A molecular calculator can reach a strange moment: the forces on the atoms approach zero, yet the calculation has not answered whether it found a resting place or a route through a reaction. Both can have a flat slope. To tell them apart, it needs to know how the surrounding energy landscape bends.[2]
HIP, published on October 13, predicts that curvature directly. In a test classifying 1,000 stationary molecular geometries, it achieved 92% accuracy, ten percentage points above the next-best comparison method.[1]
The result is easier to appreciate through the question a chemist is asking. Finding a comfortable molecular arrangement and finding the unstable arrangement a reaction passes through require different kinds of search. A system that treats every flat point as a destination can stop in the wrong place.
Why zero is an incomplete answer
Picture two imaginary landscapes: a bowl and a mountain pass. At the bowl’s bottom, moving away sends you uphill. At the centre of the pass, you can go uphill across the ridge but downhill towards either valley. Both locations can be locally level. The surrounding shape reveals whether you have found a minimum or a saddle point.
The chemical version records curvature in a Hessian matrix: the second derivatives of energy with respect to atomic positions. For N atoms, it has 3N rows and 3N columns. A first-order transition state has one direction of negative curvature; a minimum has none. That information helps an optimiser locate and distinguish the two.[2]
For a hypothetical 30-atom molecule, the matrix has 90 rows and 90 columns: 8,100 entries, calculated from the three spatial coordinates of each atom. Those entries are coupled responses. They describe how changing one coordinate alters the force associated with another. A molecule may be small enough to sketch on paper while the local geometry of its energy is already a substantial object.
The mountain analogy has a useful limit. Atoms move in three-dimensional space, but the landscape describes all their coordinates together. Its “downhill direction” can mean a coordinated motion of several atoms: one bond stretching as another forms. The search needs the right collective motion, rather than a direction drawn across an ordinary map.
Predict the curvature directly
HIP assembles a Hessian from predicted 3-by-3 blocks for atom pairs, respecting the required symmetry and behaviour under rotation. It avoids calculating the matrix by repeatedly differentiating the learned energy or forces. For molecules with 5–30 atoms, single-molecule tests on an RTX 3060 showed speedups of 10–70 times against automatic differentiation.[1]
This is a change in which mathematical object the model is asked to learn. Knowing a curve’s height approximately does not automatically give an accurate description of how sharply it bends. Small variations that barely affect the height can matter greatly after taking derivatives. Direct supervision makes curvature an explicit learning target.
The author implementation provides checkpoints and configurations for reproducing the approach. Its HORM-based training domain is primarily small, neutral organic molecules containing hydrogen, carbon, nitrogen and oxygen, including structures away from equilibrium. Researchers can therefore inspect a concrete starting point, with a stated chemical scope, before considering their own systems.[3]
Transition-state workflows also need to establish that the candidate connects the intended reactants and products. The HORM work describes tracing the reaction coordinate in both directions to check those endpoints. Correct local curvature is one component of that chemical answer.[2]
One mathematical trade-off remains: a directly predicted Hessian is not guaranteed to be exactly consistent with the model’s energy and forces, or to correspond to one globally consistent energy surface. The authors position it as useful local curvature for optimisation and analysis.[1]
For a reaction search, that local view can still be valuable. At each step, the computer must choose how to move an arrangement of atoms towards the particular saddle it seeks. Making curvature cheaper to obtain gives that decision a richer description of its surroundings. The goal is to turn a promising molecular guess into a pathway a chemist can examine.
Sources & further reading
- HIP: Hessian Interatomic Potentials without derivatives Nature Computational Science · 13 Oct 2026 · Accessed 17 Oct 2026
- HORM: A Large Scale Molecular Hessian Database for Optimizing Reactive Machine Learning Interatomic Potentials arXiv · 18 May 2025 · Accessed 17 Oct 2026
- HIP: Hessian Interatomic Potentials — author implementation and documentation GitHub / Andreas Burger and collaborators · Accessed 17 Oct 2026