Title: Why does silicon have an indirect band gap?

Abstract: Density functional theory (DFT) is a powerful tool for predicting the electronic band structures of functional materials. However, it is often difficult to intuit how major band structure features$-$such as band gap magnitude, location of band extrema, effective masses, etc.$-$arise from the underlying crystal chemistry of a material. Band structure is represented in reciprocal space, but arises from the orbital and bonding interactions between atoms in real space. Here, we present a conceptual and computable framework to extract chemical bonding origins of DFT-calculated band structure features. As a key example, we explain here the indirect band gap of silicon, which profoundly impacts its properties for use in photovoltaics and electronics. Even in this basic semiconductor material, our approach leads to new insights to understand and engineer its conduction band minimum position. These calculation techniques can be broadly applied to reveal the crystal chemistry origins of electronic structure features in other optical, electronic, and magnetic materials.