Importance of Electronic Entropy for Machine Learning Interatomic Potentials

Abstract

Machine learning interatomic potentials (MLIPs) enable large-scale atomistic simulations but remain challenged in describing mixed-valence materials where charge ordering strongly influences thermodynamic stability. Here we investigate the role of electronic entropy in MLIP structural optimization of the battery cathode material \ce{NaFePO4}. We show that conventional MLIPs fail to reproduce the correct stability of intermediate \ce{Na} concentrations because structural optimization leads to incorrect \ce{Fe^{2+}}/\ce{Fe^{3+}} charge assignments, resulting in erroneous energy ordering and convex-hull predictions. Analysis of magnetic moments during structural optimization reveals that MLIPs are unable to capture electronic entropy associated with charge ordering. To address this limitation, we introduce an approach that embeds charge-state information directly into the MLIP representation by distinguishing between \ce{Fe^{2+}} and \ce{Fe^{3+}} environments during training. Retraining CHGNet, cPaiNN, and MACE with this representation enables accurate structural optimization, correct identification of charge ordering, and improved agreement with density functional theory convex hulls. Our results demonstrate that incorporating electronic entropy into MLIP representations is essential for modeling charge-disordered materials and provide a practical framework for extending MLIP simulations to mixed-valence transition-metal systems.

Figures (9)

• Figure 1: The stable crystal structures of NaFePO4 at different concentration of sodium.
• Figure 2: a) the different GA generations and their steps. b) The total MACE MLIP driven GA convex hull based on all the GA generations c) DFT optimized structures based on the MACE MLIP driven GA convex hull
• Figure 3: The ten lowest-energy structures identified by the MACE-driven GA at 66% Na concentration, together with the experimentally verified phase used as a reference. Na atoms are shown as purple spheres, Fe as dark orange spheres, P as light orange spheres, and O as red spheres. Small red dots indicate Na vacancies.
• Figure 4: Distribution of DFT energies for 171 different Fe^2+/Fe^3+ arrangements in the orthorhombic Na_0.66FePO4 structure. The experimentally derived configuration is indicated with red and lies significantly below the mean of the sampled distribution.
• Figure 5: Comparison of Fe magnetic moments predicted by DFT and MLIPs for the eleven structures shown in \ref{['fig:GA_structures']}. (a) Magnetic moments of Fe atoms in the initial pre-optimized structures compared with those obtained after DFT structural optimization. The grey region highlights the range where the oxidation state assignment is ambiguous. (b) Magnetic moments of the pre-optimized structures predicted by CHGNet and cPaiNN compared with the corresponding DFT values. (c) Magnetic moments of the MLIP-optimized structures compared with those obtained from DFT optimization. (d) Magnetic moments from single-point MLIP calculations on the DFT-optimized structures compared with those obtained from DFT optimized structure.