# HIP: Hessian Interatomic Potentials without derivatives

arXiv: https://arxiv.org/abs/2509.21624 · HTML: https://mlsanity.com/paper/2509.21624/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 75 references that are in the map.
- First posted on arXiv 25 Sep 2025.

## Builds on (most cited first, 19 of 19)

- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction](https://mlsanity.com/paper/2502.12147/) (2025)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [Efficient and Equivariant Graph Networks for Predicting Quantum Hamiltonian](https://mlsanity.com/paper/2306.04922/) (2023)
- [Transition1x -- a Dataset for Building Generalizable Reactive Machine Learning Potentials](https://mlsanity.com/paper/2207.12858/) (2022)
- [Second-order optimization with lazy Hessians](https://mlsanity.com/paper/2212.00781/) (2022)
- [Enhancing the Scalability and Applicability of Kohn-Sham Hamiltonians for Molecular Systems](https://mlsanity.com/paper/2502.19227/) (2025)
- [Hessian QM9: A quantum chemistry database of molecular Hessians in implicit solvents](https://mlsanity.com/paper/2408.08006/) (2024)
- [Synthetic pre-training for neural-network interatomic potentials](https://mlsanity.com/paper/2307.15714/) (2023)

## Built on by (most cited first, 12 of 12)

- [PFT: Phonon Fine-tuning for Machine Learned Interatomic Potentials](https://mlsanity.com/paper/2601.07742/) (2026)
- [Projected Hessian Learning: Fast Curvature Supervision for Accurate Machine-Learning Interatomic Potentials](https://mlsanity.com/paper/2603.04523/) (2026)
- [Global Plane Waves From Local Gaussians: Periodic Charge Densities in a Blink](https://mlsanity.com/paper/2601.19966/) (2026)
- [M=oLe-Lambda: Learning the Coupled-Cluster Response State for Energies, Gradients, and Properties](https://mlsanity.com/paper/2605.29622/) (2026)
- [Hessian-informed machine learning interatomic potential towards bridging theory and experiments](https://mlsanity.com/paper/2603.25373/) (2026)
- [Hessian Matching for Machine-Learned Coarse-Grained Molecular Dynamics](https://mlsanity.com/paper/2605.12823/) (2026)
- [MALOQ: Massively Accelerated Learning of Operators for Quantum Transport](https://mlsanity.com/paper/2606.28911/) (2026)
- [Hessian-Informed Conformation Augmentation for Potential Energy Surface Curvature in Machine Learning Interatomic Potentials](https://mlsanity.com/paper/2609.05233/) (2026)
- [Truncated automatic sparse differentiation for machine learning interatomic potentials](https://mlsanity.com/paper/2609.20510/) (2026)
- [Grand Canonical Generators](https://mlsanity.com/paper/2610.00683/) (2026)
- [Physics-Aligned Electronic Ground-State Learning Improves Generalization](https://mlsanity.com/paper/2610.10298/) (2026)
- [Characterizing and Mitigating Distributional Bias in Physics-Informed Generative Models](https://mlsanity.com/paper/2610.16582/) (2026)

## Abstract (arXiv)

Molecular Hessians, the second derivatives of the potential energy, are fundamental to many workflows in computational chemistry. Usually, accurate Hessians are computationally expensive to calculate and scale poorly with system size, whether computed using quantum chemistry methods or machine-learning interatomic potentials (MLIPs). In this work, we introduce Hessian interatomic potentials (HIPs), a deep learning model that directly predicts Hessians without relying on automatic differentiation or finite differences. To do so, we construct SE(3)-equivariant, symmetric Hessians from irreducible representation (irrep) features up to degree l=2, computed by a graph neural network. HIP Hessians are one to two orders of magnitude faster, more accurate, more memory efficient, easier to train, and exhibit more favourable scaling with system size. We validate our predictions across a wide range of downstream tasks, demonstrating consistently superior performance in transition state search, geometry optimization, zero-point energy corrections, and vibrational analysis. We open-source the HIP code and model weights.