Dimensionality and electronic structure in square-net-derived bonding motifs: A tribute to Hoffmann’s orbital perspective

Square-nets of atoms, studied systematically since Tremel and Hoffmann’s 1987 survey and redefined under the hypervalent-bonding framework of Papoian and Hoffmann in 2000, appear in thousands of known intermetallics. Many of those materials are topological semimetals, charge density wave hosts, and complex magnets. In tribute to Roald Hoffmann, we extend that framework to the full family of square-net-derived bonding motifs: one-dimensional ladders, isolated and layered two-dimensional square-nets (including squished and offset-stacked variants), and simple three-dimensional cubes. For each motif, we derive a tight-binding electronic structure, identify the symmetry-enforced band crossings that arise upon unit-cell doubling, and benchmark against density-functional calculations on a representative compound. We point out which types of bands will result in topological semimetal band structures upon folding. We then search the Materials Project ( ∼ 155,000 candidates) using a home-built “Square Net Catcher” code, finding 312 isolated square nets, 30 ladders, 20 stacked, 11 squished, and only 1 simple cube. To understand the cube better, we consider the rock-salt structure, as it realizes the doubled-cube motif, to find 21 weakly-polar examples. We show that those can be viewed as doubled analogs of the simple cube, which explains how the band inversion responsible for topology arises in the PbTe family. The analysis brings Hoffmann’s orbital-chemistry perspective into contact with the modern structure-to-property language and highlights motifs which remain fertile ground for discovery.