The Local Virial Theorem in Electron Localization Function Partitions

Abstract The local form of the virial theorem provides a stringent condition for assessing whether a spatial region behaves as a mechanically closed subsystem in molecular space. While basins derived from the topology of the electron density satisfy this condition by construction, we show that ELF basins do not. The transformations leading to the ELF do not preserve the energetic scale and tensorial information required by the standard local virial relation, yielding nonzero residuals of 2G(Ωη) + V(Ωη) across all basin types and chemical environments examined. Despite this failure of local virial closure, the residual is not chemically arbitrary: across independent series of C–X bonds, it correlates strongly with the Intrinsic Bond Strength Index (IBSI) and increases systematically with formal bond order, tracking the electronic reorganization associated with stronger, shorter bonds. Furthermore, unlike the electron density, the ELF is not derived from the same variational framework associated with density-based energetic partitions. ELF basins are chemically meaningful localization patterns, but they should not be interpreted as mechanically closed energetic subsystems.