Theoretica chimica acta

, Volume 73, Issue 2–3, pp 173–200 | Cite as

Origin and meaning of the Fermi contact interaction

  • Werner Kutzelnigg


The Fermi-contact interaction (FCI) can easily be derived from 1st order perturbation theory applied to the non-relativistic wave equation for a spin-(1/2) particle of Lévy-Leblond, with the nuclear spin described by the field of an “external” magnetic dipole, and it results from the fact that the “turn-over-rule” for the operator \(\vec \sigma \vec p\)is only valid if the derivatives implicit in \(\vec p\) are taken “in the distribution sense”. If one avoids to apply the turn-over-rule, the FCI is obtained without the need to introduce a “δ-function”. It is also shown that the formulation of a magnetic point dipole as the limit of an extended nucleus directly leads to the FCI. Traditional methods of the derivation of the FCI are analyzed in the light of this new interpretation. It is then explained why the perturbation expansions in powers of the magnetic moment of the nucleus necessarily diverges, but that the expression for the 1st order energy on which the concept of the FCI is based, can nevertheless be justified by means of the Hellmann-Feynman theorem with a correction term if singular wave functions are involved. Finally some comments on a theory beyond first order are made.

Key words

Fermi contact interaction Lévy-Leblond equation Hyperfine interaction Hellmann-Feynman theorem Perturbation theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fermi E (1930) Z Phys 60:320Google Scholar
  2. 2.
    Fermi E, Segré E (1933) Z Phys 82:729Google Scholar
  3. 3.
    Abragam A, Pryce MHL (1951) Proc. Roy Soc A205:135Google Scholar
  4. 4.
    Bloch F (1936) Phys Rev 50:259; (1937) 51:994Google Scholar
  5. 5.
    Casimir HBG (1936) Physica 3:936Google Scholar
  6. 6.
    Breit F (1930) Phys Rev 35:1447; (1931) 38:463Google Scholar
  7. 7.
    Pyykkö P (1972) J Mag Res 8:15Google Scholar
  8. 8.
    Davies DW (1967) The theory of the electronic and magnetic properties of molecules. Wiley, LondonGoogle Scholar
  9. 9.
    Harriman JE (1978) Theoretical foundations of electron spin resonance. Academic Press, New YorkGoogle Scholar
  10. 10.
    Blinder SM (1960) J Mol Spectr 5:17; Blinder SM (1965) Adv Quant Chem 2:47Google Scholar
  11. 11.
    Löwdin PO (1964) J Mol Spectr 14:131Google Scholar
  12. 12.
    Ramsey NF (1956) Molecular beams. Clarendon, OxfordGoogle Scholar
  13. 13.
    Bethe HA, Salpeter EE (1957) Quantum mechanics of one and two-electron atoms. Springer, Berlin Heidelberg New YorkGoogle Scholar
  14. 14.
    Messiah A (1962) Quantum Mechanics. Wiley, New YorkGoogle Scholar
  15. 15.
    Casimir HBG (1936) On the interaction between atomic nuclei and electron. Teyler's, HaarlemGoogle Scholar
  16. 16.
    Hameka HF (1965) Advanced quantum chemistry. Addison-Wesley, Reading, MassGoogle Scholar
  17. 17.
    Nierenberg WA (1957) Ann Rev Nucl Sci 7:353Google Scholar
  18. 18.
    Das TP (1973) Relativistic quantum mechanics of electrons. Harper and Row, New YorkGoogle Scholar
  19. 19.
    Lévy-Leblond JM (1967) Commun Math Phys 6:286; Lévy-Leblond JM (1974) Riv Nuov Cim 4:99Google Scholar
  20. 20.
    Lindgren I, Morrison J (1982) Atomic many-body theory. Springer, Berlin Heidelberg New YorkGoogle Scholar
  21. 21.
    Moss RE, Watson JKG (1976) Can J Phys 54:2240Google Scholar
  22. 22.
    Armstrong jr L (1966) J Math Phys 7:1891Google Scholar
  23. 23.
    Rose EM (1961) Relativistic electron theory. Wiley, New YorkGoogle Scholar
  24. 24.
    Lighthill MJ (1958) Introduction to Fourier analysis and generalized functions. Cambridge University PressGoogle Scholar
  25. 25.
    Schwarz L (1950) Theorie des distributions. Hermann, ParisGoogle Scholar
  26. 26.
    Ferrell RA (1960) Am J Phys 28:484Google Scholar
  27. 27.
    Milford RJ (1960) Am J Phys 28:521Google Scholar
  28. 28.
    Rado GT (1962) Am J Phys 30:716Google Scholar
  29. 29.
    Schwartz C (1955) Phys Rev 97:380Google Scholar
  30. 30.
    Schwartz C (1959) Ann Phys (New York) 2:156Google Scholar
  31. 31.
    Power JD, Pitzer RM (1971) Chem Phys Lett 8:615Google Scholar
  32. 32.
    Moore EA, Moss RE (1975) Mol Phys 30:1297, 1315Google Scholar
  33. 33.
    Latvamaa E, Kurittu L, Pyykkö P, Tataru L (1973) J Phys B 6:591Google Scholar
  34. 34.
    Pyykkö P (1975) Theor Chim Acta 39:185Google Scholar
  35. 35.
    Trivedi HP (1979) Mol Phys 38:1603Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Werner Kutzelnigg
    • 1
  1. 1.Lehstuhl für Theoretische ChemieRuhr-Universität BochumBochumGermany

Personalised recommendations