Approximate spatiotemporal structure of the vertex function T(1,2;3) in many-body perturbation theory

The vertex function F(1, 2; 3) within Hedin's equations constitutes a long-standing challenge to electronic -structure theory. We propose an explicit expression for F as a function of space and time. This ansatz fulfils the equation of motion within reasonable approximations. The equation of motion is controlled by the screened Coulomb interaction W and the Green's function G. Concerning W we find that its dynamics (here approximated by plasmons) is crucial for obtaining a realistic F. Concerning G the renormalization of the quasiparticle peak and the split-off satellites are important. In combination these effects can be approximated by simple expressions that can easily be incorporated in the determination of the screening and the electronic self-energy. Similar to previous literature we find significant cancellation of such vertex corrections and electronic self-consistency. In the case of silicon as a prototypical semiconductor material we obtain realistic results for the optical spectrum, the electronic spectral function, and the fundamental band gap.