Theoretical studies of semiconductors and band insulators are usually based on variants of the $GW$ method without full self-consistency, such as single-shot $G^0{W}^0$ or quasiparticle self-consistent $GW$. Fully self-consistent $GW$ provides a poor description of the gap size and electronic structure due to the lack of vertex corrections. While it is hard to predict at which order corrections can be neglected, local vertex corrections to all orders can be consistently included by combining $GW$ with extended dynamical mean field theory (EDMFT). Here, we show that ab initio multitier $GW+\mathrm{EDMFT}$ calculations, which achieve full self-consistency in a suitably defined low-energy space, provide an accurate description of semiconductors and band insulators, comparable to the established methods which are typically used for this class of materials. Our results demonstrate that the range of applicability of $GW+\mathrm{EDMFT}$ extends to weakly correlated systems, and they imply that despite the weak correlations, local vertex corrections are an important ingredient in the diagrammatic treatment of semiconductors and band insulators.

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• Electronic structure
• First-principles calculations
• Semiconductors
• Dynamical mean field theory
• Electron-correlation calculations
• GW method