Kohn–Sham Method

  • Takao Tsuneda


To begin the topic of DFT, this chapter overviews how the Kohn–Sham method was developed historically and then introduces various extensions of this method. The Thomas–Fermi method, which is the first form of DFT, is first explained, focusing on the local density approximation of kinetic and exchange energy density functionals, in Sect.4.1. Then, the Hohenberg–Kohn theorem, which is the basic theory of DFT, is reviewed, with a mention of the constrained search formulation used to solve the V -representability problem, in Sect. 4.2. The Kohn–Sham method based on this theorem is introduced, along with the corresponding computational methods, in Sect. 4.3. As the extension of the Kohn–Sham method to include general functionals, the generalized Kohn–Sham method is surveyed in Sect. 4.4. The constrained search method, which directly constructs a Kohn–Sham potential from the electron density, is explained, and as a consequence of this method, it is clarified why the Kohn–Sham method can accurately reproduce chemical behavior in Sect. 4.5. Finally, the time-dependent and coupled-perturbed Kohn–Sham methods are reviewed as methods with which to apply the Kohn–Sham method to calculations of photoexcitation spectra and response properties, respectively, in Sects. 4.6 and 4.7.


External Potential Slater Determinant Energy Density Functional Dynamical Electron Correlation Kohn Theorem 
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.


  1. Casida, M.E.: In: Seminario, J.J. (ed.) Recent Developments and Applications of Modern Density Functional Theory. Elsevier, Amsterdam (1996)Google Scholar
  2. Dirac, P.A.M.: Camb. Phil. Soc. 26, 376–385 (1930)CrossRefGoogle Scholar
  3. Eschrig, H.: The Fundamentals of Density Functional Theory, 2nd edn. EAGLE, Leipzig (2003)Google Scholar
  4. Fermi, E.: Z. Phys. 48, 73–79 (1928)CrossRefGoogle Scholar
  5. Fermi, E., Amaldi E.: Accad. Ital. Rome 6, 117–149 (1934)Google Scholar
  6. Feynman, R.P.: Phys. Rev. 56, 340–343 (1939)CrossRefGoogle Scholar
  7. Fock V.: Z. Phys. 61, 126–148 (1930)CrossRefGoogle Scholar
  8. Gilbert, T.L.: Phys. Rev. B 12, 2111–2120 (1975)CrossRefGoogle Scholar
  9. Gross, E.K.U., Burke, K.: Lect. Notes Phys. 706, 1–17 (2006)CrossRefGoogle Scholar
  10. Gross, E.K.U., Ullrich, C.A., Gossmann, U.A.: In: Dreizler, R., Gross, E.K.U. (eds.), Density Functional Theory, NATO ASI Series B. Plenum, New York (1995)Google Scholar
  11. Hirata, S., Head-Gordon, M.: Chem. Phys. Lett. 314, 291–299 (1999)CrossRefGoogle Scholar
  12. Hohenberg, P., Kohn, W.: Phys. Rev. B 136, 864–871 (1964)CrossRefGoogle Scholar
  13. Jensen F.: Introduction to Computational Chemistry. Wiley, Chichester (2006)Google Scholar
  14. Kohn, W., Sham, L.J.: Phys. Rev. A 140, 1133–1138 (1965)CrossRefGoogle Scholar
  15. Kutzelnigg, W.: J. Mol. Struct. Theochem 768, 163–173 (2006)CrossRefGoogle Scholar
  16. Lee, A.M., Colwell, S.M.: J. Chem. Phys. 101, 9704–9709 (1994)CrossRefGoogle Scholar
  17. Levy, M.: Proc. Natl. Acad. Sci. USA 76, 6062–6065 (1979)CrossRefGoogle Scholar
  18. Lieb, E.H.: Int. J. Quantum Chem. 24, 243–277 (1983)CrossRefGoogle Scholar
  19. Marques M.A.L., Castro, A., Rubio, A.: J. Chem. Phys. 115, 3006–3014 (2001)CrossRefGoogle Scholar
  20. McWeeny, R.: Methods of Molecular Quantum Mechanics, 2nd edn. Academic Press, San Diego (1992)Google Scholar
  21. Parr, R.G., Yang, W.: Density-Functional Theory of Atoms and Molecules. Oxford University Press, New York (1994)Google Scholar
  22. Runge, E., Gross, E.K.U.: Phys. Rev. Lett. 52, 997–1000 (1984)CrossRefGoogle Scholar
  23. Schipper, P.R.T., Gritsenko, O.V., Baerends, E.J.: Phys. Rev. A 57, 1729–1742 (1998)CrossRefGoogle Scholar
  24. Schrödinger, E.: Ann. Phys. 80, 437–490 (1926)CrossRefGoogle Scholar
  25. Seidl, A., Görling, A., Vogl, P., Majewski, J.A., Levy, M.: Phys. Rev. B 53, 3764–3774 (1986)CrossRefGoogle Scholar
  26. Thomas, L.H.: Proc. Cam. Phyl. Soc. 23, 542–548 (1927)CrossRefGoogle Scholar
  27. Ullrich, C.A.: Time-Dependent Density-Functional Theory. Oxford University Press, New York (2012)Google Scholar
  28. van Leeuwen, R.: Lect. Notes Phys. 706, 17–31 (2006)CrossRefGoogle Scholar
  29. von Weizsäcker, C.F.: Z. Phys. 96, 431–458 (1935)CrossRefGoogle Scholar
  30. Yabana, K., Bertsch, G.F.: Phys. Rev. B 54, 4484–4487 (1996)CrossRefGoogle Scholar
  31. Yang, Z., Burke, K.: Phys. Rev. A 88, 042514(1–14) (2013)Google Scholar
  32. Zhao, Q., Morrison, R.C., Parr, R.G.: Phys. Rev. A 50, 2138–2142 (1994)CrossRefGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Takao Tsuneda
    • 1
  1. 1.University of YamanashiKofuJapan

Personalised recommendations