MPI - CPfS

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   ρ1(r, r') the kinetic energy is: 

T   =   −½ r=r'r'2 ρ1(r, r') dv

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

t(r)   =   −½ ∇r'2 ρ1(r, r')

can be written as: 

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

where  ψi(r) stands for the Kohn-Sham orbitals with the occupation numbers ni and  &rho(r) is the corresponding electron density. Ignoring the Pauli principle yields kinetic energy density  tw(r) defined by the electron density: 

tw(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  tP(r) = t(r) - tw(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  χKS(r) in a spin-polarized ELF formula is given by the Pauli kinetic energy density composed from the spin dependent parts  tP,α (r) and   tP,β (r) divided by the kinetic energy density of a spin-polarized uniform electron gas [KOHOUT1996]

χKS(r)   =   [tP,α (r)   +   tP,β (r)][th,α (r)   +   th,β (r)]

with 

tP,α (r)   =   ½ ∑iα |∇ψi(r)|²   −   1⁄8 |∇ρα (r)|² ⁄ ρα (r)

and 

th,α (r)   =   22/3 cF ρα(r)5/3

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

tP (r)   =   ½ ∑iα |∇ψi(r)|²   +   ½ ∑iβ |∇ψi(r)|²   −   1⁄8 |(ρα (r) + ρβ (r))|² (ρα (r) + ρβ (r))

and dividing by 

th(r)   =   cF (ρα (r) + ρβ (r))5/3

usually gives results resembling the ones using the  χKS(r) based spin-polarized formula.