Coordinate Scaling Requirements for Approximating Exchange and Correlation

  • Mel Levy
Chapter
Part of the NATO ASI Series book series (NSSB, volume 337)

Abstract

In order to continue to better approximate the exact exchange-correlation density functional, Exc[n], for electronic structure calculations, sets of exact coordinate scaling and convexity requirements are reviewed for the purpose of testing approximations. The scaling requirements dictate how Exc[n] must transform when n(x, y, z) is replaced by αβγn(αx,βy,γz). High-density and low-density limits are explored. Simple formulas are shown which isolate the kinetic component of Exc[n] and which generate Exc[n] from just its electron-electron repulsion component, or exchange-correlation hole, at a coupling constant of unity. The scaling and convexity conditions are derived with help from the constrained-search formulation and through the development of a coupling-constant perturbation expansion. The local density approximation and various gradient approximations are compared against the exact conditions, with emphasis upon where the conditions are respected and where the conditions are violated. Numerical results are presented.

Keywords

Local Density Approximation Correlation Energy Electronic Structure Calculation Virial Theorem Uniform Scaling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R.M. Dreizler and E.K.U. Gross, Density Functional Theory, Springer-Verlag, Berlin-New York, 1990.MATHCrossRefGoogle Scholar
  2. 2.
    R.G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989.Google Scholar
  3. 3.
    R.O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989).ADSCrossRefGoogle Scholar
  4. 4.
    Entire issue of Advances in Quantum Chemistry, edited by S.B. Trickey, Academic Press, San Diego, 1990.Google Scholar
  5. 5.
    E.S. Kryachho and E.V. Ludena, Density Functinal Theory of Many-Electron Systems, Kluwer Press, Dordrecht, 1990.CrossRefGoogle Scholar
  6. 6.
    N.H. March, Electron Density Theory of Atoms and Molecules, Academic Press, London, 1992.Google Scholar
  7. 7.
    J.K. Labanowski and J.W. Andzelm, editors, Density Functional Methods in Chemistry, Springer-Verlag, New York, 1991.Google Scholar
  8. 8.
    R.M. Dreizler and J. da Providencia, editors, Density Functional Methods in Physics, Plenum Press, New York, 1985.Google Scholar
  9. 9.
    N.H. March and B.M. Deb, editors, The Single-Particle Density in Physics and Chemistry, Academic Press, New York, 1987.Google Scholar
  10. 10.
    R. Erdahl and V.H. Smith, Jr., editors, Density Matrices and Density Functionals, Reidel Press, Dordrecht, 1987.Google Scholar
  11. 11.
    M. Levy and J.P. Perdew, Phys. Rev. A. 32, 2010 (1985).ADSCrossRefGoogle Scholar
  12. 12.
    M. Levy, W. Yang, and R.G. Parr, J. Chem. Phys. 83, 2334 (1985).ADSCrossRefGoogle Scholar
  13. 13.
    W. Yang, page 499 in Ref. 10.Google Scholar
  14. 14.
    M. Levy, Int. J. Quantum Chem. S23, 617 (1989).Google Scholar
  15. 15.
    H. Ou-Yang and M. Levy, Phys. Rev. A 42, 155 (1990).ADSCrossRefGoogle Scholar
  16. 16.
    M. Levy and H. Ou-Yang, Phys. Rev. A 42, 651 (1990).ADSCrossRefGoogle Scholar
  17. 17.
    M. Levy, page 69 in Ref. 4.Google Scholar
  18. 18.
    L.C. Wilson and M. Levy, Phys. Rev. B 41, 12930 (1990).ADSCrossRefGoogle Scholar
  19. 19.
    H. Ou-Yang and M. Levy, Phys. Rev. Lett. 65, 1036 (1990).ADSCrossRefGoogle Scholar
  20. 20.
