# A Recipe for Charge Density Prediction

arXiv: https://arxiv.org/abs/2405.19276 · HTML: https://mlsanity.com/paper/2405.19276/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 69 references that are in the map.
- First posted on arXiv 29 May 2024.

## Builds on (most cited first, 29 of 29)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (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)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [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)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [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)
- [The Open Catalyst 2020 (OC20) Dataset and Community Challenges](https://mlsanity.com/paper/2010.09990/) (2020)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [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)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [Forces are not Enough: Benchmark and Critical Evaluation for Machine Learning Force Fields with Molecular Simulations](https://mlsanity.com/paper/2210.07237/) (2022)
- [A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systems](https://mlsanity.com/paper/2312.07511/) (2023)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [Spherical Channels for Modeling Atomic Interactions](https://mlsanity.com/paper/2206.14331/) (2022)
- [Informing Geometric Deep Learning with Electronic Interactions to Accelerate Quantum Chemistry](https://mlsanity.com/paper/2105.14655/) (2021)
- [Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids](https://mlsanity.com/paper/2112.00652/) (2021)
- [Symphony: Symmetry-Equivariant Point-Centered Spherical Harmonics for 3D Molecule Generation](https://mlsanity.com/paper/2311.16199/) (2023)
- [EGraFFBench: Evaluation of Equivariant Graph Neural Network Force Fields for Atomistic Simulations](https://mlsanity.com/paper/2310.02428/) (2023)
- [Higher-Order Equivariant Neural Networks for Charge Density Prediction in Materials](https://mlsanity.com/paper/2312.05388/) (2023)
- [Equivariant Neural Operator Learning with Graphon Convolution](https://mlsanity.com/paper/2311.10908/) (2023)
- [Gaussian Plane-Wave Neural Operator for Electron Density Estimation](https://mlsanity.com/paper/2402.04278/) (2024)
- [Towards Combinatorial Generalization for Catalysts: A Kohn-Sham Charge-Density Approach](https://mlsanity.com/paper/2310.18702/) (2023)
- [Equivariant Scalar Fields for Molecular Docking with Fast Fourier Transforms](https://mlsanity.com/paper/2312.04323/) (2023)

## Built on by (most cited first, 9 of 9)

- [ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals](https://mlsanity.com/paper/2503.08305/) (2025)
- [Towards A Transferable Acceleration Method for Density Functional Theory](https://mlsanity.com/paper/2509.25724/) (2025)
- [Global Plane Waves From Local Gaussians: Periodic Charge Densities in a Blink](https://mlsanity.com/paper/2601.19966/) (2026)
- [ED-DiT: Physics-Guided Diffusion Pretraining for Transferable Molecular Representations from Electron Density](https://mlsanity.com/paper/2608.03260/) (2026)
- [E3STO: Orbital Inspired SE(3)-Equivariant Molecular Representation for Electron Density Prediction](https://mlsanity.com/paper/2410.06119/) (2024)
- [Multitask learning with semiempirical orbital charges enables sample-efficient MLIPs](https://mlsanity.com/paper/2605.24073/) (2026)
- [Neural-Network Solutions to Real-Space Charge Density and Generalization](https://mlsanity.com/paper/2609.14906/) (2026)
- [Complete Neural Electronic Initialization Accelerates Materials DFT](https://mlsanity.com/paper/2609.21759/) (2026)
- [Equivariant Flow Matching for Electron Density Prediction](https://mlsanity.com/paper/2610.02651/) (2026)

## Abstract (arXiv)

In density functional theory, charge density is the core attribute of atomic systems from which all chemical properties can be derived. Machine learning methods are promising in significantly accelerating charge density prediction, yet existing approaches either lack accuracy or scalability. We propose a recipe that can achieve both. In particular, we identify three key ingredients: (1) representing the charge density with atomic and virtual orbitals (spherical fields centered at atom/virtual coordinates); (2) using expressive and learnable orbital basis sets (basis function for the spherical fields); and (3) using high-capacity equivariant neural network architecture. Our method achieves state-of-the-art accuracy while being more than an order of magnitude faster than existing methods. Furthermore, our method enables flexible efficiency-accuracy trade-offs by adjusting the model/basis sizes.