# SE(3)-equivariant prediction of molecular wavefunctions and electronic densities

arXiv: https://arxiv.org/abs/2106.02347 · HTML: https://mlsanity.com/paper/2106.02347/

## Key facts

- Cited by 67 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 74 references that are in the map.
- First posted on arXiv 4 Jun 2021.

## Builds on (most cited first, 30 of 34)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [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)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [SpookyNet: Learning Force Fields with Electronic Degrees of Freedom and Nonlocal Effects](https://mlsanity.com/paper/2105.00304/) (2021)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [Unifying machine learning and quantum chemistry -- a deep neural network for molecular wavefunctions](https://mlsanity.com/paper/1906.10033/) (2019)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [OrbNet: Deep Learning for Quantum Chemistry Using Symmetry-Adapted Atomic-Orbital Features](https://mlsanity.com/paper/2007.08026/) (2020)
- [Covariant Compositional Networks For Learning Graphs](https://mlsanity.com/paper/1801.02144/) (2018)

## Built on by (most cited first, 30 of 67)

- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [Graph neural networks for materials science and chemistry](https://mlsanity.com/paper/2208.09481/) (2022)
- [Geometric Deep Learning on Molecular Representations](https://mlsanity.com/paper/2107.12375/) (2021)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Structure-based drug design with geometric deep learning](https://mlsanity.com/paper/2210.11250/) (2022)
- [So3krates: Equivariant attention for interactions on arbitrary length-scales in molecular systems](https://mlsanity.com/paper/2205.14276/) (2022)
- [Efficient and Equivariant Graph Networks for Predicting Quantum Hamiltonian](https://mlsanity.com/paper/2306.04922/) (2023)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products](https://mlsanity.com/paper/2401.10216/) (2024)
- [SchNetPack 2.0: A neural network toolbox for atomistic machine learning](https://mlsanity.com/paper/2212.05517/) (2022)
- [QH9: A Quantum Hamiltonian Prediction Benchmark for QM9 Molecules](https://mlsanity.com/paper/2306.09549/) (2023)
- [Generative Coarse-Graining of Molecular Conformations](https://mlsanity.com/paper/2201.12176/) (2022)
- [Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency](https://mlsanity.com/paper/2406.07598/) (2024)
- [Generalizing Denoising to Non-Equilibrium Structures Improves Equivariant Force Fields](https://mlsanity.com/paper/2403.09549/) (2024)
- [Enhancing the Scalability and Applicability of Kohn-Sham Hamiltonians for Molecular Systems](https://mlsanity.com/paper/2502.19227/) (2025)
- [Higher-Order Equivariant Neural Networks for Charge Density Prediction in Materials](https://mlsanity.com/paper/2312.05388/) (2023)
- [Self-Consistency Training for Density-Functional-Theory Hamiltonian Prediction](https://mlsanity.com/paper/2403.09560/) (2024)
- [Euclidean Fast Attention -- Machine Learning Global Atomic Representations at Linear Cost](https://mlsanity.com/paper/2412.08541/) (2024)
- [HIP: Hessian Interatomic Potentials without derivatives](https://mlsanity.com/paper/2509.21624/) (2025)
- [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)
- [EquiformerV3: Scaling Efficient, Expressive, and General SE(3)-Equivariant Graph Attention Transformers](https://mlsanity.com/paper/2604.09130/) (2026)
- [Knowledge-augmented Deep Learning and Its Applications: A Survey](https://mlsanity.com/paper/2212.00017/) (2022)
- [How simple can you go? An off-the-shelf transformer approach to molecular dynamics](https://mlsanity.com/paper/2503.01431/) (2025)
- [ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals](https://mlsanity.com/paper/2503.08305/) (2025)
- [Accurate Machine Learned Quantum-Mechanical Force Fields for Biomolecular Simulations](https://mlsanity.com/paper/2205.08306/) (2022)
- [Variational Monte Carlo on a Budget -- Fine-tuning pre-trained Neural Wavefunctions](https://mlsanity.com/paper/2307.09337/) (2023)
- [Transferable E(3) equivariant parameterization for Hamiltonian of molecules and solids](https://mlsanity.com/paper/2210.16190/) (2022)
- [Learning Harmonic Molecular Representations on Riemannian Manifold](https://mlsanity.com/paper/2303.15520/) (2023)

## Abstract (arXiv)

Machine learning has enabled the prediction of quantum chemical properties with high accuracy and efficiency, allowing to bypass computationally costly ab initio calculations. Instead of training on a fixed set of properties, more recent approaches attempt to learn the electronic wavefunction (or density) as a central quantity of atomistic systems, from which all other observables can be derived. This is complicated by the fact that wavefunctions transform non-trivially under molecular rotations, which makes them a challenging prediction target. To solve this issue, we introduce general SE(3)-equivariant operations and building blocks for constructing deep learning architectures for geometric point cloud data and apply them to reconstruct wavefunctions of atomistic systems with unprecedented accuracy. Our model achieves speedups of over three orders of magnitude compared to ab initio methods and reduces prediction errors by up to two orders of magnitude compared to the previous state-of-the-art. This accuracy makes it possible to derive properties such as energies and forces directly from the wavefunction in an end-to-end manner. We demonstrate the potential of our approach in a transfer learning application, where a model trained on low accuracy reference wavefunctions implicitly learns to correct for electronic many-body interactions from observables computed at a higher level of theory. Such machine-learned wavefunction surrogates pave the way towards novel semi-empirical methods, offering resolution at an electronic level while drastically decreasing computational cost. Additionally, the predicted wavefunctions can serve as initial guess in conventional ab initio methods, decreasing the number of iterations required to arrive at a converged solution, thus leading to significant speedups without any loss of accuracy or robustness.