality estimate of vertical ionization potentials and yields a high-quality zero-order estimate of excitation energies within TDDFT. 258 (3.1.13) Romaniello : The link between DFT and many-body perturbation theory (MBPT) based on Green's functions has been particularly beneficial. The Sham-Schlüter equation (SSE) , which relates the Kohn-Sham potential of DFT to the self-energy of MBPT, has given several insights into approximations to the v xc . As an example, one can easily retrieve the OEP equations from the linearized version of the SSE. Also, the time-dependent version of the SSE 262 has been very useful in the context of TDDFT. For example, one can show that the TDDFT exchange-correlation kernel f xc can be written exactly as two contributions, one responsible for the shift of the Kohn-Sham band gap to the fundamental gap and the other accounting for excitonic effects. This splitting has been recently used to calculate accurate optical spectra of semiconductors and insulators within a pure Kohn-Sham TDDFT framework -that is, without invoking empirical information nor theory beyond Kohn-Sham DFT (e.g., GW theory) to correct the Kohn-Sham gap. 264 (3.1.14) Galli : One of the outstanding open problems in defining approximate density functionals pertains to the description of the electronic properties of solid-solid and solid-liquid interfaces. When systems with different dielectric properties are interfaced -for example, a metal with an insulator or a semiconductor such as silicon with an insulating liquid such as waternone of the existing functionals can accurately describe band offsets and other electronic properties. This issue can be mitigated by carrying out GW calculations starting from DFT orbitals (for nonmetallic systems) However, this GW @DFT approach does not work when the underlying wave function provided by DFT turns out to be too inaccurate as a starting point -for example, for some transition-metal oxides.