  This paper describes an all-electron implementation of the self-consistent GW
(sc-GW) approach -- i.e. based on the solution of the Dyson equation -- in an
all-electron numeric atom-centered orbital (NAO) basis set. We cast Hedin's
equations into a matrix form that is suitable for numerical calculations by
means of i) the resolution of identity technique to handle 4-center integrals;
and ii) a basis representation for the imaginary-frequency dependence of
dynamical operators. In contrast to perturbative G0W0, sc-GW provides a
consistent framework for ground- and excited-state properties and facilitates
an unbiased assessment of the GW approximation. For excited-states, we
benchmark sc-GW for five molecules relevant for organic photovoltaic
applications: thiophene, benzothiazole, 1,2,5-thiadiazole, naphthalene, and
tetrathiafulvalene. At self-consistency, the quasi-particle energies are found
to be in good agreement with experiment and, on average, more accurate than
G0W0 based on Hartree-Fock (HF) or density-functional theory with the
Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional. Based on the
Galitskii-Migdal total energy, structural properties are investigated for a set
of diatomic molecules. For binding energies, bond lengths, and vibrational
frequencies sc-GW and G0W0 achieve a comparable performance, which is, however,
not as good as that of exact-exchange plus correlation in the random-phase
approximation (EX+cRPA) and its advancement to renormalized second-order
perturbation theory (rPT2). Finally, the improved description of dipole moments
for a small set of diatomic molecules demonstrates the quality of the sc-GW
ground state density.
