Equivariant Cellular Sheaves Model Molecular Electronic Structure

Equivariant message-passing networks are standard for predicting molecular properties, with recent work extending to E(3)-equivariant Hamiltonian prediction. Separately, topological deep learning has advanced graph networks to cellular sheaves. This paper bridges these fields by observing a structural connection: the molecular single-particle Hamiltonian, when shifted, can be represented as the Laplacian of a cellular sheaf on a molecular cell complex. By making the restriction maps O(3)-steerable two-center kernels derived from bond geometry, the model recovers the Slater-Koster form and yields an E(3)- and permutation-equivariant operator. This formulation has three significant consequences. Firstly, the zeroth sheaf cohomology (H^0) becomes a topological invariant, identifying non-bonding orbitals and providing a lower bound for their count. Secondly, the Hodge 1-Laplacian allows higher cells (rings) to encode cycle and delocalization information through H^1. Thirdly, this model strictly generalizes existing E(3)-equivariant message-passing networks and CW networks, inheriting anti-oversmoothing properties. Numerical validation confirms the exactness of the Hamiltonian-to-sheaf embedding, accurate non-bonding orbital counts, and superior error and rotation generalization on electronic targets.