  We give a detailed presentation of our recent scheme to include correlation
effects in molecular transport calculations using the GW approximation within
the non-equilibrium Keldysh formalism. We restrict the GW self-energy to the
central region, and describe the leads by density functional theory (DFT). A
minimal basis of maximally localized Wannier functions is applied both in the
central GW region and the leads. The importance of using a conserving, i.e.
fully self-consistent, GW self-energy is demonstrated both analytically and by
numerical examples. We introduce an effective spin-dependent interaction which
automatically reduces self-interaction errors to all orders in the interaction.
The scheme is applied to the Anderson model in- and out of equilibrium. In
equilibrium at zero temperature we find that GW describes the Kondo resonance
fairly well for intermediate interaction strengths. Out of equilibrium we
demonstrate that the one-shot G0W0 approximation can produce severe errors, in
particular at high bias. Finally, we consider a benzene molecule between
featureless leads. It is found that the molecule's HOMO-LUMO gap as calculated
in GW is significantly reduced as the coupling to the leads is increased,
reflecting the more efficient screening in the strongly coupled junction. For
the IV characteristics of the junction we find that HF and G0W0[G_HF] yield
results closer to GW than does DFT and G0W0[G_DFT]. This is explained in terms
of self-interaction effects and life-time reduction due to electron-electron
interactions.
