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.