Skip to main content
SpringerLink
Log in

Density functionals: Where do they come from, why do they work?

  • Chapter
  • First Online:
Density Functional Theory I

Part of the book series: Topics in Current Chemistry ((TOPCURRCHEM,volume 180))

Abstract

Gradient-corrected or semi-local functionals (GGA's) have achieved the accuracy required to make density functional theory a useful tool in quantum chemistry. We show that local (LSD) and semi-local functionals work because they usefully model the exchange-correlation hole around an average electron, rather than by yielding accurate results at all electron positions. We discuss the system-averaged hole at small interelectronic separations, where such functionals are extremely accurate, and at large interelectronic separations, where the local approximation is incorrect for finite systems. We argue that the “on-top” hole density provides the missing link between real atoms and molecules and the uniform electron gas. We show how exchange-correlation potentials can be related to energies. We also discuss how the degree of nonlocality, i.e., the error made by LSD, is related to the spatial extent of the hole. Decomposing the energy by coupling-constant and spin, we find that the deeper the on-top hole is, the smaller the error in the local approximation to the energy. We use this insight to demonstrate that Hartree-Fock hybrid functionals do not consistently improve on GGA. A different hybrid invokes wavefunction methods for exchange and parallel-spin correlation, but we show that configuration interaction wavefunction calculations with limited basis sets for the Ne atom make the same relative errors in the antiparallel- and parallel-spin correlation energies, despite the lack of a Coulomb cusp in the parallel-spin correlation hole. Finally, we review a recent reinterpretation of spin density functional theory, which is preferable to the standard interpretation in certain cases of extreme nonlocality.

preparation for Density Functional Theory, ed. R. Nalewajski, Springer-Verlag, Berlin, 1996

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

6 References

  1. Jones RO, Gunnarsson O (1989) Rev Mod Phys 61: 689

    Article  CAS  Google Scholar 

  2. Kohn W, Sham LJ (1965) Phys Rev 140: A 1133

    Article  Google Scholar 

  3. Vosko SH, Wilk L, Nusair M (1980) Can J Phys 58: 1200

    Article  CAS  Google Scholar 

  4. Perdew JP, Wang Y (1992) Phys Rev B 45: 13244

    Google Scholar 

  5. Fulde P (1991) Electron Correlations in Molecules and Solids. Springer, Berlin Heidelberg New York

    Google Scholar 

  6. Langreth DC, Mehl MJ (1983) Phys Rev B 28: 1809

    Google Scholar 

  7. Perdew JP (1986) Phys Rev B 33: 8822; 34: 7406 (E)

    Google Scholar 

  8. Perdew JP, Wang Y (1986) Phys Rev B 33: 8800 (1989); 40: 3399 (E)

    Google Scholar 

  9. Becke AD (1988) Phys Rev A 38: 3098

    Google Scholar 

  10. Lee C, Yang W, Parr RG (1988) Phys Rev B 37: 785

    Google Scholar 

  11. Perdew JP (1991) in: Electronic Structure of Solids '91, edited by P. Ziesche and H. Eschrig (Akademie Verlag, Berlin)

    Google Scholar 

  12. Perdew JP, Burke K, in: Proceedings of the 8th International Congress of Quantum Chemistry, 19–24 June, 1994, Prague, to appear in Int. J. Quantum Chem.

    Google Scholar 

  13. Perdew JP, Chevary JA, Vosko SH, Jackson KA, Pederson MR, Singh DJ, Fiolhais C (1992) Phys Rev B 46: 6671 (1993); 48: 4978 (E)

    Google Scholar 

  14. Burke K, Perdew JP, Levy M (1995) in Modern Density Functional Theory: A Tool for Chemistry, edited by J. M. Seminario and P. Politzer (Elsevier, Amsterdam)

    Google Scholar 

  15. Hammer B, Jacobsen KW, Nørskov JK (1993) Phys Rev Lett 70: 3971

    Article  CAS  Google Scholar 

  16. Stixrude L, Cohen RE, Singh DJ (1994) Phys Rev B 50: 6442

    Google Scholar 

  17. Burke K, Perdew JP, Ernzerhof M, Accuracy of density functionals and system-averaged exchange-correlation holes, in preparation for Phys Rev Lett

    Google Scholar 

  18. Burke K, Perdew JP, Ernzerhof M, Why semilocal functionals work: Accuracy of the on-top hole density, in preparation for J Chem Phys

    Google Scholar 

  19. Ernzerhof M, Burke K, Perdew JP, Long-range asymptotic behavior of ground-state wavefunctions, one-matrices, and pair densities, submitted to J Chem Phys

    Google Scholar 

  20. Umrigar CJ, Gonze X, in High Performance Computing and its Application to the Physical Sciences, Proceedings of the Mardi Gras 1993 Conference, edited by D. A. Browne et al. (World Scientific, Singapore, 1993)

