From spectra to total energy: beyond the GW approximation designing effective interactions

The total energy and electron addition and removal spectra can in principle be calculated exactly from the one-body Green\textquoterights function (GF). In practice, the GF is most often obtained from an approximate self-energy. For the band structure of solids and energy levels of molecules, the GW approximation has become the state-of-the-art approach. This approximation is a first-order expression of the self-energy in terms of the screened Coulomb interaction. By the way of contrast, it is not clear what is the best framework to access the ground state total energy. Indeed, most total energy calculations are today performed using density-functional theory (DFT), not Green\textquoterights functions. This is due both to the fact that GF calculations have usually a higher computational cost than DFT calculations, and to the fact that there is today no well established approximation for the total energy in the GF framework. Still, there are good reasons to investigate ways to use Green\textquoterights functions to calculate the total energy. First, exact expressions for the total energy as functional of GF and/or the self-energy are known in principle. Second, the GF framework suggests powerful and systematic approximations. However, the GW approximation, while suitable for spectra, is in practice not satisfactory for the total energy, where usually a high precision is required. Moreover, the validity of the GW approximation is limited to weakly to moderately correlated systems. In this thesis, we embark on a journey to overcome the limitations of GW . We first make a comprehensive exploration to understand the failures and constraints of GW. Subsequently, we present general theoretical developments and suggest new approximations based on the use of effective interactions in order to improve total energy calculations based on Green\textquoterights functions. The theoretical developments and the quality of the approximations are illustrated through applications to an exactly solvable model, the Hubbard dimer. The results confirm our conjectures and motivate future applications to real materials.