Counting electrons in electrides

The wave nature of electrons makes the quantification of charge fundamentally challenging. In complex materials like electrides, this challenge is amplified by the small charge and complex shape of electride wavefunctions. For these reasons, popular integration methods like the Bader method usually fail to assign any charge to the bare electrons in an electride. To address this challenge, we develop an algorithm that instead partitions the charge based on the electron localization function (ELF), a popular scheme for visualizing chemically important features in molecules and solids. The algorithm uses Bader segmentation of the ELF to find the electride electrons and Voronoi segmentation of the ELF to identify atoms. We apply this method, “BadELF”, to the quantification of atomic radii and oxidation states in both ionic compounds and electrides. For ionic compounds, we find that the BadELF method yields radii that agree closely with Shannon crystal radii, while the oxidation states agree closely with the Bader method. When applied to electrides, however, only the BadELF algorithm yields chemically meaningful charges. We argue that the BadELF method provides a useful strategy to identify electrides and obtain new insight about their most essential property: the quantity of electrons within them.