# 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" *property of the same-spin pair density*.

The same-spin pair probability density P_{2}(**r**, **r'**)
is the probability density to simulaneously find two like-spin electrons at positions **r** and **r'**.
In Hartree-Fock (HF) approximation:

_{2}(

**r**,

**r'**) = &rho(

**r**) ρ(

**r'**) − |ρ

_{1}(

**r**,

**r'**)|²

The conditional same-spin pair probability density P_{cond}(**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:

_{cond}(

**r**,

**r'**) = ρ(

**r'**) − |ρ

_{1}(

**r**,

**r'**)|² ⁄ ρ(

**r**)

with the electron densities ρ(**r**) and ρ(**r'**), and
the σ-spin one-particle density matrix ρ_{1}(**r**, **r'**)
of the HF determinant:

_{1}(

**r**,

**r'**) = ∑

_{i}

^{σ}ψ

_{i}

^{*}(

**r'**)ψ

_{i}(

**r**)

where the summation runs over all occupied σ-spin (i.e. either up or
down spin) orbitals ψ_{i}(**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
P_{cond}(**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]:

_{cond}(

**r**,

*s*) = 1⁄3 [∑

_{i}

^{σ}|∇ψ

_{i}(

**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 **r** as shown by Dobson

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

**r**) = ∑

_{i}

^{σ}|∇ψ

_{i}(

**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 D_{h}(**r**)
Becke and Edgecombe defined ELF as follows:

**r**) = 1 ⁄ [1 + χ²

_{BE}(

**r**)]

with

_{BE}(

**r**) = D(

**r**) ⁄ D

_{h}(

**r**)

where

_{h}(

**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
χ_{BE}(**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
χ_{BE}(**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
D_{h}(**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*.