Machine Learning Difference Charge Density

TL;DR

The paper introduces Δ-SAED, a Δ-learning approach that trains neural networks to predict the difference charge density $ρ_d = ρ_t - ρ_a$, where $ρ_a$ is the superposition of atomic densities (SAED). By embedding physical priors through SAED, Δ-SAED improves charge-density predictions across multiple benchmarks and enhances transferability to non-self-consistent DFT properties, including Si allotropes, often with simpler core-region radial/angular dependencies. The results show significant reductions in $ε_{mae}$ and high rates of improvement across datasets, supporting the method's robustness and potential for cross-DFT-code foundation modeling. The work also discusses the broader implications for charge-density modeling and provides datasets and model parameters for reproducibility.

Abstract

In density functional theory (DFT), the ground state charge density is the fundamental variable which determines all other ground state properties. Many machine learning charge density models are developed by prior efforts, which have been proven useful to accelerate DFT calculations. Yet they all use the total charge density (TCD) as the training target. In this work, we advocate predicting difference charge density (DCD) instead. We term this simple technique by $Δ$-SAED, which leverages the prior physical information of superposition of atomic electron densities (SAED). The robustness of $Δ$-SAED is demonstrated through evaluations over diverse benchmark datasets, showing an extra accuracy gain for more than 90% structures in the test sets. Using a Si allotropy dataset, $Δ$-SAED is demonstrated to advance model's transferability to chemical accuracy for non-self-consistent calculations. By incorporating physical priors to compensate for the limited expressive power of machine learning models, $Δ$-SAED offers a cost-free yet robust approach to improving charge density prediction and enhancing non-self-consistent performance.

Figures (5)

• Figure 1: Workflow of $\Delta$-SAED. Starting from a structure specified by atomic numbers and coordinates $\{(\mathbf{r}_i, Z_i)\}$, green arrows represent dataset preparation, blue arrows represent predictions using neural network $\mathcal{F}$ with parameters $\mathcal{W}$, and red arrows represent network training with the loss function $\mathcal{L}(\rho_d - \hat{\rho}_d)$.
• Figure 2: $\Delta$-SAED performance on benchmark datasets with $\Delta_{rel}\varepsilon_{mae}$ distributions across test structures. (a) Distributions of $\Delta_{rel}\varepsilon_{mae}$ via violin plots for the NMC, QM9, MP test set. (b) $\Delta_{rel}\varepsilon_{mae}$ distribution for metals and insulators in the MP test set.
• Figure 3: MAEs of per-atom energies (a), forces (b), band gaps (c), band energies (d) and charge density (e) for 18 Si allotropes from Materials Project to evaluate transferability of the Si DCD and TCD models. The MP ids are shown in the right box. Blue bars correspond to non-self-consistent calculations derived from $\hat{\rho_d} + \rho_a$, while green bars from $\hat{\rho_t}$. All MAEs use self-consistent results as the ground truth. The red lines are chemical accuracies, with 1 meV/atom for per-atom energy, 30 meV/Å for force, and 43 meV for band energy and band gap.
• Figure 4: Radial distributions of density functions for 4 Si allotropes in Fig. \ref{['fig:mae_compare_nscf']}. Blue denotes the target density, red denotes the absolute density error, and darker regions indicate higher grid density. TCD refers to total charge density, with $\rho_t$ shown in blue and $|\rho_t-\hat{\rho}_t|$ in red, while DCD refers to difference charge density, with $\rho_d$ shown in blue and $|\rho_d-\hat{\rho}_d|$ in red.
• Figure 5: Per-atom $\varepsilon_{mae}$ and the count of each element type of the MP test set.