Rotation Invariant Machine Learning for Highly Symmetric Molecular Configurations

Atomistic machine learning can approximate expensive quantum mechanical calculations at a fraction of their computational cost, but accuracy depends strongly on how the three-dimensional atomic environments are represented. Invariant representations enable fast computation but can fail to distinguish geometrically distinct configurations, especially in highly symmetric environments. We introduce a light-weight benchmark to evaluate expressivity on $n$-fold symmetric configurations, and use it to expose degeneracies in both invariant and equivariant architectures. Guided by these results, we construct a new invariant basis for HIP-HOP which resolves most of the identified failures. On a methane simulation dataset, the extended basis achieves the lowest error among the evaluated HIP-HOP variants while using $31\times$ fewer parameters and running $~20\times$ faster than Equiformer. These results show that carefully constructed invariant representations can mitigate key expressivity limitations while balancing accuracy and computational efficiency.