Skip to main content
SpringerLink
Log in

The Hellmann-Feynman theorem: a perspective

  • Original Paper
  • Published:
Journal of Molecular Modeling Aims and scope Submit manuscript

Abstract

The Hellmann-Feynman theorem has, with a few exceptions, not been exploited to the degree that it merits. This is due, at least in part, to a widespread failure to recognize that its greatest value may be conceptual rather than numerical, i.e., in achieving insight into molecular properties and behavior. In this brief overview, we shall discuss three examples of significant concepts that have come out of the Hellmann-Feynman theorem: (1) The forces exerted upon the nuclei in molecules are entirely Coulombic in nature. (2) The total energies of atoms and molecules can be expressed rigorously in terms of just the electrostatic potentials at their nuclei that are produced by the electrons and other nuclei. (3) Dispersion forces are due to the attractions of nuclei to their own polarized electronic densities. To summarize, energy and force analyses should not be viewed as competitive but rather as complementary.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Schrödinger E (1926) Quantisierung als Eigenwertproblem. (Dritte Mitteilung: Störungstheorie, mit Anwerdung auf dem Starkeffekt der Balmerlinean). Ann Phys 80:437–490

    Article  Google Scholar 

  2. Güttinger P (1932) Das Verhalten von Atomen in magnetischen Drehfeld. Z Phys 73:169–184

    Article  Google Scholar 

  3. Pauli W (1933) Principles of wave mechanics, Handbuch der Physik, vol 24. Springer, Berlin, p 162

    Google Scholar 

  4. Hellmann H (1933) Zur Rolle der kinetischen Elektronenenergie für die zwischenatomaren Krӓfte. Z Phys 85:180–190

    Article  CAS  Google Scholar 

  5. Feynman RP (1939) Forces in molecules. Phys Rev 56:340–343

    Article  CAS  Google Scholar 

  6. Hellmann H (1937) Einführung in die Quantenchemie. Deuticke, Leipzig

    Google Scholar 

  7. Coulson CA, Bell RP (1945) Kinetic energy, potential energy and force in molecule formation. Trans Faraday Soc 41:141–149

  8. Berlin T (1951) Binding regions in diatomic molecules. J Chem Phys 19:208–213

    Article  CAS  Google Scholar 

  9. Bader RFW, Jones GA (1961) The Hellmann-Feynman theorem and chemical binding. Can J Chem 39:1253–1265

    Article  CAS  Google Scholar 

  10. Bader RFW, Henneker WH, Cade PE (1967) Molecular charge distributions and chemical binding. J Chem Phys 46:3341–3363

    Article  CAS  Google Scholar 

  11. Bader RFW (1981) The nature of chemical binding. In: Deb BM (ed) The force concept in chemistry. Reinhold, New York, pp 39–136

    Google Scholar 

  12. Wilson Jr EB (1962) Four-dimensional electron density function. J Chem Phys 36:2232–2233

    Article  CAS  Google Scholar 

  13. Musher JI (1966) Comment on some theorems of quantum chemistry. Am J Phys 34:267–268

    Article  CAS  Google Scholar 

  14. Slater JC (1972) Hellmann-Feynman and virial theorems in the Xɑ method. J Chem Phys 57:2389–2396

    Article  CAS  Google Scholar 

  15. Fernández Rico J, López R, Ema I, Ramírez G (2005) Chemical notions from the electron density. J Chem Theory Comput 1:1083–1095

    Article  CAS  Google Scholar 

  16. Deb BM (1981) Preface. In: Deb BM (ed) The force concept in chemistry. Reinhold, New York, p ix

  17. Isaacson W (2007) Einstein: His life and universe. Simon and Schuster, New York, p 549

    Google Scholar 

  18. Levine IN (2000) Quantum chemistry5th edn. Prentice Hall, Upper Saddle River

    Google Scholar 

  19. Deb BM (1973) The force concept in chemistry. Rev Mod Phys 45:22–43

    Article  CAS  Google Scholar 

  20. Epstein ST (1974) Generalized Hellmann-Feynman theorems and hypervirial theorems for Xɑ-like methods. J Chem Phys 60:3328–3329

