# Transferable E(3) equivariant parameterization for Hamiltonian of molecules and solids

arXiv: https://arxiv.org/abs/2210.16190 · HTML: https://mlsanity.com/paper/2210.16190/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 85 references that are in the map.
- First posted on arXiv 28 Oct 2022.

## Builds on (most cited first, 14 of 14)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [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)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Crystal Diffusion Variational Autoencoder for Periodic Material Generation](https://mlsanity.com/paper/2110.06197/) (2021)
- [GeoMol: Torsional Geometric Generation of Molecular 3D Conformer Ensembles](https://mlsanity.com/paper/2106.07802/) (2021)
- [SpookyNet: Learning Force Fields with Electronic Degrees of Freedom and Nonlocal Effects](https://mlsanity.com/paper/2105.00304/) (2021)
- [Unifying machine learning and quantum chemistry -- a deep neural network for molecular wavefunctions](https://mlsanity.com/paper/1906.10033/) (2019)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [Equivariant representations for molecular Hamiltonians and N-center atomic-scale properties](https://mlsanity.com/paper/2109.12083/) (2021)
- [Heterogeneous relational message passing networks for molecular dynamics simulations](https://mlsanity.com/paper/2109.00711/) (2021)

## Built on by (most cited first, 8 of 8)

- [Enhancing the Scalability and Applicability of Kohn-Sham Hamiltonians for Molecular Systems](https://mlsanity.com/paper/2502.19227/) (2025)
- [Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Frames](https://mlsanity.com/paper/2506.09398/) (2025)
- [Universal Machine Learning Kohn-Sham Hamiltonian for Materials](https://mlsanity.com/paper/2402.09251/) (2024)
- [General time-reversal equivariant neural network potential for magnetic materials](https://mlsanity.com/paper/2211.11403/) (2022)
- [nabla2DFT: A Universal Quantum Chemistry Dataset of Drug-Like Molecules and a Benchmark for Neural Network Potentials](https://mlsanity.com/paper/2406.14347/) (2024)
- [Deep Learning Accelerated Quantum Transport Simulations in Nanoelectronics: From Break Junctions to Field-Effect Transistors](https://mlsanity.com/paper/2411.08800/) (2024)
- [Graph Transformer Networks for Accurate Band Structure Prediction: An End-to-End Approach](https://mlsanity.com/paper/2411.16483/) (2024)
- [Equivariant Flow Matching for Electron Density Prediction](https://mlsanity.com/paper/2610.02651/) (2026)

## Abstract (arXiv)

Using the message-passing mechanism in machine learning (ML) instead of self-consistent iterations to directly build the mapping from structures to electronic Hamiltonian matrices will greatly improve the efficiency of density functional theory (DFT) calculations. In this work, we proposed a general analytic Hamiltonian representation in an E(3) equivariant framework, which can fit the ab initio Hamiltonian of molecules and solids by a complete data-driven method and are equivariant under rotation, space inversion, and time reversal operations. Our model reached state-of-the-art precision in the benchmark test and accurately predicted the electronic Hamiltonian matrices and related properties of various periodic and aperiodic systems, showing high transferability and generalization ability. This framework provides a general transferable model that can be used to accelerate the electronic structure calculations on different large systems with the same network weights trained on small structures.