Many-body theory beyond GW: Towards a complete description of two-body correlated propagation

Starting with Hedin's equations, simple expressions for the irreducible self-energy are derived. The derivation with vertex effects included in the self-energy results in a number of terms beyond GW such as second-order screened exchange (the term that also gives rise to vertex/excitonic effects in the polarization) and an infinite series describing correlations between the added-particle (removed-hole) and the excited electrons and holes: the screened T-matrix channels. Two-body correlated propagation is considered, with the third propagating freely, however, three-body correlations are discussed and these can be added hierarchically to the method. For electron-hole propagation, the reducible polarization is calculated, which results in an expression for the self-energy that can be solved analytically without the need for widely used expensive numerical methods for frequency integration or having to adopt the plasmon-pole approximation. The method requires diagonalization of the usual particle-hole matrix-derived from the Bethe-Salpter equation-and also a particle-particle+holehole matrix that has a similar structure; with the second-order exchange present in all channels. The method goes beyond current widely used methods that adopt the GW approximation, whilst being ab initio in nature and requiring relatively modest extensions to existing functionality, e.g., it could be implemented in the sophisticated quasiparticle self-consistent framework. It is anticipated the method will significantly improve on the accuracy of current state-of-the-art calculations, improving our understanding of processes in simple and strongly correlated systems, with the additional contributions demanding a similar cost to methods that include vertex effects in the polarization only.