# Scaling Density Functional Theory with Gaussian Splatting

arXiv: https://arxiv.org/abs/2609.31483 · HTML: https://mlsanity.com/paper/2609.31483/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 93 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 18 of 18)

- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [OrbNet: Deep Learning for Quantum Chemistry Using Symmetry-Adapted Atomic-Orbital Features](https://mlsanity.com/paper/2007.08026/) (2020)
- [A practical guide to machine learning interatomic potentials -- Status and future](https://mlsanity.com/paper/2503.09814/) (2025)
- [DeePKS: a comprehensive data-driven approach towards chemically accurate density functional theory](https://mlsanity.com/paper/2008.00167/) (2020)
- [ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals](https://mlsanity.com/paper/2503.08305/) (2025)
- [DQC: a Python program package for Differentiable Quantum Chemistry](https://mlsanity.com/paper/2110.11678/) (2021)
- [D4FT: A Deep Learning Approach to Kohn-Sham Density Functional Theory](https://mlsanity.com/paper/2303.00399/) (2023)
- [MESS: Modern Electronic Structure Simulations](https://mlsanity.com/paper/2406.03121/) (2024)
- [Diagonalization without Diagonalization: A Direct Optimization Approach for Solid-State Density Functional Theory](https://mlsanity.com/paper/2411.05033/) (2024)
- [Self-Refining Training for Amortized Density Functional Theory](https://mlsanity.com/paper/2506.01225/) (2025)

## Abstract (arXiv)

Density functional theory (DFT) strikes a practical balance between accuracy and computational cost in many problems of computational chemistry and materials science. However, many DFT calculations are limited by fixed atom-centered basis sets, which dictate how accuracy and cost scale with system size. We propose Gaussian Splatting for Density Functional Theory (GS-DFT), which represents molecular orbitals as a cloud of Gaussians whose positions, shapes, and mixing coefficients are optimized jointly by gradient descent to minimize the energy without training data. Conceptually, GS-DFT is 3D Gaussian splatting with the renderer replaced by quantum mechanics. We introduce two key solver components: adaptive density fitting with screening for efficient evaluation of two-electron integrals, and a regularized differentiable orthogonalization of the molecular orbitals. Empirically, the optimized basis reaches the accuracy of the largest conventional basis sets with a fraction of the parameters, converging systematically in energy, density, and nuclear forces. At equal parameter count, it captures the stretched-bond and anion physics that fixed bases only recover with specialized basis augmentation. The resulting solver exhibits quadratic peak memory scaling in the cloud size, allowing us to simulate systems of up to 2,742 atoms (10,406 electrons) without any modifications at triple-zeta scale using a single four-GPU node.