    Article  CAS  Google Scholar 

  21. Gáspár R, Nagy Á (1987). Int J Quantum Chem 31:639–647

    Article  Google Scholar 

  22. Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev B 136:864–871

    Article  Google Scholar 

  23. Epstein ST (1981) The Hellmann-Feynman theorem. In: Deb BM (ed) The force concept in chemistry. Van Nostrand Reinhold, New York, pp 1–38

    Google Scholar 

  24. Møller C, Plesset MS (1934) Note on an approximation treatment for many-elecron systems. Phys Rev 46:618–622

    Article  Google Scholar 

  25. Pople JA, Seeger R (1975) Electron density in Møller-Plesset theory. J Chem Phys 62:4566

    Article  CAS  Google Scholar 

  26. Autschbach J, Schwarz WHE (2000) Where do the forces in molecules come from? A density functional study of N2 and HCl. J Phys Chem A 104:6039–6046

    Article  CAS  Google Scholar 

  27. Deb BM (1981) Miscellaneous applications of the Hellmann-Feynman theorem. In: Deb BM (ed) The force concept in chemistry. Reinhold, New York, pp 388–417

    Google Scholar 

  28. Kern CW, Karplus M (1964) Analysis of charge distributions: hydrogen fluoride. J Chem Phys 40:1374–1389

    Article  CAS  Google Scholar 

  29. Hirschfelder JO, Curtiss CF, Bird RB (1954) Molecular theory of gases and liquids. Wiley, New York

    Google Scholar 

  30. Bader RFW (2006) Pauli repulsions exist only in the eye of the beholder. Chem Eur J 12:2896–2901

    Article  CAS  PubMed  Google Scholar 

  31. Scerri ER (2000) Have orbitals really been observed? J Chem Ed 77:1492–1494

    Article  CAS  Google Scholar 

  32. Schrödinger E (1926) Quantisierung als Eigenwertproblem. (Vierte Mitteilung). Ann Phys 81:109–139

    Article  Google Scholar 

  33. Bader RFW (2010) The density in density functional theory. J Mol Struct (THEOCHEM) 943:2–18

    Article  CAS  Google Scholar 

  34. Fernández Rico J, López R, Ema I, Ramírez G (2002) Density and binding forces in diatomics. J Chem Phys 116:1788–1799

    Article  CAS  Google Scholar 

  35. Hirshfeld FL, Rzotkiewicz S (1974) Electrostatic binding in the first-row AH and A2 diatomic molecules. Mol Phys 27:1319–1343

    Article  CAS  Google Scholar 

  36. Herzberg G (1950) Molecular Spectra and Molecular Structure, vol I. Reinhold, New York

    Google Scholar 

  37. Politzer P (1965) The electrostatic forces within the carbon monoxide molecule. J Phys Chem 69:2132–2134

    Article  CAS  Google Scholar 

  38. Parr RG, Donnelly RA, Levy M, Palke WE (1978) Electronegativity: the density functional viewpoint. J Chem Phys 68:3801–3807

    Article  CAS  Google Scholar 

  39. Levy M, Clement SC, Tal Y (1981) Correlation energies from Hartree-Fock electrostatic potentials at nuclei and generation of electrostatic potentials from asymptotic and zero-order information. In: Politzer P, Truhlar DG (eds) Chemical applications of atomic and molecular electrostatic potentials. Plenum, New York, pp 29–50

    Chapter  Google Scholar 

  40. Politzer P (1987) Atomic and molecular energy and energy difference formulae based upon electrostatic potentials at nuclei. In: March NH, Deb BM (eds) The single-particle density in physics and chemistry. Academic, San Diego, pp 59–72

