# QH9: A Quantum Hamiltonian Prediction Benchmark for QM9 Molecules

arXiv: https://arxiv.org/abs/2306.09549 · HTML: https://mlsanity.com/paper/2306.09549/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 66 references that are in the map.
- First posted on arXiv 15 Jun 2023.

## Builds on (most cited first, 28 of 28)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [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)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [The Open Catalyst 2020 (OC20) Dataset and Community Challenges](https://mlsanity.com/paper/2010.09990/) (2020)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [The Open Catalyst 2022 (OC22) Dataset and Challenges for Oxide Electrocatalysts](https://mlsanity.com/paper/2206.08917/) (2022)
- [Periodic Graph Transformers for Crystal Material Property Prediction](https://mlsanity.com/paper/2209.11807/) (2022)
- [DIG: A Turnkey Library for Diving into Graph Deep Learning Research](https://mlsanity.com/paper/2103.12608/) (2021)
- [Unifying machine learning and quantum chemistry -- a deep neural network for molecular wavefunctions](https://mlsanity.com/paper/1906.10033/) (2019)
- [ComENet: Towards Complete and Efficient Message Passing for 3D Molecular Graphs](https://mlsanity.com/paper/2206.08515/) (2022)
- [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)
- [Efficient Approximations of Complete Interatomic Potentials for Crystal Property Prediction](https://mlsanity.com/paper/2306.10045/) (2023)
- [A deep neural network for molecular wave functions in quasi-atomic minimal basis representation](https://mlsanity.com/paper/2005.06979/) (2020)
- [Graph Neural Network for Hamiltonian-Based Material Property Prediction](https://mlsanity.com/paper/2005.13352/) (2020)
- [Equivalence Between SE(3) Equivariant Networks via Steerable Kernels and Group Convolution](https://mlsanity.com/paper/2211.15903/) (2022)

## Built on by (most cited first, 26 of 26)

- [Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products](https://mlsanity.com/paper/2401.10216/) (2024)
- [Enhancing the Scalability and Applicability of Kohn-Sham Hamiltonians for Molecular Systems](https://mlsanity.com/paper/2502.19227/) (2025)
- [Self-Consistency Training for Density-Functional-Theory Hamiltonian Prediction](https://mlsanity.com/paper/2403.09560/) (2024)
- [Efficient and Scalable Density Functional Theory Hamiltonian Prediction through Adaptive Sparsity](https://mlsanity.com/paper/2502.01171/) (2025)
- [High-order Equivariant Flow Matching for Density Functional Theory Hamiltonian Prediction](https://mlsanity.com/paper/2505.18817/) (2025)
- [Autonomous Agents for Scientific Discovery: Orchestrating Scientists, Language, Code, and Physics](https://mlsanity.com/paper/2510.09901/) (2025)
- [DA-MoE: Addressing Depth-Sensitivity in Graph-Level Analysis through Mixture of Experts](https://mlsanity.com/paper/2411.03025/) (2024)
- [Learning from the electronic structure of molecules across the periodic table](https://mlsanity.com/paper/2510.00224/) (2025)
- [Infusing Self-Consistency into Density Functional Theory Hamiltonian Prediction via Deep Equilibrium Models](https://mlsanity.com/paper/2406.03794/) (2024)
- [Equivariant Graph Network Approximations of High-Degree Polynomials for Force Field Prediction](https://mlsanity.com/paper/2411.04219/) (2024)
- [Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Frames](https://mlsanity.com/paper/2506.09398/) (2025)
- [Reducing the Cost of Quantum Chemical Data By Backpropagating Through Density Functional Theory](https://mlsanity.com/paper/2402.04030/) (2024)
- [TraceGrad: a Framework Learning Expressive SO(3)-equivariant Non-linear Representations for Electronic-Structure Hamiltonian Prediction](https://mlsanity.com/paper/2405.05722/) (2024)
- [Towards A Transferable Acceleration Method for Density Functional Theory](https://mlsanity.com/paper/2509.25724/) (2025)
- [Transferable SCF-Acceleration through Solver-Aligned Initialization Learning](https://mlsanity.com/paper/2604.21657/) (2026)
- [Self-consistent Validation for Machine Learning Electronic Structure](https://mlsanity.com/paper/2402.10186/) (2024)
- [nabla2DFT: A Universal Quantum Chemistry Dataset of Drug-Like Molecules and a Benchmark for Neural Network Potentials](https://mlsanity.com/paper/2406.14347/) (2024)
- [EDBench: Large-Scale Electron Density Data for Molecular Modeling](https://mlsanity.com/paper/2505.09262/) (2025)
- [TITAN: A Trajectory-Informed Technique for Adaptive Parameter Freezing in Large-Scale VQE](https://mlsanity.com/paper/2509.15193/) (2025)
- [Advancing Universal Deep Learning for Electronic-Structure Hamiltonian Prediction of Materials](https://mlsanity.com/paper/2509.19877/) (2025)
- [Inductive Graph Representation Learning with Quantum Graph Neural Networks](https://mlsanity.com/paper/2503.24111/) (2025)
- [How Far Can You Grow? Characterizing the Extrapolation Frontier of Graph Generative Models for Materials Science](https://mlsanity.com/paper/2602.09309/) (2026)
- [Neural Polarization: Toward Electron Density for Molecules by Extending Equivariant Networks](https://mlsanity.com/paper/2406.00441/) (2024)
- [A Unified Benchmark for Evaluating Knowledge Graph Construction Methods and Graph Neural Networks](https://mlsanity.com/paper/2605.05476/) (2026)
- [A Fixed-Point Neural Operator for Size- and Functional-Transferable Hamiltonian Prediction](https://mlsanity.com/paper/2606.14498/) (2026)
- [MALOQ: Massively Accelerated Learning of Operators for Quantum Transport](https://mlsanity.com/paper/2606.28911/) (2026)

## Abstract (arXiv)

Supervised machine learning approaches have been increasingly used in accelerating electronic structure prediction as surrogates of first-principle computational methods, such as density functional theory (DFT). While numerous quantum chemistry datasets focus on chemical properties and atomic forces, the ability to achieve accurate and efficient prediction of the Hamiltonian matrix is highly desired, as it is the most important and fundamental physical quantity that determines the quantum states of physical systems and chemical properties. In this work, we generate a new Quantum Hamiltonian dataset, named as QH9, to provide precise Hamiltonian matrices for 999 or 2998 molecular dynamics trajectories and 130,831 stable molecular geometries, based on the QM9 dataset. By designing benchmark tasks with various molecules, we show that current machine learning models have the capacity to predict Hamiltonian matrices for arbitrary molecules. Both the QH9 dataset and the baseline models are provided to the community through an open-source benchmark, which can be highly valuable for developing machine learning methods and accelerating molecular and materials design for scientific and technological applications. Our benchmark is publicly available at https://github.com/divelab/AIRS/tree/main/OpenDFT/QHBench.