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We present fully self-consistent results for the self-energy of the electron gas within the GW approximation. This means that the self-consistent Green's function G, as obtained from Dyson's equation, is used not only for obtaining the self-energy but also for constructing the screened interaction W within the random-phase approximation. Such a theory is particle and energy conserving in the sense of Kadanoff and Baym. We find an increase in the weight of the quasiparticle as compared to ordinary non-self-consistent calculations but also to calculations with partial self-consistency using a fixed W. The quasiparticle bandwidth is larger than that of free electrons and the satellite structure is broad and featureless; both results clearly contradict the experimental evidence. The total energy, though, is as accurate as that from quantum Monte Carlo calculations, and its derivative with respect to particle number agrees with the Fermi energy as obtained directly from the pole of the Green's function at the Fermi level. Our results indicate that, unless vertex corrections are included, non-self-consistent results are to be preferred for most properties except for the total energy.

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We have calculated total energies of atoms and diatomic molecules from the Luttinger-Ward functional, using self-energy approximations to second order as well as the GW approximation. In order to assess the variational quality of this functional, we have also solved the Dyson equation self-consistently. The Luttinger-Ward functional is compared to the variational functional due to Klein, and we demonstrate that the variational property of the latter functional is inferior to that of the Luttinger-Ward functional. We also show how to obtain variational density functionals from the functionals of the Green function. These orbital functional schemes are important for systems where density-functional theory using local functionals of the density necessarily fails. We derive an optimized effective potential ͑OEP͒ scheme that is based on the Luttinger-Ward functional and, unlike the conventional OEP schemes, produces energies in good agreement with the values obtained from the self-consistent Green function. Our calculations show that, when applied to molecules, the Luttinger-Ward functional is more sensitive to the quality of the input Green function than when applied to atoms, but the energies are remarkably close to the self-consistent values when the Hartree-Fock Green function is used as input. This Luttinger-Ward functional is therefore a simple and efficient method for studying the merits of various self-energy approximations while avoiding the computationally demanding task of solving the Dyson equation self-consistently.