Meaning and Functional Form of the Electron Localization Function

The electron localization function (ELF) is an important tool to study electronic structure. In this article, the concept of electron localization is introduced and the overlap between the physical meaning and functional form of the ELF is discussed in detail from two points of view: electron pair density and kinetic energy density. Additionally, we extend the ELF to a generalized spin-polarized form. From case studies we found that the reference term plays an important role in the ELF. A comparison between the two spin-polarized forms of ELF shows that the form used in the CheckDen and TopMoD programs is unreasonable as they obviously underestimate the degree of localization in the single-electron region. Finally, we point out some mistakes in the literature because of a misunderstanding of the ELF.