MPI - CPfS

Original definition of ELF

The electron localization function (ELF) was introduced by Becke and Edgecombe as a "simple measure of electron localization in atomic and molecular systems" [BECKE1990]. The original formula is based on the Taylor expansion of the spherically averaged conditional same-spin pair probability density to find an electron close to a same-spin reference electron. The main aspect of this formulation is that thus defined ELF is a property of the same-spin pair density

The same-spin pair probability density  P2(r, r') is the probability density to simulaneously find two like-spin electrons at positions r and r'. In Hartree-Fock (HF) approximation: 

P2(r, r')  =   &rho(r) ρ(r')  −    |ρ1(r, r')|²

The conditional same-spin pair probability density  Pcond(r, r') is the probability density to find an electron at some position r' if a like-spin reference electron is located with certainty at position r. In Hartree-Fock (HF) approximation: 

Pcond(r, r')  =   ρ(r')  −    |ρ1(r, r')|² ⁄ ρ(r)

with the electron densities  ρ(r) and  ρ(r'), and the σ-spin one-particle density matrix  ρ1(r, r') of the HF determinant: 

ρ1(r, r')  =   ∑iσ ψi*(r'i(r)

where the summation runs over all occupied  σ-spin (i.e. either up or down spin) orbitals  ψi(r).

The probability density to find a like-spin electron at a distance s from the reference point  r can be found by a Taylor expansion of the spherically averaged conditional same-spin probability density  Pcond(r, s) (the spherical average is on a shell of radius s around the reference point  r). The first (s independent) term of the Taylor expansion vanishes, because the conditional probability density to find two like-spin electrons at the same position  r is, as a direct consequence of the Pauli principle, equal to zero. The linear term is dependent on the gradient of the HF Fermi hole at  r - thus it vanishes as well. The leading (quadratical) term of the Taylor expansion of the spherically averaged conditional same-spin probability density is [BECKE1990]: 

Pcond(r, s)  =   1⁄3 [iσ |∇ψi(r)|²   −   ¼ |∇ρ(r)|² ⁄ ρ(r)] s²   +   ...

The expression in the brackets is besides a ρ factor proportional to the Fermi hole mobility function of Luken and Culberson [LUKEN1982] and is related to the curvature of the HF Fermi hole at  r as shown by Dobson [DOBSON1991]

Becke and Edgecombe associated the localization of an electron with the probability density to find a second like-spin electron near the reference point. The smaller this probability density, i.e. the smaller the expression 

D(r)   =   ∑iσ |∇ψi(r)|²   −   ¼ |∇ρ(r)|² ⁄ ρ(r)

of the quadratic term, the higher localized an electron is. Thus, the Pauli repulsion between two like-spin electrons, described by the smallness of  D(r), is taken as a measure of the electron localization. Using the corresponding factor found for uniform electron gas  Dh(r) Becke and Edgecombe defined ELF as follows: 

η(r)  =   1 ⁄ [1 + χ²BE(r)]

with 

χBE(r)  =   D(r) ⁄ Dh(r)

where 

Dh(r)  =   3/5 (6π²)2/3 ρ(r)5/3

Given by the definition, ELF values are bound between 0 and 1. 

In the seminal paper of Becke and Edgecombe the ratio  χBE(r) was attributed to a dimensionless localization index calibrated with respect to the uniform electron gas as a reference. Nevertheless, it should be mentioned that this reference was chosen arbitrarily (originally, Luken and Culberson had defined a function similar to χBE(r), but instead of a division they preferred a subtraction, again arbitrarily choosing the uniform electron gas as a reference). The only measure of the electron localization, as described by the two authors, is the expression  D(r). However, ELF cannot yield the value of  D(r) - i.e. the actual measure of the electron localization - because it depends, through  Dh(r), on the electron density as well. In this sense, ELF is a relative measure of the electron localization. 

High ELF values show that at the examined position the electrons are more localized than in a uniform electron gas of the same density.  η(r) = 1⁄2 indicates that the effect of the Pauli repulsion is the same as in the uniform electron gas of the same density. Of course, it cannot be compared with the uniform electron gas with respect to other properties (it is obvious that the electron density gradient in an atom, molecule or solid differs from zero almost everywhere). See also section How to interpret.