# ELF for density functionals

In density functional theory the pair density is not explicitly
defined. Thus, the original formulation of ELF derived from the pair density
is not applicable. Searching for a possibility to use ELF in density functional
calculations, Savin et al. **r**) of Becke and
Edgecombe. The main aspect of Savin's formulation is that thus defined ELF
is a *property based on the diagonal elements of the one-particle density matrix, i.e. the electron density*.

In the Kohn-Sham method the kinetic energy of N noninteracting electrons is:

_{s}= ½

`∫`∑

_{i}

^{N}|∇ψ

_{i}(

**r**)|² dv

with the Kohn-Sham orbitals `ψ`_{i}(**r**). The positive
definite kinetic energy density
t(**r**) = ½ ∑_{i}^{N} |∇ψ_{i}(**r**)|²
is bounded by a minimum value:

**r**) ≥ 1⁄8 |∇ρ(

**r**)|² ⁄ ρ(

**r**)

when all orbitals are proportional to √ρ (i.e. like in a bosonic system) . The Pauli kinetic energy is the energy due to the redistribution of the electrons in accordance with the Pauli principle. It is the integral of the Pauli kinetic energy density:

_{P}(

**r**) = t(

**r**) − 1⁄8 |∇ρ(

**r**)|² ⁄ ρ(

**r**)

The Pauli kinetic energy density itself does not resolve the bonding
situation. It is the more or less arbitrary division of
t_{P}(**r**) by the kinetic energy density of a uniform
electron gas of the same electron density (with the Fermi constant
c_{F} = 3⁄10 (3π²)^{2/3})

_{h}(

**r**) = c

_{F}ρ(

**r**)

^{5/3}

that yields all the information. For a closed shell system the ratio

_{S}(

**r**) = t

_{P}(

**r**) ⁄ t

_{h}(

**r**)

is formally identical with the ratio χ_{BE}(**r**)
in the HF approximation. This identity holds also for an open shell system,
when the kinetic energy densities are computed for the corresponding spin
part only. Then also the ELF formulas based on χ_{BE}(**r**)
and χ_{S}(**r**) respectively, are identical. In the
interpretation of Savin et al. ELF is a measure of the influence of Pauli
principle as given by the Pauli kinetic energy density, *relative* to
a uniform electron gas of the same density. Similarly to the original
definition, ELF does not mirror t_{P}(**r**).

An expression equivalent to χ_{S}(**r**) of Savin et
al. was found already 1983 by Deb and Ghosh

**r**) = −¼ ∇²ρ(

**r**) + 1⁄8 |∇ρ(

**r**)|² ⁄ ρ(

**r**) + c

_{F}f(

**r**) ρ(

**r**)

^{5/3}

The right hand side of the above equation consists, besides the density
Laplacian that vanishes by an integration over the whole space, of the full
Weizsäcker term 1⁄8 |∇ρ(**r**)|² ⁄ ρ(**r**)
and a modified Thomas-Fermi term with a correction factor f(**r**).
Substituting for the left hand side the Hartree-Fock expression for the kinetic
energy density:

**r**) = ½ ∑

_{i}|∇ψ

_{i}(

**r**)|² − ¼ ∇²ρ(

**r**)

unveils the correction factor f(**r**) of Deb and Ghosh as the
ratio χ_{S}(**r**) of Savin. Besides calculating
f(**r**) for noble gas atoms (revealing the atomic shell structure)
Deb and Ghosh did not further exploit this function.