Electron localization function

In quantum chemistry, the electron localization function (ELF) is a method for the mapping of electron pair probability in multielectronic systems, defined by Becke and Combe in 1990. ELF's usefulness stems from the way it allows the analysis of electron localization in a "chemically intuitive" way. For example, the shell structure of heavy atoms is obvious when plotting ELF vs r; the ELF for radon, for example, has six clear maxima, whereas the electron density decreases almost monotonically and doesn't have such a clear structure. When applied to molecules, an analysis of the ELF shows a clear separation between the core and valence electron, and also shows covalent bonds and lone pairs, in what has been called "a faithful visualization of valence shell electron pair repulsion theory (VSEPR) in action". Another feature of ELF is that it is invariant with respect to transformation of the molecular orbitals.

ELF is defined as

$\mathrm {ELF} =(1+\chi _{\sigma }^{2})^{-1}$

where

$\chi _{\sigma }=D_{\sigma }/D_{\sigma }^{0}$,

$D_{\sigma }^{0}={\tfrac {3}{5}}(6\pi ^{2})^{2/3}\rho _{\sigma }^{5/3}$,

$D_{\sigma }=\tau _{\sigma }-{\frac {1}{4}}{\frac {(\nabla \rho _{\sigma })^{2}}{\rho _{\sigma }}}$,

and

$\tau _{\sigma }=\sum _{i}^{\sigma }{|\nabla \psi _{i}|^{2}}$,

where ρ is the electron density, τ is the kinetic energy density, D⁰ corresponds to D for a uniform electron gas with spin density equal to ρ(r). χ is a dimensionless localization index that expresses electron localization with respect to the uniform electron gas. ELF is defined such that its values are in the range 0 ≤ ELF ≤ 1, with ELF = 1 corresponding to perfect localization and ELF = ½ corresponding to the electron gas.

References

• A. D. Becke and K. E. Edgecombe. A simple measure of electron localization in atomic and molecular systems. J. Chem. Phys. 1990, 92, 5397-5403. doi:10.1063/1.458517