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. [SAVIN1992A] utilized the observation that the Kohn-Sham orbital representation of the Pauli kinetic energy density has the same formal structure as the expression  D(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: 

Ts   =   ½ iN |∇ψi(r)|² dv

with the Kohn-Sham orbitals  ψi(r). The positive definite kinetic energy density  t(r) = ½ ∑iN |∇ψi(r)|²  is bounded by a minimum value:

t(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: 

tP(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  tP(r) by the kinetic energy density of a uniform electron gas of the same electron density (with the Fermi constant  cF = 3⁄10 (3π²)2/3)

th(r)  =   cF ρ(r)5/3

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

χS(r)  =   tP(r) ⁄ th(r)

is formally identical with the ratio χBE(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 χBE(r) and χS(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  tP(r). 

An expression equivalent to χS(r) of Savin et al. was found already 1983 by Deb and Ghosh [DEB1983]. Deb and Ghosh were searching for a "proper local description" of the kinetic energy density. They proposed the following formulation of the kinetic energy density: 

t(r)  =   −¼ ∇²ρ(r)   +   1⁄8 |∇ρ(r)|² ⁄ ρ(r)   +   cF 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: 

t(r)  =   ½ ∑i |∇ψi(r)|²   −   ¼ ∇²ρ(r)

unveils the correction factor f(r)  of Deb and Ghosh as the ratio  χS(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.