Importance of Electronic Entropy for Machine Learning Interatomic Potentials

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.