Skip to main content
SpringerLink
Log in

Classical kinetic energy, quantum fluctuation terms and kinetic-energy functionals

  • Regular Article
  • Published:
Theoretical Chemistry Accounts Aims and scope Submit manuscript

Abstract

We employ a recently formulated dequantization procedure to obtain an exact expression for the kinetic energy which is applicable to all kinetic-energy functionals. We express the kinetic energy of an N-electron system as the sum of an N-electron classical kinetic energy and an N-electron purely quantum kinetic energy arising from the quantum fluctuations that turn the classical momentum into the quantum momentum. This leads to an interesting analogy with Nelson’s stochastic approach to quantum mechanics, which we use to conceptually clarify the physical nature of part of the kinetic-energy functional in terms of statistical fluctuations and in direct correspondence with Fisher Information Theory. We show that the N-electron purely quantum kinetic energy can be written as the sum of the (one-electron) Weizsäcker term and an (N−1)-electron kinetic correlation term. We further show that the Weizsäcker term results from local fluctuations while the kinetic correlation term results from the nonlocal fluctuations. We then write the N-electron classical kinetic energy as the sum of the (one-electron) classical kinetic energy and another (N−1)-electron kinetic correlation term. For one-electron orbitals (where kinetic correlation is neglected) we obtain an exact (albeit impractical) expression for the noninteracting kinetic energy as the sum of the classical kinetic energy and the Weizsäcker term. The classical kinetic energy is seen to be explicitly dependent on the electron phase, and this has implications for the development of accurate orbital-free kinetic-energy functionals. Also, there is a direct connection between the classical kinetic energy and the angular momentum and, across a row of the periodic table, the classical kinetic energy component of the noninteracting kinetic energy generally increases as Z increases. Finally, we underline that, although our aim in this paper is conceptual rather than practical, our results are potentially useful for the construction of improved kinetic-energy functionals.

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

Similar content being viewed by others

References

  1. Parr RG and Yang W (1989). Density functional theory of atoms and molecules. Oxford University Press, New York

    Google Scholar 

  2. Dreizler RM and Gross EKU (1990). Density functional theory: an approach to the quantum many body problem. Springer, Berlin

