ELF Basins Have no Local Virial Theorem

The local virial theorem provides a rigorous criterion for identifying mechanically closed sub systems in molecular space. While basins derived from the topology of the electron density satisfy this condition by construction, we demonstrate that ELF basins do not. The chain of transformations defining the ELF irreversibly destroys the energetic scale and tensorial information required for a local virial relation, yielding non-zero residuals of 2G(Ω)+V(Ω) across all basin types and chemical environments examined. Furthermore, unlike the electron density, the ELF is not connected to any variational or energetic principle, which precludes the existence of an analogous relation for its basins. ELF basins are chemical patterns, not physical subsystems, and their integrated properties lack a transferable energetic identity.