Hartree–Fock Method

  • Takao Tsuneda


This chapter gives a detailed description of the Hartree–Fock method, which is the fundamental method of quantum chemistry, and the computational tools for efficiently solving this method. First, the Hartree method for solving the three-body problem, in order to calculate multi-electron systems, is explained in Sect. 2.1, and the molecular orbital theory, which uses the LCAO–MO approximation, is presented in Sect. 2.2. After introducing the Slater determinant in Sect. 2.3, the Hartree–Fock method is described, including the formulation of the equation and the SCF process to solve this equation, in Sect. 2.4. Then, the Roothaan method for casting the Hartree–Fock equation as a matrix equation and the basis functions used in this method are surveyed in Sect. 2.5 and Sect. 2.6, respectively. The methods used for the efficient two-electron integral calculation, which is the bottleneck in the Hartree–Fock calculation, are concretely explained in Sect. 2.7. As an extension of the Hartree–Fock method to the open-shell case, the unrestricted Hartree–Fock (UHF) method is reviewed in Sect. 2.8. The electronic states of atoms are finally discussed as an achievement of the Hatree–Fock method, with reference to the periodical table, in Sect. 2.9.


Atomic Orbital Orbital Energy Hamiltonian Operator Fast Multipole Method Core Orbital 
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  1. Becke, A.D., Dickson, R.M.: J. Chem. Phys. 89, 2993–2997 (1988)CrossRefGoogle Scholar
  2. Born, M., Oppenheimer, R.: Ann. Phys. 389, 457–484 (1927)CrossRefGoogle Scholar
  3. Boys, S.F.: Proc. R. Soc. Lond. A. 200, 542–554 (1950)CrossRefGoogle Scholar
  4. Burant, J.C., Scuseria, G.E., Frisch, M.J.: J. Chem. Phys. 105, 8969–8972 (1996)CrossRefGoogle Scholar
  5. Coulson, C.A.: Proc. Camb. Philos. Soc. 34, 204–212 (1938)CrossRefGoogle Scholar
  6. Delley, B.: J. Chem. Phys. 92, 508–517 (1990)CrossRefGoogle Scholar
  7. Delley, B.: J. Phys. Chem. 100, 6107–6110 (1996)CrossRefGoogle Scholar
  8. Dirac, P.A.M.: Proc. R. Soc. Lond. A 112, 661–677 (1926)CrossRefGoogle Scholar
  9. Dupuis, M., Rys, J., King, H.F.: J. Chem. Phys. 65, 111–116 (1976)CrossRefGoogle Scholar
  10. Ekeland, I.: Le Meilleur des Mondes Possibles (Japanese). Editions du Seuil (2009)Google Scholar
  11. Eliav, E., Kaldor, U., Ishikawa, Y.: Phys. Rev. Lett. 74, 1079–1082 (1995)CrossRefGoogle Scholar
  12. Fock, V.: Z. Phys. 61, 126–148 (1930)CrossRefGoogle Scholar
  13. Fukui, K., Yonezawa, T., Shingu, H.: J. Chem. Phys. 20, 722–725 (1952)CrossRefGoogle Scholar
  14. Gill, P.M.W., Pople, J.A.: Int. J. Quantum Chem. 40, 753–772 (1991)CrossRefGoogle Scholar
  15. Greengard, L., Rokhlin, V.: J. Comput. Phys. 73, 325–348 (1987)CrossRefGoogle Scholar
  16. Hall, G.G.: Proc. R. Soc. Lond. A 205, 541–552 (1951)CrossRefGoogle Scholar
  17. Hartree, D.R.: Math. Proc. Camb. Philos. Soc. 24, 89–132; 426–437 (1928)Google Scholar
  18. Heisenberg, W.: Z. Phys. 38, 411–426 (1926)CrossRefGoogle Scholar
  19. Hund, F.: Z. Phys. 33, 345–374 (1925a)CrossRefGoogle Scholar
  20. Hund, F.: Z. Phys. 34, 296–308 (1925b)CrossRefGoogle Scholar
  21. Hund, F.: Z. Phys. 36, 657–674 (1926)CrossRefGoogle Scholar
  22. Jensen, F.: Introduction to Computational Chemistry. Wiley, Chichester (2006)Google Scholar
  23. Küchle, W., Dolg, M., Stoll, H., Preuss, H.: J. Chem. Phys. 100, 7535–7542 (1994)CrossRefGoogle Scholar
  24. Landau, L.D.: Z. Phys. 45, 430–441 (1927)CrossRefGoogle Scholar
  25. Lennard-Jones, J.E.: Trans. Faraday Soc. 25, 668–676 (1929)CrossRefGoogle Scholar
  26. Manby, F.R., Knowles, P.J., Lloyd, A.W.: J. Chem. Phys. 115, 9144–9148 (2001)CrossRefGoogle Scholar
  27. Mulliken, R.S.: Phys. Rev. 29, 637–649 (1927)CrossRefGoogle Scholar
  28. Obara, S., Saika, A.: J. Chem. Phys. 84, 3963–3974 (1986)CrossRefGoogle Scholar
  29. Pauli, W.: Z. Phys. 31, 765–783 (1925)CrossRefGoogle Scholar
  30. Pople, J.A., Hehre, W.J.: J. Comput. Chem. 27, 161–168 (1978)Google Scholar
  31. Pople, J.A., Nesbet, R.K.: J. Chem. Phys. 22, 571–572 (1954)CrossRefGoogle Scholar
  32. Pulay, P.: Mol. Phys. 17, 197–204 (1969)CrossRefGoogle Scholar
  33. Roothaan, C.C.J.: Rev. Mod. Phys. 23, 69–89 (1951)CrossRefGoogle Scholar
  34. Schwegler, E., Challacombe, M., Head-Gordon, M.: J. Chem. Phys. 106, 9703–9707 (1997a)CrossRefGoogle Scholar
  35. Schwegler, E., Challacombe, M., Head-Gordon, M.: J. Chem. Phys. 106, 9708–9717 (1997b)CrossRefGoogle Scholar
  36. Slater, J.C.: Phys. Rev. 34, 1293–1322 (1929)CrossRefGoogle Scholar
  37. Slater, J.C.: Phys. Rev. 35, 210–211 (1930)CrossRefGoogle Scholar
  38. Szabo, A., Ostlund, N.S.: Modern Quantum Chemistry Introduction to Advanced Electronic Structure Theory. Dover, New York (1996)Google Scholar
  39. van Duijneveldt, F.B., van Duijneveldt-van de Rijdt, J.G.C.M., van Lenthe, J.H.: Chem. Rev. 94, 1873–1885 (1994)Google Scholar
  40. von Neumann, J.: Göttinger Nachr. 1, 245–272 (1927)Google Scholar
  41. Watson, M.A., Kurashige, Y., Nakajima, T., Hirao, K.: J. Chem. Phys. 128, 054105(1–7) (2008)Google Scholar
  42. White, C.A., Johnson, B.G., Gill, P.M.W., Head-Gordon, M.: Chem. Phys. Lett. 230, 8–16 (1994)CrossRefGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

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

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