Bridging Bonds and Bands: A Toolkit for Interpreting the Crystal Chemistry of Electronic Structure

All materials are composed of atoms held together by bonds. Except, not really. Quantum mechanics distributes the electron density to minimize the total energy, without regard for which atom or bond that energy originates from. Yet, a local decomposition of interactions persists; in some regions, the atomic nucleus dominates the energy, while in others, it arises from wavefunction overlap, electron–electron correlations, or even influences from distant atomic cores. Unfortunately, in our pursuit of ever-more accurate electronic structure methods, the computational materials community has lost touch with the atom- and bond-based scientific interpretation that allows insight into the heart of quantum mechanics. This dissertation aims to recover that physical intuition first by leveraging existing computational tools—and when they fall short—by developing a new one: Crystal Orbital Guided Iteration To atomic-Orbitals (COGITO). My goal is to show that COGITO provides the chemically grounded, physically faithful, and deeply interpretable understanding of electronic structure that has long been missing. I begin by exploring how chemical bonds shape not only a material’s energy but also its electronic band structure—governing optical, transport, and magnetic properties. To isolate which orbitals control specific features, I develop a strategy to distill and interpret sparse tight-binding models from DFT with maximally localized Wannier functions (MLWFs). In silicon, this reveals that the indirect gap emerges from competing interactions: first nearest-neighbors raise the conduction band near X, while second nearest-neighbors pull it down in a cosine-like shape—together forming the conduction band dip near X. In contrast to silicon, we identify changes in orbital size alone spark shifts in the band position and width to cause the anomalously large band gap of α-PbO. While these studies reveal the potential of using Wannier orbitals to rationalize band structures, existing orbital-based methods—including MLWF, LOBSTER, and quasi-atomic orbitals—do not simultaneously provide strict atomic localization, adaptability and uniqueness, completeness of the DFT wavefunctions, and high-quality tight-binding interpolation. Recognizing this gap, I developed Crystal Orbital Guided Iteration To atomic-Orbitals (COGITO), a rigorous computational framework that maps the Kohn-Sham DFT wavefunctions into an analytically defined nonorthogonal atomic orbitals. COGITO uniquely addresses two key mathematical barriers previously limiting nonorthogonal approaches: uncontrolled orbital mixing and an inherent fixed-overlap constraint. Through an iterative, chemically guided procedure, COGITO refines orbital projections while simultaneously adjusting the atomic basis to achieve self-consistent completeness of the DFT wavefunctions. With this robust orbital foundation, COGITO enables unprecedented visualization and quantification of covalent bonds within solids—providing orbital-resolved bond densities, single-particle covalent bond energies, and even magnetic interactions at the bond level. To demonstrate its transformative potential, I apply COGITO to outstanding challenges in solid-state chemistry and physics: elucidating the chemical origins of structural stability in perovskites and revealing how various density functional approximations distinctly alter local electronic structure and bonding characteristics. Ultimately, COGITO redefines our capacity to chemically interpret electronic structures of solids, marking a fundamental paradigm shift toward chemically intuitive, predictive, and design-oriented electronic structure analysis.