# Highly Accurate Real-space Electron Densities with Neural Networks

arXiv: https://arxiv.org/abs/2409.01306 · HTML: https://mlsanity.com/paper/2409.01306/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 69 references that are in the map.
- First posted on arXiv 2 Sep 2024.

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

- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [A Self-Attention Ansatz for Ab-initio Quantum Chemistry](https://mlsanity.com/paper/2211.13672/) (2022)
- [Learning deep kernels for exponential family densities](https://mlsanity.com/paper/1811.08357/) (2018)
- [Ab-initio quantum chemistry with neural-network wavefunctions](https://mlsanity.com/paper/2208.12590/) (2022)
- [Better, Faster Fermionic Neural Networks](https://mlsanity.com/paper/2011.07125/) (2020)
- [Gold-standard solutions to the Schr"odinger equation using deep learning: How much physics do we need?](https://mlsanity.com/paper/2205.09438/) (2022)
- [Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation](https://mlsanity.com/paper/2112.03491/) (2021)
- [The Optimal Noise in Noise-Contrastive Learning Is Not What You Think](https://mlsanity.com/paper/2203.01110/) (2022)
- [Neural Pfaffians: Solving Many Many-Electron Schr"odinger Equations](https://mlsanity.com/paper/2405.14762/) (2024)

## Built on by (most cited first, 4 of 4)

- [An ab initio foundation model of wavefunctions that accurately describes chemical bond breaking](https://mlsanity.com/paper/2506.19960/) (2025)
- [Accurate and scalable exchange-correlation with deep learning](https://mlsanity.com/paper/2506.14665/) (2025)
- [Learning Equivariant Non-Local Electron Density Functionals](https://mlsanity.com/paper/2410.07972/) (2024)
- [Excited Pfaffians: Generalized Neural Wave Functions Across Structure and State](https://mlsanity.com/paper/2603.14515/) (2026)

## Abstract (arXiv)

Variational ab-initio methods in quantum chemistry stand out among other methods in providing direct access to the wave function. This allows in principle straightforward extraction of any other observable of interest, besides the energy, but in practice this extraction is often technically difficult and computationally impractical. Here, we consider the electron density as a central observable in quantum chemistry and introduce a novel method to obtain accurate densities from real-space many-electron wave functions by representing the density with a neural network that captures known asymptotic properties and is trained from the wave function by score matching and noise-contrastive estimation. We use variational quantum Monte Carlo with deep-learning ans"atze (deep QMC) to obtain highly accurate wave functions free of basis set errors, and from them, using our novel method, correspondingly accurate electron densities, which we demonstrate by calculating dipole moments, nuclear forces, contact densities, and other density-based properties.