    Google Scholar 

  21. Umrigar CJ, Gonze X (1994) Phys Rev A 50: 3827

    Google Scholar 

  22. Filippi C, Umrigar CJ, Taut M (1994) J Chem Phys 100: 1290

    Article  CAS  Google Scholar 

  23. Perdew JP, Parr RG, Levy M, Balduz JL Jr (1982) Phys Rev Lett 49: 1691

    Article  CAS  Google Scholar 

  24. Perdew JP, in: Density Functional Methods in Physics, edited by R. M. Dreizler and J. da Providencia (Plenum, NY, 1985), p. 265

    Google Scholar 

  25. Harbola MK, Sahni V (1989) Phys Rev Lett 62: 489

    Article  CAS  Google Scholar 

  26. Sahni V, Harbol MK (1990) Int J Quantum Chem S 24: 569

    Article  Google Scholar 

  27. Wang Y, Perdew JP, Chevary JA, MacDonald LD, Vosko SH (1990) Phys Rev A 41: 78

    Google Scholar 

  28. Holas A, March NH (1995) Phys Rev A 51: 2040

    Google Scholar 

  29. Levy M, March NH, Line-integral formulas for exchange and correlation potentials separately, submitted to Phys. Rev. A.

    Google Scholar 

  30. Becke AD (1993) J Chem Phys 98: 1372

    Article  CAS  Google Scholar 

  31. Barone V (1994) Chem Phys Lett 226: 392

    Article  CAS  Google Scholar 

  32. Kutzelnigg W, Klopper W (1991) J Chem Phys 94: 1985

    Article  CAS  Google Scholar 

  33. Termath V, Klopper W, Kutzelnigg W (1991) J Chem Phys 94: 2002

    Article  CAS  Google Scholar 

  34. Klopper W, Kutzelnigg W (1991) J Chem Phys 94: 2020

    Article  CAS  Google Scholar 

  35. Perdew JP (1993) Int J Quantum Chem S 27: 93

    Article  Google Scholar 

  36. Perdew JP, Savin A, Burke K (1995) Phys Rev A 51: 4531

    Google Scholar 

  37. Parr RG, Yang W (1989) Density Functional Theory of Atoms and Molecules (Oxford, New York)

    Google Scholar 

  38. Langreth DC, Perdew JP (1975) Solid State Commun 17: 1425

    Article  Google Scholar 

  39. Levy M, Perdew JP (1985) Phys Rev A 32: 2010

    Google Scholar 

  40. Görling A, Ernzerhof M (1995) Phys Rev A 51: 4501

    Google Scholar 

  41. Gunnarsson O, Lundqvist BI (1976) Phys Rev B 13: 4274

    Google Scholar 

  42. Gunnarsson O, Jonson M, Lundqvist BI (1979) Phys Rev B 20: 3136

    Google Scholar 

  43. Burke K, Perdew JP, in: Thirty Years of Density Functional Theory, 13–16 June, 1994, Carcow, to appear in Int J Quantum Chem

    Google Scholar 

  44. Perdew JP, Wang Y (1992) Phys Rev B 46: 12947

    Google Scholar 

  45. Levy M, in: Density Functional Theory, eds. R. Dreizler and E. K. U. Gross, NATO ASI Series (Plenum, New York, 1995)

    Google Scholar 

  46. Kimball JC (1973) Phys Rev A 7: 1648

    Google Scholar 

  47. Davidson ER (1976) Reduced Density Matrices in Quantum Chemistry (Academic Press, New York)

    Google Scholar 

  48. Löwdin PO (1955) Phys Rev 97: 1490

    Article  Google Scholar 

  49. Ziegler T, Rauk A, Baerends EJ (1977) Theoret Chim Acta 43: 261

    Article  CAS  Google Scholar 

  50. Harris J (1984) Phys Rev A 29: 1648

    Google Scholar 

  51. Burke K, Perdew JP (1995) Mod Phys Lett B 9: 829

    Google Scholar 

  52. Burke K, Perdew JP, Langreth DC (1994) Phys Rev Lett 73: 1283

    Article  CAS  Google Scholar 

  53. Perdew JP, Burke K, Wang Y, Real space cutoff construction of a generalized gradient approximation: derivation of the PW91 functional, submitted to Phys Rev B

    Google Scholar 

  54. Perdew JP (1994) Int J Quantum Chem 49: 539

    Article  Google Scholar 

  55. Yasuhara H (1972) Solid State Commun 11: 1481

    Article  Google Scholar 

  56. Taut M (1993) Phys Rev A 48: 3561

    Google Scholar 

  57. Perdew JP, Zunger A (1981) Phys Rev B 23: 5048

    Google Scholar 

  58. Colle R, Salvetti O (1975) Theoret Chim Acta 37: 329

    Article  CAS  Google Scholar 

  59. McWeeny R (1976) in: The New World of Quantum Chemistry: Proceedings of the Second International Congress of Quantum Chemistry, eds. B. Pullman and R. G. Parr (Reidel, Dordrecht)