    H. Ou-Yang and M. Levy, Phys. Rev. A 44, 54 (1991).ADSCrossRefGoogle Scholar
  21. 21.
    M. Levy, Phys. Rev. A 43, 4637 (1991).ADSCrossRefGoogle Scholar
  22. 22.
    A. Görling and M. Levy, Phys. Rev. A 45, 1509 (1992).ADSCrossRefGoogle Scholar
  23. 23.
    M. Levy and J.P. Perdew, Phys. Rev. B 48, 11638 (1993).ADSCrossRefGoogle Scholar
  24. 24.
    A. Görling and M. Levy, Phys. Rev. B 47, 13105 (1993).ADSCrossRefGoogle Scholar
  25. 25.
    A. Görling, M. Levy, and J.P. Perdew, Phys. Rev. B 47, 1167 (1993).ADSCrossRefGoogle Scholar
  26. 26.
    Q. Zhao, M. Levy and R.G. Parr, Phys. Rev. A 47, 918 (1993).ADSCrossRefGoogle Scholar
  27. 27.
    M. Levy and A. Görling, Phil. Mag., in press.Google Scholar
  28. 28.
    M. Levy and J.P. Perdew, Int. J. Quantum Chem. 49, 539 (1994).CrossRefGoogle Scholar
  29. 29.
    J.P. Perdew, in Electronic Structure of Solids ‘91, edited by P. Ziesche and H. Eschrig, Academie Verlag, Berlin, 1991.Google Scholar
  30. 30.
    E.H. Lieb and S. Oxford, Int. J. Quantum Chem. 19, 427 (1981).CrossRefGoogle Scholar
  31. 31.
    D.C. Langreth and J.P. Perdew, Solid State Commun. 17, 1425 (1975).ADSCrossRefGoogle Scholar
  32. 32.
    O. Gunnarsson and B.I. Lundqvist, Phys. Rev. B 13, 4274 (1976).ADSCrossRefGoogle Scholar
  33. 33.
    D.C. Langreth and J.P. Perdew, Phys. Rev. B 15, 2884 (1976).ADSCrossRefGoogle Scholar
  34. 34.
    See Appendix A in Ref. [25].Google Scholar
  35. 35.
    Q. Zhao and R.G. Parr, Phys. Rev. A 46, 5320 (1992).ADSCrossRefGoogle Scholar
  36. 36.
    P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    M. Levy, Proc. Natl. Acad. Sci (U.S.A.) 76, 6062 (1979).ADSCrossRefGoogle Scholar
  38. 38.
    M. Levy, Phys. Rev. A 26, 1200 (1982).ADSCrossRefGoogle Scholar
  39. 39.
    J.E. Harriman, Phys. Rev. A 24, 680 (1981).ADSCrossRefGoogle Scholar
  40. 40.
    E.H. Lieb, Int. J. Quantum Chem. 24, 224 (1983).CrossRefGoogle Scholar
  41. 41.
    J.K. Percus, Int. J. Quantum Chem. 13, 89 (1978).CrossRefGoogle Scholar
  42. 42.
    M. Gell-Mann and K.A. Brueckner, Phys. Rev. 106, 364 (1957).MathSciNetADSMATHCrossRefGoogle Scholar
  43. 43.
    W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965).MathSciNetADSCrossRefGoogle Scholar
  44. 44.
    E. Engel and S.H. Vosko, Phys. Rev. A 47, 2800 (1993).ADSCrossRefGoogle Scholar
  45. 45.
    F.W. Averill and G.S. Painter, Phys. Rev. B 24, 6795 (1981).ADSCrossRefGoogle Scholar
  46. 46.
    S.K. Ghosh and R.G. Parr, J. Chem. Phys. 82, 3307 (1985).ADSCrossRefGoogle Scholar
  47. 47.
    D.J. Lacks and R.G. Gordon, Phys. Rev. A 47, 4681 (1993).ADSCrossRefGoogle Scholar
  48. 48.
    A.D. Becke, Phys. Rev. A 33, 3098 (1988).ADSCrossRefGoogle Scholar
  49. 49.
    J.P. Perdew and Y. Wang, Phys. Rev. B 33, 8800 (1986).ADSCrossRefGoogle Scholar
  50. 50.
    D.C. Langreth and M.J. Mehl, Phys. Rev. B 28, 1809 (1993).ADSCrossRefGoogle Scholar
  51. 51.
    C. Lee, W. Yang, and R.G. Parr, Phys. Rev. B 37, 785 (1988).ADSCrossRefGoogle Scholar
  52. 52.
    C.J. Umrigar and X. Gonze, published in “High Performance Computing and its Application to the Physical Sciences”, proceedings of the Mardi Gras’ 93 Conference, edited by D.A. Browne et al., (World Scientific, 1993).Google Scholar
  53. 53.
    C.J. Umrigar and X. Gonze, to appear in Phys. Rev. A.Google Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Mel Levy
    • 1
  1. 1.Department of Chemistry and Quantum Theory GroupTulane UniversityNew OrleansUSA

Personalised recommendations