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Approximations to the exchangecorrelation (XC) functional of the KohnSham (KS) Density Functional Theory (DFT) can be
classified by the socalled â€œJacobâ€™s ladder.â€ The ground
on which the ladder lies is the Hartree approximation (XC energy is zero), and the first rung is the local
density approximation (LDA) in which the XC energy density is a simple local function of the density.
The second rung of the Jacobâ€™s ladder is the generalized
gradient approximation (GGA): in this case the XC energy density depends also on the gradient of the density.
In the third rung (metaGGA) an additional variable is used, the KohnSham kinetic energy
density which allows, e.g., to construct selfcorrelationfree functionals.
Functionals in the above rungs can have high accuracy for different class of problems in chemistry and solidstate physics, but
their main limitation is the selfinteraction error (SIE)[156,157,158,159].
To avoid the SIE the exchange must be treated exactly
and this can be achieved by functionals in the fourth rung which depend explicitly on all the occupied KS orbitals.
In the KS formalism the EXX (exactexchange) energy is (for closedshell systems n_{s} = 2
)[156,157,158,159]:
i.e. the same functional form of the HartreeFock (HF) exchange but computed with KS orbitals
which are obtained using a selfconsistent local EXX potential.
At this point we should recall that hybrid DFT functionals (including HF exchange), doesn't belong
to the KS formalism: in hybrid DFT, in fact, the nonlocal HF exchange operator
NLx() = 
is employed
in the selfconsistent (Generalized KohnSham) equations determining the orbitals.
While LDA, GGA, metaGGA and hybrid functionals are implemented (for groundstate calculations) in
the dscf and ridft, the odft module
considers functionals of the fourth rung.
Currently exchangeonly orbitaldependent approaches are implemented in the odft module.
The EXX KS local potential (
vxEXX()
) can be obtained using the optimized effective potential (OEP) method
(in each selfconsistent step):[157,160,158,159]:
dχ_{s}()vxEXX() 
= 
2n_{s}φ_{a}NLxφ_{s} 
(16.2) 
where
χ_{s}() = 2n_{s}
is the noninteracting density response.
An effective approximation to the OEPEXX potential is given by the Localized HartreeFock (LHF) potential[156] which is given by
vxLHF() 
= 
 n_{s}d 


+ 
n_{s}φ_{i}vxLHFNLxφ_{j} 
(16.3) 
where the first term is called Slater potential and the second term correction term.
If terms i≠j
are neglected in the correction term, the KriegerLiIafrate (KLI) potential[161] is obtained.
Note that the Eq. (16.3) depends only on occupied orbitals, whereas Eq. (16.2) depends also on virtual orbitals.
The LHF total energy is assumed to be the EXX total energy, even if LHF is not variational (although the deviation from the EXX energy
is very small, usually below 0.01%
).
The LHF potential is equivalent to the Common Energy Denominator Approximation (CEDA) [162] and to the
Effective Local Potential (ELP) [163].
Both OEPEXX and LHF (in contrast to functionals of the first three rungs) satisfy the HOMO condition[161]
φ_{HOMO}vxφ_{HOMO} = φ_{HOMO}NLxφ_{HOMO} , 
(16.4) 
and the asymptotic relation[164,165]
vx()φ_{M}vxNLxφ_{M}  . 
(16.5) 
where φ_{M}
is the highest occupied orbital which do not have a nodal surface
in the asymptotic region along direction
.
Considering together with condition (16.4), we finally obtain that:
 

vx()
will approach 1/r
along all directions
where
φ_{HOMO}()
does not have a nodal surface in the asymptotic region (e.g. this is the case of atoms);
 
 on directions which belong to the nodal surface
of the HOMO, the
vx()
will approach
φ_{M}vxNLxφ_{M}  1/r
.
Both OEPEXX and LHF gives total energies very close to the HartreeFock one (actually
ELHF > EEXX > EHF
), thus, without
an appropriate correlation functional, these
methods are not suitable for thermochemistry.
On the other hand OEPEXX and LHF give very good KS orbital spectra.
In fact the eigenvalues of the HOMO is very close to the HartreeFock and
to exact ionization potential (I.P): this is in contrast to functional of the first three rungs which underestimate the
HOMO energy by several eVs.
In addition a continuum set of bound unoccupied orbitals are obtained.
Thus OEPEXX or LHF KS orbitals are very good input quantities for computing
NMR shielding constants[166], energylevels in hybrid interfaces[167] and TDDFT excitation energies [168]
(the latter using LDA/GGA kernels, not the hybrid ones).
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