We perform GW calculations on atoms and diatomic molecules at different levels of selfconsistency and investigate the effects of self-consistency on total energies, ionization potentials and on particle number conservation. We further propose a partially self-consistent GW scheme in which we keep the correlation part of the self-energy fixed within the self-consistency cycle. This approximation is compared to the fully self-consistent GW results and to the GW0 and the G0W0 approximations. Total energies, ionization potentials and two-electron removal energies obtained with our partially self-consistent GW approximation are in excellent agreement with fully selfconsistent GW results while requiring only a fraction of the computational effort. We also find that self-consistent and partially self-consistent schemes provide ionization energies of similar quality as the G0W0 values but yield better total energies and energy differences.

We implement time propagation of the nonequilibrium Green function for atoms and molecules by solving the Kadanoff-Baym equations within a conserving self-energy approximation. We here demonstrate the usefulness of time propagation for calculating spectral functions and for describing the correlated electron dynamics in a nonperturbative electric field. We also demonstrate the use of time propagation as a method for calculating charge-neutral excitation energies, equivalent to highly advanced solutions of the Bethe-Salpeter equation.

We have calculated the self-consistent Green's function for a number of atoms and diatomic molecules. This Green's function is obtained from a conserving self-energy approximation, which implies that the observables calculated from the Green's functions agree with the macroscopic conservation laws for particle number, momentum, and energy. As a further consequence, the kinetic and potential energies agree with the virial theorem, and the many possible methods for calculating the total energy all give the same result. In these calculations we use the finite temperature formalism and calculate the Green's function on the imaginary time axis. This allows for a simple extension to nonequilibrium systems. We have compared the energies from self-consistent Green's functions to those of nonselfconsistent schemes and also calculated ionization potentials from the Green's functions by using the extended Koopmans' theorem.

Abstract. -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 (GW A) [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 GW A 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

We have developed a time propagation scheme for the Kadanoff-Baym equations for general inhomogeneous systems. These equations describe the time evolution of the nonequilibrium Green function for interacting many-body systems in the presence of time-dependent external fields. The external fields are treated nonperturbatively whereas the many-body interactions are incorporated perturbatively using Φ-derivable self-energy approximations that guarantee the satisfaction of the macroscopic conservation laws of the system. These approximations are discussed in detail for the time-dependent Hartree-Fock, the second Born and the GW approximation.