Hartree-Fock and lowest-order vertex-correction contribution to the direct gap of the semiconductor silicon

We have calculated the contribution of the second-order vertex-correction diagram to the direct gap of silicon at the {Γ} point and have compared it with the Hartree-Fock contribution. Both contributions have been calculated by using the Monte Carlo method for the involved three- and six-dimensional integrations. Our results show that the second-order contribution is much smaller than the first-order Hartree-Fock contribution. Although we have calculated the vertex correction diagram using the bare instead of the screened Coulomb interaction, our results give a first indication that vertex corrections can indeed be neglected, an assumption which is inherent in the GW approximation to the electron self-energy in a semiconductor.