Topological Descriptors for the Electron Density of Inorganic Solids

Descriptors play an important role in data-driven materials design. While most descriptors of crystalline materials emphasize structure and composition, they often neglect the electron density─a complex yet fundamental quantity that governs material properties. Here, we introduce Betti curves as topological descriptors that compress electron densities into compact representations. Derived from persistent homology, Betti curves capture bonding characteristics by encoding components, cycles, and voids across varied electron density thresholds. Machine learning models trained on Betti curves outperform those trained on raw electron densities by an average of 33 percentage points in classifying structure prototypes, predicting thermodynamic stability, and distinguishing metals from nonmetals. Shannon entropy calculations reveal that Betti curves retain comparable information content to electron density while requiring 2 orders of magnitude less data. By combining expressive power with compact representation, Betti curves highlight the potential of topological data analysis to advance materials design.