Learning the Electronic Hamiltonian of Large Atomic Structures

Graph neural networks (GNNs) have shown promise in learning the ground-state electronic properties of materials, subverting *ab initio* density functional theory (DFT) calculations when the underlying lattices can be represented as small and/or repeatable unit cells (i.e., molecules and periodic crystals). Realistic systems are, however, non-ideal and generally characterized by higher structural complexity. As such, they require large (10+ Å) unit cells and thousands of atoms to be accurately described. At these scales, DFT becomes computationally prohibitive, making GNNs especially attractive. In this work, we present a strictly local equivariant GNN capable of learning the electronic Hamiltonian (**H**) of realistically extended materials. It incorporates an *augmented partitioning* approach that enables training on arbitrarily large structures while preserving local atomic environments beyond boundaries. We demonstrate its capabilities by predicting the electronic Hamiltonian of various systems with up to 3,000 nodes (atoms), 500,000+ edges, ~ 28 million orbital interactions (nonzero entries of **H**), and $\leq$0.53\% error in the eigenvalue spectra. Our work expands the applicability of current electronic property prediction methods to some of the most challenging cases encountered in computational materials science, namely systems with disorder, interfaces, and defects.

Our work focuses on the machine learning of Hamiltonian matrices for large systems of atoms. The Hamiltonian matrix is an operator that allows key information about a material to be extracted, including how well it conducts electricity. It is therefore an integral part of material/device research and is normally computed from the ground up using physics approaches.Previous work on this topic is largely restricted to either small, isolated groups of atoms (molecules) or atomic structures with repeating patterns (e.g. crystals). In real life, however, materials are rarely perfect, often containing defects, disorder, and interfaces. To capture these details, the Hamiltonian of large numbers of atoms (1000+) in various arrangements needs to be computed, making previous ground-up computational methods unfeasible. In this work, we aim to overcome this open problem by proposing a local graph network combined with an approach that breaks down large atomic graphs into small, independent slices. Besides overcoming memory limitations of hardware, it also allows for flexible, efficient parallel training while maintaining the correct atomic neighborhoods. In short, our work bridges the gap between ML methods and real-life large scale materials applications.