Determining Principles for Electronic Band Structures Design Yielding Desired Properties 

Understanding how atomic-level interactions tune electronic bands is key to designing band structures that result in desired electronic, optical, and topological properties of materials. However, it is challenging to develop such understanding because even small changes in atomic-level structure or compositions can modify inter-orbital interactions, crystal-field splittings, and spin-orbit effects, all of which can affect electronic band structure significantly. Density functional theory can yield highly accurate electronic band structures, but it often does not explicitly reveal how the bands emerge from the constituent atomic orbitals. In this work, we combine analytical molecular orbital (MO) theory¹ with numerical DFT and tight binding (TB) model² to determine physical principles that describe how atomic orbitals interact to form electronic bands in complex tetravalent materials, with a mixed ionic binding environment. To demonstrate the effectiveness of the combined approach, we choose Bi₂O₂Ch (Ch = S, Se, Te) as an example and analyze how the mixing of atomic orbitals and band structures changes upon substituting different chalcogens. MO theory analysis provides a qualitative understanding of how contributions from different orbitals to the VBM and CBM vary upon Ch substitution, accounting for the change in band gap character. DFT study confirms that O-2p and Ch np states form the VBM and Bi 6p form the CBM in these compounds, resulting in a decrease of indirect band gap from Bi₂O₂S to Bi₂O₂Te⁴. A further analysis with the Slater–Koster³ TB model reproduces DFT trends but, more importantly, reveals how different orbitals interact to form specific bands and their curvature (e.g., flatness). This combined approach uncovers physical insights for band formation and establishes a foundation for the development of data-driven approaches. The extracted interatomic interactions, bond lengths, coordination number, and TB parameters from the combined method become the input feature for a machine learning (ML) model. We train the ML model to learn band structure character based on these inputs. Our approach bypasses conventional trial-and-error methods by integrating MO theory, DFT, TB modeling, and ML to enable data-driven inverse design of materials with desired electronic and optical properties.