    Google Scholar 

  3. Koch W and Holthausen MC (2000). A chemist’s guide to density functional theory. Wiley-VCH, Weinheim

    Google Scholar 

  4. Nelson E (1966). Phys Rev 150: 1079

    Article  CAS  Google Scholar 

  5. Nelson E (1967). Dynamical theories of Brownian motion. Princeton University Press, Princeton

    Google Scholar 

  6. Fényes I (1952). Z Physik 132: 81

    Article  Google Scholar 

  7. Weizel W (1953). Z Physik 134: 264

    Article  Google Scholar 

  8. Weizel W (1953). Z Physik 135: 270

    Article  Google Scholar 

  9. Weizel W (1954). Z Physik 136: 582

    Article  Google Scholar 

  10. Mosna RA, Hamilton IP and Delle Site L (2006). J Phys A 39: L229 quantph/0511068

    Article  Google Scholar 

  11. Thomas LH (1927). Proc Camb Philos Soc 23: 542

    Article  CAS  Google Scholar 

  12. Fermi E (1927). Rend Accad Lincei 6: 602

    CAS  Google Scholar 

  13. v Weizsäcker CF (1935). Z Phys 96: 431

    Article  Google Scholar 

  14. March NH and Young WH (1958). Proc Phys Soc 72: 182

    Article  Google Scholar 

  15. Acharya PK, Bartolotti LJ, Sears SB and Parr RG (1980). Proc Nat Acad Sci 77: 6978

    Article  CAS  Google Scholar 

  16. Gázquez JL and Ludeña EV (1981). Chem Phys Lett 83: 145

    Article  Google Scholar 

  17. Gázquez JL and Robles J (1982). J Chem Phys 76: 1467

    Article  Google Scholar 

  18. Acharya PK (1983). J Chem Phys 78: 2101

    Article  CAS  Google Scholar 

  19. Kompaneets AS and Pavlovski ES (1957). Sov Phys JETP 4: 328

    CAS  Google Scholar 

  20. Kirzhnits PA (1957). Sov Phys JETP 5: 64

    Google Scholar 

  21. Yang W (1986). Phys Rev A 34: 4575

    Article  Google Scholar 

  22. Herring C (1986). Phys Rev A 34: 2614

    Article  Google Scholar 

  23. Dirac PAM (1930). Proc Camb Philos Soc 26: 376

    CAS  Google Scholar 

  24. Spruch L (1991). Rev Mod Phys 63: 151

    Article  Google Scholar 

  25. Scott JMC (1952). Philos Mag 43: 859

    Google Scholar 

  26. Fisher RA (1925). Proc Camb Philos Soc 22: 700

    Google Scholar 

  27. Nagy A (2003). J Chem Phys 119: 9401

    Article  CAS  Google Scholar 

  28. Hohenberg P and Kohn W (1964). Phys Rev 136: B864

    Article  Google Scholar 

  29. Kohn W and Sham LJ (1965). Phys Rev 140: A1133

    Article  Google Scholar 

  30. Chan GK, Cohen AJ and Handy NC (2001). J Chem Phys 114: 631

    Article  CAS  Google Scholar 

  31. Iyengar SS, Ernzerhof M, Maximoff SN and Scuseria GE (2001). Phys Rev A 63: 052508

    Article  CAS  Google Scholar 

  32. Tran F and Wesołowski TA (2002). Chem Phys Lett 360: 209

    Article  CAS  Google Scholar 

  33. Sim E, Larkin J, Burke K and Bock CW (2003). J Chem Phys 118: 8140

    Article  CAS  Google Scholar 

  34. Jiang H and Yang WT (2004). J Chem Phys 121: 2030

    Article  CAS  Google Scholar 

  35. Chai JD and Weeks JA (2004). J Phys Chem B 108: 6870

    Article  CAS  Google Scholar 

  36. Blanc X and Cances E (2005). J Chem Phys 122: 214106

    Article  CAS  Google Scholar 

  37. Ovchinnikov IV, Neuhauser D (2006) J Chem Phys 124:024105

    Google Scholar 

  38. Zhou B and Wang YA (2006). J Chem Phys 124: 081107

    Article  CAS  Google Scholar 

  39. Mosna RA, Hamilton IP, Delle Site L (2005) J Phys A 38:3869 quant-ph/0504124

    Google Scholar 

  40. Goldstein H (1980). Classical mechanics. Addison-Wesley, Reading

    Google Scholar 

  41. Holland PR (1993). The quantum theory of motion. Cambridge University Press, Cambridge

    Google Scholar 

  42. Sears SB, Parr RG and Dinur U (1980). Isr J Chem 19: 165

    CAS  Google Scholar 

  43. Miao MS (2001). J Phys A 34: 8171

    Article  CAS  Google Scholar 

  44. Ayers PW (2005). J Math Phys 46: 062107

    Article  CAS  Google Scholar 

  45. Delle Site L (2006). J Phys A 39: 3047

    Article  CAS  Google Scholar 

  46. Gadre SR (1991). Adv Quantum Chem 22: 1

    Article  Google Scholar 

  47. Romera E and Dehesa JS (1994). Phys Rev A 50: 256

    Article  CAS  Google Scholar 

  48. Shannon CE (1948). Bell Syst Tech J 27: 623

    Google Scholar 

  49. Romera E and Dehesa JS (2004). J Chem Phys 120: 8906

    Article  CAS  Google Scholar 

  50. Sagar RP and Guevara NL (2005). J Chem Phys 123: 044108

    Article  CAS  Google Scholar 

  51. Ghosh SK and Deb BM (1982). Phys Rep 92: 1

    Article  CAS  Google Scholar 

  52. Luo S (2002). J Phys A 35: 5181

    Article  Google Scholar 

  53. Delle Site L (2005). J Phys A 38: 7893

    Article  CAS  Google Scholar 

  54. Jensen JHD and Luttinger JM (1952). Phys Rev 86: 907

    Article  CAS  Google Scholar 

  55. Oliphant TA Jr (1956). Phys Rev 104: 954

    Article  CAS  Google Scholar 

  56. Hellman H (1936). Acta Physicochem USSR 4: 225

    Google Scholar 

  57. Kemister G and Nordholm S (1982). J Chem Phys 76: 5043

    Article  CAS  Google Scholar 

  58. Englert B-G and Schwinger J (1984). Phys Rev A 29: 2339

    Article  CAS  Google Scholar 

  59. Englert B-G and Schwinger J (1985). Phys Rev A 32: 47

    Article  CAS  Google Scholar 

  60. Herring C and Chopra M (1988). Phys Rev A 37: 31

    Article  Google Scholar 

  61. Lacks DJ and Gordon RG (1994). J Chem Phys 100: 4446

    Article  CAS  Google Scholar 

  62. García-González P, Alvarellos JE and Chacón E (1996). Phys Rev A 54: 1897

    Article  Google Scholar 

  63. March NH (2003). Int J Quantum Chem 92:1

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemistry, Wilfrid Laurier University, Waterloo, ON, Canada, N2L 3C5

    I. P. Hamilton

  2. Instituto de Matemática, Estatí stica e Computação Cientí fica, Universidade Estadual de Campinas, C. P. 6065, 13083-859, Campinas, SP, Brazil

    Ricardo A. Mosna

  3. Max-Planck-Institute for Polymer Research, Ackermannweg 10, 55021, Mainz, Germany

    L. Delle Site

Authors
  1. I. P. Hamilton
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ricardo A. Mosna
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Delle Site
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. P. Hamilton.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hamilton, I.P., Mosna, R.A. & Site, L.D. Classical kinetic energy, quantum fluctuation terms and kinetic-energy functionals. Theor Chem Account 118, 407–415 (2007). https://doi.org/10.1007/s00214-007-0279-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00214-007-0279-5

Keywords

Search

Navigation