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The conceptual power of the Hellmann–Feynman theorem

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Abstract

The power of the Hellmann–Feynman theorem is primarily conceptual. It provides insight and understanding of molecular properties and behavior. In this overview, we discuss several examples of concepts coming out of the theorem. (1) It shows that the forces exerted upon the nuclei in a molecule, which hold the molecule together, are purely Coulombic in nature. (2) It indicates whether the role of the electronic charge in different portions of a molecule’s space is bond-strengthening or bond-weakening. (3) It demonstrates the importance of the electrostatic potentials at the nuclei of a molecule, and that the total energies of atoms and molecules can be expressed rigorously in terms of just these potentials, with no explicit reference to electron–electron interactions. (4) It shows that dispersion forces arise from the interactions of nuclei with their own polarized electronic densities. Our discussion focuses particularly upon the contributions of Richard Bader in these areas.

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References

  1. Schrődinger E (1926) Quantisierung als Eigenwertproblem. Ann Phys 80:437–490

  2. Gűttinger 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 24. Springer, Berlin, p 162

    Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  8. Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ

    Google Scholar 

  9. Politzer P, Murray JS (2018) The Hellmann-Feynman theorem - a perspective. J Mol Model 24:266

    Article  Google Scholar 

  10. Lennard-Jones J, Pople JA (1951) The molecular orbital theory of chemical valency. IX. The interaction of paired electrons in chemical bonds, Proc Royal Soc, London, Series A 210:190–206

    CAS  Google Scholar 

  11. Matta CF, Bader RFW (2006) An experimentalist’s reply to “what is an atom in a molecule?” J Phys Chem A 110:6365–6371

    Article  CAS  Google Scholar 

  12. London F (1928) Zur Quantentheorie der Homőopolaren Valenzzahlen 46:455–477

    CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  18. Bader RFW, Keaveny I, Cade PE (1967) Molecular charge distributions and chemical binding. II. First-row diatomic hydrides AH. J Chem Phys 47:3381–3402

    Article  CAS  Google Scholar 

  19. Cade PE, Bader RFW, Henneker WH, Keaveny I (1969) Molecular charge distributions and chemical binding. IV. The second-row diatomic hydrides AH. J Chem Phys 50:5313–5333

    Article  CAS  Google Scholar 

  20. Bader RFW, Bandrauk AD (1968) Molecular charge distributions and chemical binding. III. The isoelectronic series N2, CO, BF, and C2. BeO and LiF. J Chem Phys 49:1653–1665

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  24. Deb BM (1981) Preface to: the force concept in chemistry. In: Deb BM (ed). Van Nostrand Reinhold, New York, p ix

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

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  28. Bachrach SM (1994) Population analysis and electron densities from quantum mechanics. In: Lipkowitz KB, Boyd DB (eds) Reviews in computational chemistry, vol 5. VCH Publishers, New York, ch 3:171–227

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

    Article  CAS  Google Scholar 

  30. Cramer CJ (2002) Essentials of computational chemistry. Wiley, Chichester, UK, p 102

    Google Scholar 

  31. Bader RFW (2007) Everyman’s derivation of the theory of atoms in molecules. J Phys Chem A 111:7966–7972

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  33. Herzberg G (1950) Molecular spectra and molecular structure, vol I. Van Nostrand, New York

    Google Scholar 

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

    Article  CAS  Google Scholar 

  35. Bader RFW, Carroll MT, Cheeseman JR, Chang C (1987) Properties of atoms in molecules. J Am Chem Soc 109:7968–7979

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  38. 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 

  39. Silberbach H (1991) The electron density and chemical bonding: a reinvestigation of Berlin’s theorem. J Chem Phys 94:2977–2985

    Article  CAS  Google Scholar 

  40. Bader RFW (2010) Definition of molecular structure: by choice or by appeal to observation. J Phys Chem A 114:7431–7444

    Article  CAS  Google Scholar 

  41. Hurley AC (1962) Virial theorem for polyatomic molecules. J Chem Phys 37:449–450

    Article  CAS  Google Scholar 

  42. Bader RFW (1960) The use of the Hellmann-Feynman theorem to calculate molecular energies. Can J Chem 38:2117–2127

    Article  CAS  Google Scholar 

  43. Politzer P, Parr RG (1974) Some new energy formulas for atoms and molecules. J Chem Phys 61:4258–4262

    Article  CAS  Google Scholar 

  44. Politzer P (2004) Atomic and molecular energies as functionals of the electrostatic potential. Theor Chem Accts 111:395–399

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  46. 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 Press, New York, pp 29–50

    Chapter  Google Scholar 

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

    Google Scholar 

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

    Article  CAS  Google Scholar 

  49. Moller C, Plesset MS (1934) Note on an approximate treatment for many-electron systems. Phys Rev 46:618–622

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  51. 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 

  52. Politzer P, Murray JS (2021) Electrostatic potentials at the nuclei of atoms and molecules. Theor Chem Accts 140:7

    Article  CAS  Google Scholar 

  53. Eisenschitz R, London F (1930) Űber das Verhältnis der van der Waalsschen Kräfte zu den homőopolaren Bindungskräften. Z Phys 60:491–527

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  57. 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 

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

    Google Scholar 

  59. 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 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  62. Murray JS, Zadeh DH, Lane P, Politzer P (2019) The role of “excluded” electronic charge in noncovalent interactions. Mol Phys 117:2260–2266

    Article  CAS  Google Scholar 

  63. Bader RFW (2009) Bond paths are not chemical bonds. J Phys Chem A 113:10391–10396

    Article  CAS  Google Scholar 

  64. Ruedenberg K (1962) The physical nature of the chemical bond. Rev Mod Phys 34:326–376

    Article  CAS  Google Scholar 

  65. Gordon MS, Jensen JH (2000) Perspective on “the physical nature of the chemical bond.” Theor Chem Acc 103:248–251

    Article  CAS  Google Scholar 

  66. Ruedenberg K, Schmidt MW (2009) Physical understanding through variational reasoning: electron sharing and covalent bonding. J Phys Chem A 113:1954–1968

    Article  CAS  Google Scholar 

  67. Jacobsen H (2009) Chemical bonding in view of electron charge density and kinetic energy density descriptors. J Comput Chem 30:1093–1102

    Article  CAS  Google Scholar 

  68. Zhao L, Schwarz WHE, Frenking G (2019) The Lewis electron-pair bonding model: the physical background, one century later. Nature Rev Chem 3:35–47

    Article  Google Scholar 

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

    Peter Politzer & Jane S. Murray

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  1. Peter Politzer
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  2. Jane S. Murray
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Calculations: This is a conceptual paper. Writing: P. Politzer, 75% and J. S. Murray, 25%. Draft revision: P. Politzer, 50% and J. S. Murray, 50%.

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Correspondence to Peter Politzer.

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Politzer, P., Murray, J.S. The conceptual power of the Hellmann–Feynman theorem. Struct Chem 34, 17–21 (2023). https://doi.org/10.1007/s11224-022-01961-9

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