Charge density-supervised real-space representations for atomistic energy prediction

Ground-state electron density contains sufficient information to determine ground-state energy, while providing spatial electronic information. This study developed a charge-density-informed graph neural network (hereafter CDGNN) that jointly learns energy and valence electron density from equivariant atomic graphs. This approach incorporates density-supervised real-space probe representations into the global energy regressor, allowing density-related spatial information to contribute to energy prediction. On the QM9 dataset, CDGNN reduced the energy prediction errors relative to an equivariant GNN baseline for interpolation and extrapolation to larger molecules. For small H-containing Fe and Fe-C datasets, CDGNN performed similarly to the baseline, indicating the dataset-dependent benefits of density supervision. CDGNN also accurately predicted the charge density distributions for molecular and Fe-based systems. The attention analysis showed higher weights for probe representations near the H atoms in Fe-based structures. These findings highlight the potential of integrating charge-density information for energy prediction and its task-dependent benefits.