Unified description of ground and excited states of finite systems: the self-consistent \textit{GW} approach

Fully self-consistent $GW$ calculations -- based on the iterative solution of the Dyson equation -- provide an approach for consistently describing ground and excited states on the same quantum mechanical level. Based on our implementation in the all-electron localized basis code FHI-aims [1], we show that for finite systems self-consistent $GW$ reaches the same final Green function regardless of the starting point. The results show that self-consistency systematically improves ionization energies and total energies of closed shell systems compared to perturbative $GW$ calculations ($G_0W_0$) based on Hartree-Fock or (semi)local density-functional theory. These improvements also translate to the electron density as demonstrated by a better description of the dipole moments of hetero-atomic dimers and the similarity with the coupled cluster singles doubles (CCSD) density. The starting-point independence of the self-consistent Green function facilitates a systematic and unbiased assessment of the performance of the $GW$ approximation for finite systems. It therefore constitutes an unambiguous reference for the future development of vertex corrections and beyond $GW$ schemes. [1] V. Blum \textit{et al.}, Comp. Phys. Comm. {\bf 180}, 2175 (2009).