Self-consistent solutions of the Dyson equation are obtained using a plane-wave basis set for seven small molecules. Such self-consistent solutions can help to unify the different
GW self-consistent schemes, reduce the scatter of results in current
GW calculations, and shed light on the true effects of
GW self-consistency. Unlike other works of self-consistent
GW calculations, in the present work the Green's function is expressed as a matrix under the plane-wave basis set. The algorithmic details which enable such calculations are presented. The ability to solve the full Green's function using a plane-wave basis set may open the door for future beyond-
GW many-body perturbation theory calculations.