Adaptive Vibrational Coordinates via Symmetry-Aware Normalizing Flows

Normalizing flows have recently been proposed to learn optimal vibrational coordinates tailored for a given molecule. It was demonstrated that such learned coordinates achieve substantial improvements in the accuracy of vibrational spectral calculations and capture essential vibrational behavior of molecules. We extend this framework to incorporate molecular symmetry by introducing a novel G-equivariant invertible residual network architecture. Exact equivariance with respect to a chosen discrete molecular symmetry group is enforced by construction, ensuring that the learned coordinate transformation respects the imposed symmetry for all network parameters. When combined with a symmetry-adapted basis, this equivariance preserves the symmetry-induced block-diagonal structure of the Hamiltonian throughout the optimization. We show that optimizing the coordinates either for a set of states spanning multiple irreducible representations or for a subset transforming according to a single irreducible representation yields comparable results. This result suggests that optimal coordinates are shared across irreducible representations and provides insight into how optimized coordinates accelerate basis convergence. The utility of symmetry-aware normalizing flows is demonstrated for H₂CO and NH₃.