Many-body perturbation theory vs. density functional theory: A systematic benchmark for band gaps of solids

We benchmark many-body perturbation theory against density functional theory (DFT) for the band gaps of solids. We systematically compare four $GW$ variants $-$ $G_{0}W_{0}$ using the Godby-Needs plasmon-pole approximation ($G_{0}W_{0}$-PPA), full-frequency quasiparticle $G_{0}W_{0}$ (QP$G_{0}W_{0}$), full-frequency quasiparticle self-consistent $GW$ (QS$GW$), and QS$GW$ augmented with vertex corrections in $W$ (QS$G\hat{W}$) $-$ against the currently best performing and popular density functionals mBJ and HSE06. Our results show that $G_{0}W_{0}$-PPA calculations offer only a marginal accuracy gain over the best DFT methods, however at a higher cost. Replacing the PPA with a full-frequency integration of the dielectric screening improves the predictions dramatically, almost matching the accuracy of the QS$G\hat{W}$. The QS$GW$ removes starting-point bias, but systematically overestimates experimental gaps by about $15\%$. Adding vertex corrections to the screened Coulomb interaction, i.e., performing a QS$G\hat{W}$ calculation, eliminates the overestimation, producing band gaps that are so accurate that they even reliably flag questionable experimental measurements.