Exchange Potential: LDA and GGA versus Exact Potential
Exchange Potential: LDA and GGA versus Exact Potential
The exact multiplicative exchange potential of density functional theory
can be obtained numerically via the optimized potential
method (OPM) [1].
The OPM results can be used to analyze the properties of approximate
density functionals for the exchange energy, as the
local density approximation (LDA) and the
generalized gradient approximation (GGA).
Below these approximations are compared with the exact potentials for
a number of systems, ranging from atoms to solids.
Sometimes also the KriegerLiIafrate (KLI) approximation [2] to the
exact exchange potential is displayed.
The units of the plots are as follows:
All length scales are given in Bohr, the potentials are in Hartree.
II.1) Neon (GGA = PW91 [3])

The atomic shell structure is obvious in the exact exchange potential
(OPM).

The exact exchange potential falls off like 1/r for large
r, i.e. in the asymptotic region in which the density
decays exponentially.

The KLI approximation also shows the shell structure, although
much less pronounced.
It is exact in the asymptotic regime.

Neither the LDA nor the GGA reproduce the shell structure.
The LDA and GGA potentials decay exponentially for large r
(with the exception of the B88GGA potential [4] which formally
decays as r^{2} [5]).

The GGA potential depends on the Laplacian of the density, which
diverges like 1/r at the position of the nucleus
(cusp condition).
II.2) Chromium, Palladium, Radon
see
E. Engel and S. H. Vosko,
Phys. Rev. B 47, 13164 (1993).
II.3) Zinc
see
E. Engel and R. M. Dreizler,
J. Comput. Chem. 20, 31 (1999).
II.4) Spindifference potentials for
Manganese, Arsenic, Copper, Europium
see
E. Engel and S. H. Vosko,
Phys. Rev. A 47, 2800 (1993).
III.1) H_{2} (GGA = B88 [4])

The plots show the potential in the xzplane.
The atoms are sitting on the xaxis at z = +/0.7
Bohr.

The exact exchange potential of H_{2} only
cancels the selfinteraction component in the Hartree potential.
III.2) N_{2} (GGA = B88 [4])
see
E. Engel and S. H. Vosko,
Phys. Rev. B 50, 10498 (1994).
V.1) Silicon

GGA = PW91 [3] and B88 [4]

Planewave pseudopotential calculation with a cutoff energy of
25 Rydberg, kpoint sampling based on 19 special
kpoints (a = 10.2 Bohr).

The plots show the potential along [111] direction of the diamond
structure.
V.2) Aluminum

GGA = PW91 [3]

Planewave pseudopotential calculation with a cutoff energy of
100 Rydberg, kpoint sampling based on 44 special
kpoints, 750 KohnSham states included in evaluation
of KohnSham response function (a = 7.6 Bohr).

The plots show the potential along [100] direction of the fcc structure.

KLI approximation reasonably close to exact OPM result.

GGA deviates much more from exact potential than LDA.

J. D. Talman and W. F. Shadwick,
Phys. Rev. A 14, 36 (1976).

J. B. Krieger, Y. Li, and G. J. Iafrate,
Phys. Lett. A 146, 256 (1990).

J. P. Perdew, in
Electronic Structure of Solids 1991,
ed. by P. Ziesche and H. Eschrig (Akademie Verlag, Berlin, 1991),
p.11.

A. D. Becke,
Phys. Rev. A 38, 3098 (1988).

E. Engel, J. A. Chevary, L. D. Macdonald, and S. H. Vosko,
Z. Phys. D 23, 7 (1992).
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