AI Model Predicts Molecular Curvature Directly, Skipping Costly Derivatives

Every molecule sits on a landscape. Stretch a bond, bend an angle, twist a dihedral, and the electronic energy rises or falls across a high-dimensional surface that chemists call the potential energy surface. The gradient of that surface tells a simulation which way is downhill, but the curvature tells it something far richer: how sharply the energy bends, whether a structure is a true minimum, a saddle point on the path between reactants and products, or something stranger. That curvature is encoded in the Hessian, the matrix of second derivatives of the energy with respect to every pair of nuclear coordinates, and it has long been one of the most expensive quantities in computational chemistry to obtain. A team of researchers led by Andreas Burger and Luca Thiede at the University of Toronto, working with colleagues at the Technical University of Denmark, now reports in Nature Computational Science a way to sidestep that cost almost entirely.

Their method, called Hessian Interatomic Potentials, or HIP, is a deep learning model that predicts the full Hessian matrix directly, in a single forward pass, without resorting to automatic differentiation, finite differences, or the coupled-perturbed equation solvers that quantum chemistry packages traditionally require. For a molecule with N atoms, the Hessian is a 3N by 3N matrix, and each of its entries is a mixed second derivative of the electronic energy with respect to two nuclear coordinates. Conventional approaches compute these entries either analytically, which is notoriously complicated to implement and often unavailable for modern electronic structure methods, or numerically, by nudging each atom along each coordinate and re-evaluating forces. Both routes scale poorly. The researchers show that HIP instead delivers Hessians ten to seventy times faster than differentiation-based machine learning approaches on small molecules, cuts peak memory use by a factor of two to three, and roughly halves the prediction error.

The key insight lies in how the model is built. HIP rests on an equivariant graph neural network, in this case the EquiformerV2 architecture, in which each atom carries features that transform predictably under rotations of the molecule. These features are decomposed into irreducible representations, the angular momentum channels familiar from quantum mechanics, indexed by a degree l. The Hessian readout head that the team designed takes features up to degree l equals 2 and, using Clebsch-Gordan tensor products, reassembles them into the Cartesian 3 by 3 blocks that make up the full Hessian. Because this change of basis is exact, the resulting matrix is guaranteed by construction to transform correctly when the molecule is rotated and to be symmetric between its upper and lower triangles. In other words, the physics of the Hessian is baked into the architecture rather than hoped for from the training data.

The practical consequence of this design is a dramatic change in scaling. When a conventional machine learning potential computes a Hessian by automatic differentiation, it must effectively backpropagate through the entire network once for each row of the matrix, inflating the cost by roughly a factor of 3N relative to a plain energy calculation. HIP, by contrast, reuses the message-passing machinery of the underlying network and adds only about twenty-five percent overhead, roughly the cost of one extra layer. Because the Hessian interaction cutoff can be set independently of the backbone, the method inherits the favorable linear scaling with system size that makes modern machine learning potentials so powerful. The team demonstrated this on 710 molecules drawn from PubChem containing between thirty and one hundred atoms, well beyond the sizes seen during training, and found that HIP’s eigenvalue errors stayed essentially flat as the molecules grew, while the errors and costs of differentiation-based predictions climbed.

Accuracy was assessed on the HORM-T1x validation set, a large benchmark of reactive chemistry built on more than 1.7 million training geometries labeled with density functional theory calculations at the omega-B97X/6-31G* level. Compared with an EquiformerV2 model that obtained Hessians through automatic differentiation, HIP achieved a Hessian mean absolute error 2.5 times lower and an eigenvalue error 3.8 times lower, while running sixteen times faster. End-to-end training, in which the whole network learns from energies, forces, and Hessians simultaneously, pushed the Hessian error down further, to 0.020 electronvolts per square angstrom at just over thirty milliseconds per prediction. Notably, the study found that models trained only on energies and forces produce substantially worse Hessians even when their energies and forces are excellent, confirming that second derivatives, like forces before them, need explicit supervision to be predicted reliably.

To probe what those aggregate numbers mean physically, the researchers examined a single, well-studied reaction: the intramolecular proton transfer of glycine. They mapped a two-dimensional surface defined by the distances between the transferring proton and its neighboring nitrogen and oxygen atoms, relaxing every other degree of freedom with density functional theory at each grid point, and counted the negative Hessian eigenvalues across the surface. HIP reproduced the DFT topology closely, correctly distinguishing the minimum regions from the transition-state ridge. The differentiation-based model, by contrast, introduced spurious negative modes in regions where the true surface is locally stable. Tracking the six lowest vibrational eigenvalues along the minimum-energy path revealed the same story: HIP overlapped the quantum mechanical reference across all six modes, while the differentiated Hessians captured only the lowest mode and systematically underestimated the curvature of the rest.

The authors trace these failures to a subtle source. When they projected the forces of two different machine learning potentials onto the direction of the reaction path, both followed the overall DFT trend, but both oscillated around it. Automatic differentiation amplifies those small force fluctuations into large Hessian errors, suggesting that the noise inherent in a learned potential energy surface is what corrupts its second derivatives. Direct prediction, by learning the curvature as a first-class output rather than extracting it from a noisy surface, appears to sidestep this amplification. The result is a more self-consistent description of the potential energy landscape, one in which force errors and curvature errors rise and fall together rather than independently.

The downstream payoffs span the standard toolkit of computational chemistry. In geometry optimization, second-order rational-function optimization driven by HIP Hessians converged in the fewest steps on average across eighty perturbed molecular geometries, while equivalent workflows built on finite-difference or differentiated Hessians often failed to converge within a 150-step budget and ran one to two orders of magnitude slower in wall-clock time. In transition state searches on the ReactBench benchmark, where a growing-string guess is refined by Hessian-driven optimization and verified against explicit DFT calculations, HIP improved the rate of fully verified, tightly converged transition states by twelve percentage points over the differentiation approach. For thermochemistry, the errors in absolute zero-point energy corrections dropped by one to two orders of magnitude, and the error in reaction zero-point energy differences fell by more than ninety-seven percent. Perhaps most strikingly, when asked to classify one thousand stationary points by counting their negative eigenvalues, the task that distinguishes minima from transition states from higher-order saddles, HIP achieved ninety-two percent accuracy, ten percentage points ahead of the best competing model.

The team has released the HIP code under an open-source license along with trained model weights, datasets, and all reference calculations, a move that could see curvature-based workflows, long reserved for small systems and generous compute budgets, become routine across molecular design, catalysis, and materials discovery. The authors are candid about limitations: direct prediction does not guarantee that the learned Hessian is exactly consistent with the predicted energies and forces, accuracy still depends on the chemical coverage of training data, and the finite interaction cutoff may miss long-range curvature contributions in polar or extended systems. But they also point outward. The same strategy of dedicated equivariant readouts for expensive tensorial quantities could extend to elastic tensors, higher-order force constants, and the dynamical matrices of periodic crystals. If second derivatives can be learned this cheaply, the question becomes which other derivatives of the quantum world are waiting to be predicted directly.

Subject of Research: Direct machine learning prediction of molecular Hessians for computational chemistry

Article Title: HIP: Hessian Interatomic Potentials without derivatives

Article References: HIP: Hessian Interatomic Potentials without derivatives. (n.d.). https://doi.org/10.1038/s43588-026-01066-x

Image Credits: AI Generated

DOI: 10.1038/s43588-026-01066-x

Keywords: machine learning, interatomic potentials, Hessian, computational chemistry, graph neural networks, equivariance, transition state search, geometry optimization, zero-point energy, vibrational analysis, density functional theory, SE(3) symmetry