# Elaboration

The ELF approach of Savin et al. can of course be used for Hartree-Fock
or configuration interaction (CI) wavefunctions as well, because it is
based on the kinetic energy density. However, as Savin's ELF definition is intended for the
Pauli kinetic energy density of non-interacting electrons, in the case of CI wavefunction
the resulting natural orbitals should not be used directly, but instead, they must be transformed
into Kohn-Sham orbitals before.
For given one-particle density matrix
ρ_{1}(**r**, **r'**) the kinetic energy is:

`∫`

_{r=r'}∇

_{r'}

^{2}ρ

_{1}(

**r**,

**r'**) dv

The integrand of the above equation - the kinetic energy density:

**r**) = −½ ∇

_{r'}

^{2}ρ

_{1}(

**r**,

**r'**)

can be written as:

**r**) = ½ ∑

_{i}n

_{i}|∇ψ

_{i}(

**r**)|² − ¼ ∇²ρ(

**r**)

where ψ_{i}(**r**) stands for the Kohn-Sham orbitals
with the occupation numbers n_{i}
and &rho(**r**) is the corresponding electron density. Ignoring the
Pauli principle yields kinetic energy density t_{w}(**r**)
defined by the electron density:

_{w}(

**r**) = 1⁄8 |∇ρ(

**r**)|² ⁄ ρ(

**r**) − ¼ ∇²ρ(

**r**)

The density Laplacian in both above formulas for the kinetic energy
density does not contribute to the kinetic energy value, as the integral
of the density Laplacian over the whole space amounts to zero. Thus, multiplying
the density Laplacian by a constant (whereby changing the local behavior
of the kinetic energy density) has no influence on the integrated value
- the kinetic energy itself. This possible local change of the kinetic
energy density does not affect the ELF formula of Savin, because the density
Laplacian vanishes in the expression t_{P}(**r**) =
t(**r**) - t_{w}(**r**) for the increase of the kinetic energy
density due to the Pauli principle.

ELF describes the interaction between the same-spin electrons. The
original ELF formula was designed for one spin part only. In case of
spin polarized calculation this implies two separate diagrams - one
for each spin. The ELF approach of Savin et al. opens the possibility
to define ELF for spin polarized systems. According to Kohout and
Savin, the ratio χ_{KS}(**r**) in a spin-polarized
ELF formula is given by the Pauli kinetic energy density composed from
the spin dependent parts t_{P,α} (**r**) and
t_{P,β} (**r**) divided by the kinetic energy
density of a spin-polarized uniform electron gas

_{KS}(

**r**) = [t

_{P,α}(

**r**) + t

_{P,β}(

**r**)]

`⁄`[t

_{h,α}(

**r**) + t

_{h,β}(

**r**)]

with

_{P,α}(

**r**) = ½ ∑

_{i}

^{α}|∇ψ

_{i}(

**r**)|² − 1⁄8 |∇ρ

_{α}(

**r**)|² ⁄ ρ

_{α}(

**r**)

and

_{h,α }(

**r**) = 2

^{2/3}c

_{F}ρ

_{α}(

**r**)

^{5/3}

for the α-spin dependent parts (similarly for the
β-spin parts). It should be mentioned that using the ratio
χ_{S}(**r**) of Savin et al. even for the spin-polarized
systems, i.e. computing the Pauli kinetic energy density as:

_{P}(

**r**) = ½ ∑

_{i}

^{α}|∇ψ

_{i}(

**r**)|² + ½ ∑

_{i}

^{β}|∇ψ

_{i}(

**r**)|² − 1⁄8 |∇(ρ

_{α}(

**r**) + ρ

_{β}(

**r**))|² ⁄ (ρ

_{α}(

**r**) + ρ

_{β}(

**r**))

and dividing by

_{h}(

**r**) = c

_{F}(ρ

_{α}(

**r**) + ρ

_{β}(

**r**))

^{5/3}

usually gives results resembling the ones using the
χ_{KS}(**r**) based spin-polarized formula.