EACO: Predicting $\mathrm{O}(3)$-Equivariant Quantum Operators via Edge-Aligned $\mathrm{O}(2)$ Convolution

Predicting quantum operator matrices from atomic structures with group symmetries enables large-scale electronic-structure calculations and high-throughput materials screening. Focusing on this task, current edge-aligned $\mathrm{SO}(2)$ convolutions guarantee global $\mathrm{SO}(3)$ equivariance. Still, under reflections or spatial inversion, they cannot force particular operator components to vanish or protect electronic degeneracies. To overcome this issue, we propose Edge-Aligned $\mathrm{O}(2)$ Convolution (EACO), a message-passing layer that achieves local $\mathrm{O}(2)$ equivariance with strict locality and high computational efficiency. In particular, EACO assigns each $\mathrm{O}(3)$ irreducible representation (irrep) of angular degree $\ell$ and inversion parity $p$ the sign ${\tau }_{\ell ,p}=p(-1{)}^{\ell }$, indicating how it transforms under a reflection in the edge-aligned local frame and thus distinguishing the irreps with the same angular degree but opposite inversion parities. We prove that local $\mathrm{O}(2)$ equivariance is necessary and sufficient for global $\mathrm{O}(3)$ equivariance within this construction, so the model built on EACO preserves the transformation laws of the full $\mathrm{O}(3)$ group. Compared with SLEM, EACO reduces Hamiltonian MAE by up to 22.1% across five materials, with 17.2%$-$29.6% fewer parameters. It also reduces equivariance errors under reflections and spatial inversion by three to four orders of magnitude. On graphene, EACO recovers the horizontal-mirror (${\sigma }_h$) selection rule for flexural phonons, suppressing forbidden electron–phonon coupling while accurately reproducing allowed coupling.