    Google Scholar 

  60. Grossman JC, Mitas L, Raghavachari K (1995) Phys Rev Lett 75: 3870

    Article  CAS  Google Scholar 

  61. Buijse MA, Baerends EJ (1995) in: Density Functional Theory of Molecules, Clusters, and Solids, ed. D. E. Ellis (Kluwer Academic Publishers, Amsterdam)

    Google Scholar 

  62. Vosko SH, Lagowski JB (1986) in: Density Matrices and Density Functionals, edited by R. M. Erdahl and V. H. Smith Jr (Reidel, Dordrecht)

    Google Scholar 

  63. Handy NC, Toser DJ, Laming GJ, Murray CW, Amos RD (1994) Isr J Chem 33: 331

    Google Scholar 

  64. Perdew JP (1992) Phys Lett A 165: 79

    Google Scholar 

  65. Görling A, Levy M, Perdew JP (1993) Phys Rev B 47: 1167

    Google Scholar 

  66. Becke AD (1996) J Chem Phys 104: 1040

    Article  CAS  Google Scholar 

  67. Grev RS, Schaefer III HF (1992) J Chem Phys 96: 6854

    Article  Google Scholar 

  68. Fuentealba P, Savin A (1994) Chem Phys Lett 217: 566

    Article  CAS  Google Scholar 

  69. Stoll H, Golka E, Preuß H (1980) Theoret Chim Acta 55: 29

    Article  CAS  Google Scholar 

  70. Proynov EI, Salahub DR (1994) J Chem Phys 49: 7874 (1994)

    CAS  Google Scholar 

  71. Møller C, Plessett MS (1934) Phys Rev 46: 618

    Article  Google Scholar 

  72. Eggarter E, Eggarter TP (1978) J Phys B 11: 2069

    Google Scholar 

  73. Davidson ER, Hagstrom SA, Chakravorty SJ (1991) Phys Rev A 44: 7071

    Google Scholar 

  74. Jankowski K, Malinowski P (1980) Phys Rev A 21: 45

    Google Scholar 

  75. Jankowski K, Malinowski P, Polasik M (1979) J Phys B: Atom Molec Phys 12: 3157

    Article  CAS  Google Scholar 

  76. Rajagopal AK, Kimball JC, Banerjee M (1978) Phys Rev A 18: 2339

    Google Scholar 

  77. Ashcroft NW, Mermin ND (1976) Solid State Physics (Holt, Rinehart, Winston NY), problem 2 of Chapter 2

    Google Scholar 

  78. Shepard R, Shavitt I, Pitzer RM, Comeau DC, Pepper M, Lischka H, Szalay PG, Ahlrichs R, Brown FB, Zhoa J-G (1988) Int J Quantum Chem 142: 22

    Google Scholar 

  79. Shepard R, Lischka H, Szalay PG, Kovar T, Ernzerhof M (1992) J Chem Phys 96: 2085

    Article  CAS  Google Scholar 

  80. MOLCAS version 2, 1991, Andersson K, Flüscher MP, Lindh R, Malmqvist P-Å, Olsen J, Roos BO, Sadlej A, University of Lund, Sweden, and Widmark P-O. IBM Sweden

    Google Scholar 

  81. Szabo A, Ostlund NS (1982) Modern Quantum Chemistry (MacMillan, New York)

    Google Scholar 

  82. Ahlrichs R, Bär M, Häser M, Horn H, Kölnel C (1992) Chem Phys Lett 94: 2978

    Google Scholar 

  83. Burke K, Perdew JP, Levy M (1996) Phys Rev A April 1.

    Google Scholar 

  84. Perdew JP, Ernzerhof M, Burke K, Savin A, On-top pair-density interpretation of spin-density functional theory, with applications to magnetism to appear in Int. J. Quantum Chem.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Quantum Theory Group, Tulane University, 70118, New Orleans, LA, USA

    Matthias Ernzerhof, John P. Perdew & Kieron Burke

Authors
  1. Matthias Ernzerhof
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John P. Perdew
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kieron Burke
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

R. F. Nalewajski

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag

About this chapter

Cite this chapter

Ernzerhof, M., Perdew, J.P., Burke, K. (1996). Density functionals: Where do they come from, why do they work?. In: Nalewajski, R.F. (eds) Density Functional Theory I. Topics in Current Chemistry, vol 180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61091-X_1

Download citation

  • DOI: https://doi.org/10.1007/3-540-61091-X_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61091-5

  • Online ISBN: 978-3-540-49945-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation