Paired-Isotopologue Hessian Benchmarking for Second-Order Validation of Machine-Learned Water Potentials

Machine-learned interatomic potentials (MLIPs) are commonly validated against energies and forces, yet vibrational predictions depend on the curvature and mode information encoded in the Hessian of the potential-energy surface. Here, we introduce a paired-isotopologue Hessian benchmark for MLIPs using hydrogen-bonded water as a test system. For each MD-sampled water-cluster geometry, a common electronic Hessian is combined with matched HH, DH, and DD mass patterns, enabling the controlled evaluation of vibrational frequencies, normal modes, and isotope responses across nine universal, molecular, and water-specific potentials. Model rankings differ across force-, Hessian-, and vibration-based evaluations, while the paired isotopologue response follows the broad vibrational ordering and remains consistent across structural sources. Beyond global error measures, local coordinate analysis distinguishes Hessian errors with markedly different relevance to specific vibrational observables. Along isotope paths, near-degenerate modes can lose individual mode overlap without a corresponding loss of the vibrational subspace. Together, these results establish Hessian and mode-space validation as necessary complements to energy- and force-based evaluations of MLIPs intended for vibrational calculations.