Distribution of Charge Centers in Matter from Geometric Phases of Electrons

We develop a formalism to partition the total electronic charge in matter, derived from the geometric phases of Bloch states. The first-moment matrices of the position operators along the three Cartesian directions do not commute, preventing a simultaneous definition of well-localized charge centers. To address this issue, we construct correlated hermaphrodite Wannier functions by tracing the evolution of Bloch electrons through the Brillouin zone, whose centers capture real-space charge localization with minimal spread. The resulting charge-center maps directly expose inter- and intra-atomic hybridization, revealing diverse bonding behaviors—from localized covalent to delocalized multicentered bonds—and provide a unified geometric framework for understanding chemical bonding in molecules and solids.