Equivariant Cellular Sheaves for Molecular Electronic Structure: Bridging Sheaf Cohomology and E(3)-Equivariant Hamiltonian Learning

Abstract:Equivariant message-passing networks are the standard model for molecular property and interatomic-potential prediction, and recent work predicts the electronic Hamiltonian itself in an E(3)-equivariant way. Separately, topological deep learning has extended graph networks to cellular sheaves. Our central observation is structural: in a localized atomic-orbital basis, the molecular single-particle Hamiltonian, after a constant shift that makes it positive semidefinite, is the Laplacian of a cellular sheaf on a regular cell complex built from the molecule. Making the restriction maps O(3)-steerable two-center kernels from bond geometry recovers the Slater-Koster form as a special case and yields an E(3)- and permutation-equivariant operator. Three consequences follow. First, the zeroth sheaf cohomology H^0 = ker L is a topological invariant equal to the non-bonding (zero-mode) orbitals, recovering the classical alternant non-bonding-orbital count as a lower bound. Second, the Hodge 1-Laplacian lets higher cells (rings) carry cycle and delocalization information through H^1. Third, the model strictly generalizes E(3)-equivariant message-passing networks and CW networks, and inherits the anti-oversmoothing of non-trivial sheaf diffusion. We prove equivariance, expressivity, and cohomological-correspondence results for the Equivariant Cellular Sheaf Networks, and validate them numerically: the Hamiltonian-to-sheaf embedding is exact to machine precision, the cohomology dimension reproduces non-bonding-orbital counts across eleven conjugated molecules, the sheaf Laplacian is O(3)-equivariant to machine precision, and the equivariant model attains lower error and rotation generalization on a directional electronic target. Our contribution is this sheaf-theoretic formalization and its invariants, not equivariant Hamiltonian prediction itself.

Machine Learning (cs.LG); Chemical Physics (physics.chem-ph)

arXiv:2608.23571 [cs.LG]

(or arXiv:2608.23571v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.23571 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.