Lowest-order corrections to the RPA polarizability and GW self-energy of a semiconducting wire

We present the results of the addition of lowest-order vertex and self-consistency corrections to the RPA polarizability and the GW self-energy for a semiconducting wire. It is found that, when starting from a local density approximation zeroth-order Green function and systematically including these corrections in both the polarizability and the self-energy, the correction to the non-self-consistent RPA-GW band gap is small. Partial inclusion of these corrections leads to very different band gaps. This sheds new light on the puzzling question why non-self-consistent RPA-GW calculations of band gaps have been so very successful.