First-order corrections to random-phase approximation $\mathrm{GW}$ calculations in silicon and diamond

We report on ab initio calculations of the first-order corrections in the screened interaction $W$ to the random-phase approximation polarizability and to the $\mathrm{GW}$ self-energy, using a noninteracting Green’s function, for silicon and diamond. It is found that the first-order vertex and self-consistency corrections to the polarizability largely compensate each other. This does not hold, however, for the first-order corrections to the $\mathrm{GW}$ gap. For silicon the compensation between the first-order vertex and self-consistency correction contributions to the gap is only about 35%, while for diamond it is even absent. The resulting gap values are significantly and systematically too large, the direct gaps for silicon and diamond being 0.4 eV and 0.7 eV larger than their $\mathrm{GW}$ values, respectively. The success of $\mathrm{GW}$ in predicting electronic properties of, e.g., silicon and diamond can therefore apparently not be understood in terms of “small” corrections to $\mathrm{GW}$ to first order in $W$ using a noninteracting Green’s function.