A Chemically Meaningful Measure of Electron Localization

Electron localization and delocalization are commonly invoked in the day-to-day rationalization of chemistry. This work addresses the challenges of quantifying this elusive concept in a chemically useful manner. A general principle, requiring the simultaneous quantification of (1) a limited physical volume (classical criterion) and (2) same-spin loneliness (quantum criterion), is introduced. It is demonstrated how, by beginning with the Electron Localization Function (ELF) scalar field, one can choose to discard all points in space where the same-spin loneliness is lower than a certain value. Such a partitioning approach ensures that both criteria for quantifying localization (1 and 2) are simultaneously met. The most chemically instructive results arise when the dividing boundary condition is set by the local behavior of a homogeneous electron gas. The High Electron Localization domain Population (HELP) is introduced and applied for quantifying the localization of individual domains within molecules, as well as a measure of total electron localization in atoms and molecules. Several striking agreements with chemical intuition, experimental measurable quantities, and quantum chemical constructs are demonstrated along with understandable differences. Studies of diatomic molecules agree with current ideas on chemical bonding. The size-dependence and magnitude of localization in linear hydrocarbons is studied and compared to cyclic systems, such as benzene. The proposed methodology offers a straightforward measure for direct and quantitative comparisons between atoms, molecules, and extended condensed matter.