Why does silicon have an indirect band gap?

It is difficult to intuit how electronic structure features$-$such as band gap magnitude, location of band extrema, effective masses, etc.$-$arise from the underlying crystal chemistry of a material. Here we present a strategy to distill sparse and chemically-interpretable tight-binding models from density functional theory calculations, enabling us to interpret how multiple orbital interactions in a 3D crystal conspire to shape the overall band structure. Applying this process to silicon, we show that its indirect gap arises from a competition between first and second nearest-neighbor bonds$-$where second nearest-neighbor interactions pull the conduction band down from $\Gamma$ to $X$ in a cosine shape, but the first nearest-neighbor bonds push the band up near $X$, resulting in the characteristic dip of the silicon conduction band. By identifying the essential orbital interactions that shape the conduction band, we can further rationally tune bond strengths to morph the silicon band structure into the germanium band structure. Our computational approach serves as a general framework to extract the crystal chemistry origins of electronic structure features from density functional theory calculations, enabling a new paradigm of bonding-by-design.