A Transferable Machine-Learning Model of the Electron Density

TL;DR

This work tackles the computational bottleneck of obtaining the ground-state electron density by introducing a transferable, local ML model that predicts the valence density from atomic coordinates. It combines an atom-centered basis with symmetry-adapted Gaussian Process Regression to learn density components in a rotation-covariant manner, achieving linear-scaling predictions that transfer from small hydrocarbons (C2, C4) to larger ones (C8). The approach attains ~1% density accuracy and meaningful XC-energy predictions, with demonstrated transferability via extrapolation from butadiene/butane to octatetraene/octane. The framework offers a path to faster initialization and interpretation of electronic structure calculations and density-based fingerprints, with room for improvements in basis optimization and self-consistent extensions.

Abstract

The electronic charge density plays a central role in determining the behavior of matter at the atomic scale, but its computational evaluation requires demanding electronic-structure calculations. We introduce an atom-centered, symmetry-adapted framework to machine-learn the valence charge density based on a small number of reference calculations. The model is highly transferable, meaning it can be trained on electronic-structure data of small molecules and used to predict the charge density of larger compounds with low, linear-scaling cost. Applications are shown for various hydrocarbon molecules of increasing complexity and flexibility, and demonstrate the accuracy of the model when predicting the density on octane and octatetraene after training exclusively on butane and butadiene. This transferable, data-driven model can be used to interpret experiments, initialize electronic structure calculations, and compute electrostatic interactions in molecules and condensed-phase systems.

Figures (4)

• Figure 1: Density errors at different level of representation: (left) superposition of isolated atomic densities, (right) optimized basis set. Red and blue isosurfaces refer to an error of $\pm$0.005 Bohr$^{-3}$ respectively. The density errors for the structure depicted are reported in the two panels, while the table reports the mean errors over the whole training set for the C2 and C4 molecules.
• Figure 2: (top) representation of the angular momentum decomposition of the electron density. Red and blue isosurfaces refer to $\pm 0.01$ Bohr$^{-3}$ respectively. (bottom) angular momentum spectrum of the valence electron density of C$_2$ and C$_4$ datasets. The isotropic contributions $l=0$ express the collective variations with respect to the dataset's mean value, while the mean is statistically zero for $l>0$.
• Figure 3: Learning curves for C$_2$ and C$_4$ molecules. (left) % mean absolute error of the predicted SA-GPR densities as a function of the number of training molecules. The error normalization is provided by the total number of valence electrons. (right) root mean square errors of the exchange-correlation energies indirectly predicted from the SA-GPR densities and directly predicted via a scalar SOAP kernel, as a function of the number of training molecules. Dashed lines refer to the error carried by the basis set representation.
• Figure 4: Extrapolation results for the valence electron density of one octane (left) and one octatetraene (right) conformer. (top) DFT/PBE density isosurface at 0.25, 0.1, 0.01 Bohr$^{-3}$, (middle) machine-learning prediction isosurface at 0.25, 0.1, 0.01 Bohr$^{-3}$, (bottom) machine-learning error, red and blue isosurfaces refer to $\pm$ 0.005 Bohr$^{-3}$ respectively. Relative mean absolute errors averaged over 100 conformers are also reported for both cases.