GW+2SOSEX Self-Energy Made Positive Semidefinite

The formulation of vertex corrections beyond the GW approximation within the framework of the perturbation theory is a subtle and challenging task, which accounts for the wide variety of schemes proposed over the years. Exact self-energies are required to satisfy the mathematical condition of positive semidefiniteness. The GW self-energy fulfills this property, but the vast majority of the vertex-corrected self-energy approximations do not. In this study, we devise a positive semidefinite extension to the GW+2SOSEX self-energy that we name GW+2SOSEX-psd. To reach this goal, we demonstrate the cancellation of the bare energy poles that are contained in the fully dynamic second-order in W self-energy (G3W2). We then demonstrate, on molecular examples, the correct positive semidefiniteness of the proposed self-energy approximation and its good accuracy in predicting accurate quasiparticle energies for valence and core states.