Learning Equivariant Non-Local Electron Density Functionals (2410.07972v3)

Abstract: The accuracy of density functional theory hinges on the approximation of non-local contributions to the exchange-correlation (XC) functional. To date, machine-learned and human-designed approximations suffer from insufficient accuracy, limited scalability, or dependence on costly reference data. To address these issues, we introduce Equivariant Graph Exchange Correlation (EG-XC), a novel non-local XC functional based on equivariant graph neural networks (GNNs). Where previous works relied on semi-local functionals or fixed-size descriptors of the density, we compress the electron density into an SO(3)-equivariant nuclei-centered point cloud for efficient non-local atomic-range interactions. By applying an equivariant GNN on this point cloud, we capture molecular-range interactions in a scalable and accurate manner. To train EG-XC, we differentiate through a self-consistent field solver requiring only energy targets. In our empirical evaluation, we find EG-XC to accurately reconstruct `gold-standard' CCSD(T) energies on MD17. On out-of-distribution conformations of 3BPA, EG-XC reduces the relative MAE by 35% to 50%. Remarkably, EG-XC excels in data efficiency and molecular size extrapolation on QM9, matching force fields trained on 5 times more and larger molecules. On identical training sets, EG-XC yields on average 51% lower MAEs.

• The paper introduces the EG-XC functional, which leverages equivariant graph neural networks to improve the modeling of non-local exchange-correlation effects in DFT.
• It utilizes nuclei-centered equivariant embeddings and message passing to effectively capture long-range electron interactions while reducing computational complexity.
• Empirical results on datasets like MD17, 3BPA, and QM9 demonstrate significant error reductions, enhanced extrapolation, and superior data efficiency compared to traditional methods.

Learning Equivariant Non-Local Electron Density Functionals

This paper presents a novel approach to enhancing the accuracy of Density Functional Theory (DFT) through the introduction of the Equivariant Graph Exchange Correlation (EG-XC) functional. The research addresses key challenges associated with existing DFT methodologies, particularly the approximation of non-local contributions to the exchange-correlation (XC) functional.

Problem Overview

The central problem in DFT is the approximation of the exchange-correlation functional, which significantly impacts its accuracy. Traditional machine-learned and manually designed approximations often encounter issues such as limited accuracy, scalability, and reliance on expensive reference data. This paper introduces a new method utilizing Equivariant Graph Neural Networks (GNNs) to construct an advanced non-local XC functional.

Methodology

EG-XC integrates semi-local functionals with a non-local feature density, which is parameterized through an equivariant nuclei-centered point cloud representation of the electron density. The method captures long-range interactions that are typically challenging for conventional approaches.

Key components of the methodology include:

1. Nuclei-Centered Equivariant Embeddings: The electron density is compressed into a finite point cloud centered around nuclei, which enables handling symmetry through equivariant features and reduces computational complexity.
2. Equivariant Message Passing: By leveraging message-passing operations in GNNs, the model effectively captures long-range dependencies within the electron density.
3. Non-Local Reweighted Meta-GGA: This component computes a non-local feature density to refine the meta-generalized gradient approximation (meta-GGA), thus enhancing the accuracy of XC energy density calculations.
4. Graph Readout: The use of a graph readout of the invariant features efficiently captures additional non-local interactions.

These components enable EG-XC to train on energy targets alone by differentiating through the self-consistent field (SCF) solver.

Empirical Evaluation

EG-XC demonstrates superior performance when compared to existing methods, including semi-local and fully non-local machine-learned functionals, traditional force fields, and $\Delta$-ML approaches:

• Interpolation Accuracy: On the MD17 dataset, which comprises molecular dynamics trajectories, EG-XC achieved substantial improvements, reducing errors by a factor of 2 to 3 compared to semi-local XC-functionals.
• Extrapolation Efficiency: When extrapolating to unseen molecular conformations and larger molecular systems, EG-XC maintained high accuracy, outperforming force fields particularly in the 3BPA dataset, where it reduced mean absolute errors significantly.
• Data Efficiency: On the QM9 dataset, EG-XC demonstrated remarkable data efficiency, often achieving accuracy levels comparable to competing methods while using significantly less data.

Practical and Theoretical Implications

EG-XC represents a promising advancement in computational chemistry and material science, providing a scalable, accurate approach for modeling complex molecular interactions. The integration of equivariant neural networks into DFT computations could potentially revolutionize how non-local interactions are modeled, contributing to more precise and computational-resource-efficient simulations.

Future Directions

Future research might explore further integration with other computational models, such as plane wave basis sets in periodic systems or orbital-free DFT applications. Additional improvements could be achieved through multimodal training involving not only energy targets but also electron densities and atomic forces. Incorporating physical constraints into the non-local component of EG-XC might further enhance its generalization and accuracy.

This work lays a solid foundation for employing advanced machine learning techniques within quantum chemistry, marking a step forward in achieving more accurate and efficient material and molecular simulations.