Fully self-consistent GW calculations for atoms and molecules

– We solve the Dyson equation for atoms and diatomic molecules within the GW approximation, in order to elucidate the effects of self-consistency on the total energies and ionization potentials. We find GW to produce accurate energy differences although the selfconsistent total energies differ significantly from the exact values. Total energies obtained from the Luttinger-Ward functional ELW[G] with simple, approximate Green functions as input, are shown to be in excellent agreement with the self-consistent results. This demonstrates that the Luttinger-Ward functional is a reliable method for testing the merits of different self-energy approximations without the need to solve the Dyson equation self-consistently. Self-consistent GW ionization potentials are calculated from the Extended Koopmans Theorem, and shown to be in good agreement with the experimental results. We also find the self-consistent ionization potentials to be often better than the non-self-consistent G0W0 values. We conclude that GW calculations should be done self-consistently in order to obtain physically meaningful and unambiguous energy differences. Introduction. – Green function methods have been used with great success to calculate a wide variety of properties of electronic systems, ranging from atoms and molecules to solids. One of the most successful and widespread methods has been the GW approximation (GWA) [1], which has produced excellent results for band gaps and spectral properties of solids [2, 3], but so far has not been explored much for atoms and molecules, although it has been known that for atoms the core-valence interactions are described much more accurately by GW than Hartree-Fock (HF) [4]. Moreover, the GW calculations are rarely carried out in a self-consistent manner, and the effect of self-consistency is for this reason still a topic of considerable debate [5, 6]. In this paper we present self-consistent all-electron GW (SC-GW ) calculations for atoms and diatomic molecules. The reason for doing these calculations is two-fold: Firstly we want to study the importance of self-consistency within the GW scheme. Such calculations are usually avoided due to the rather large computational effort involved. It has been suggested that self-consistency will in fact worsen the spectral properties, though calculations on silicon and germanium crystals indicate that this is not always the case [5]. The second reason is that we aim to study transport through large molecules and molecular chains, where it is essential to account for the screening of the long range of the Coulomb interaction. The calculations on diatomic molecules are the first step in this direction. The GWA is obtained by replacing the bare Coulomb interaction v in the exchange selfenergy with the dynamically screened interaction W , such that Σ = −GW . The screened