Exchange Potential: LDA and GGA versus Exact Potential

Exchange Potential: LDA and GGA versus Exact Potential

I) Brief introduction

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 Krieger-Li-Iafrate (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) Results for atoms

II.1) Neon (GGA = PW91 [3])


neon: exchange potential



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) Spin-difference potentials for Manganese, Arsenic, Copper, Europium

see E. Engel and S. H. Vosko, Phys. Rev. A 47, 2800 (1993).




III) Results for molecules

III.1) H2 (GGA = B88 [4])


H_2: exact exchange potential
H_2: LDA
H_2: GGA



III.2) N2 (GGA = B88 [4])


N_2: exact exchange potential
N_2: LDA
N_2: GGA





IV) Results for jellium spheres

see E. Engel and S. H. Vosko, Phys. Rev. B 50, 10498 (1994).




V) Results for solids

V.1) Silicon


silicon: exchange potential




V.2) Aluminum


aluminum: exchange potential






References

  1. J. D. Talman and W. F. Shadwick, Phys. Rev. A 14, 36 (1976).
  2. J. B. Krieger, Y. Li, and G. J. Iafrate, Phys. Lett. A 146, 256 (1990).
  3. J. P. Perdew, in Electronic Structure of Solids 1991, ed. by P. Ziesche and H. Eschrig (Akademie Verlag, Berlin, 1991), p.11.
  4. A. D. Becke, Phys. Rev. A 38, 3098 (1988).
  5. E. Engel, J. A. Chevary, L. D. Macdonald, and S. H. Vosko, Z. Phys. D 23, 7 (1992).


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