    Google Scholar 

  41. Politzer P, Murray JS (2002) The fundamental nature and role of the electrostatic potential in atoms and molecules. Theor Chem Accounts 108:134–142

    Article  CAS  Google Scholar 

  42. Cohen M (1979) On the systematic linear variation of atomic expectation values. J Phys B 12:L219–L221

    Article  CAS  Google Scholar 

  43. Levy M, Tal Y (1980) Atomic binding energies from fundamental theorems involving the electron density, <r−1>, and the Z−1 perturbation expansion. J Chem Phys 72:3416–3417

    Article  CAS  Google Scholar 

  44. Eisenschitz R, London F (1930) Über das Verhӓltnis der van der Waalsschen Krӓfte zu den homöopolaren Bindungskrӓften. Z Physik 60:491–527

    Article  CAS  Google Scholar 

  45. London F (1937) The general theory of molecular forces. Trans Faraday Soc 33:8–26

    Article  CAS  Google Scholar 

  46. Salem L, Wilson Jr EB (1962) Reliability of the Hellmann-Feynman theorem for approximate charge densities. J Chem Phys 36:3421–3427

    Article  CAS  Google Scholar 

  47. Hirschfelder JO, Eliason MA (1967) Electrostatic Hellmann-Feynman theorem applied to the long-range interaction of two hydrogen atoms. J Chem Phys 47:1164–1169

    Article  CAS  Google Scholar 

  48. Bader RFW, Chandra AK (1968) A view of bond formation in terms of molecular charge distributions. Can J Chem 46:953–966

    Article  CAS  Google Scholar 

  49. Hunt KLC (1990) Dispersion dipoles and dispersion forces: proof of Feynman’s “conjecture” and generalization to interacting molecules of arbitrary symmetry. J Chem Phys 92:1180–1187

    Article  CAS  Google Scholar 

  50. Thonhauser T, Cooper VR, Li S, Puzder A, Hyldgaard P, Langreth DC (2007) Van der Waals density functional: self-consisent potential and the nature of the van der Waals bond. Phys Rev B 76:–125112(1-11)

  51. Clark T (2017) Halogen bonds and σ-holes. Faraday Discuss. 203:9–27

    Article  CAS  PubMed  Google Scholar 

  52. Murray JS, Zadeh DH, Lane P, Politzer P (2018) The role of “excluded” electronic charge in noncovalent interactions, Mol Phys, in press

  53. Bonaccorsi R, Scrocco E, Tomasi J (1970) Molecular SCF calculations for the ground state of some three-membered ring molecules. J Chem Phys 52:5270–5284

    Article  CAS  Google Scholar 

  54. Scrocco E, Tomasi J (1973) The electrostatic molecular potential as a tool for the interpretation of molecular properties. Topics Curr Chem 42:95–170

    CAS  Google Scholar 

  55. Murray JS, Sen K (eds) (1996) Molecular electrostatic potentials: concepts and applications. Elsevier, Amsterdam

    Google Scholar 

  56. Murray JS, Politzer P (2011) The electrostatic potential: an overview. WIREs Comp Mol Sci 1:153–163

    Article  CAS  Google Scholar 

Download references

Acknowledgments

It is our pleasure to join in honoring Professor Pratim K. Chattaraj, an esteemed friend and scientist.

Author information

Authors and Affiliations

  1. Department of Chemistry, University of New Orleans, New Orleans, LA, 70148, USA

    Peter Politzer & Jane S. Murray

Authors
  1. Peter Politzer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jane S. Murray
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Politzer.

Additional information

This paper belongs to Topical Collection International Conference on Systems and Processes in Physics, Chemistry and Biology (ICSPPCB-2018) in honor of Professor Pratim K. Chattaraj on his sixtieth birthday

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Politzer, P., Murray, J.S. The Hellmann-Feynman theorem: a perspective. J Mol Model 24, 266 (2018). https://doi.org/10.1007/s00894-018-3784-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00894-018-3784-7

Keywords

Search

Navigation