Journal of Mathematical Chemistry

, Volume 52, Issue 9, pp 2322–2329 | Cite as

On the crossing of the energy levels of a parameter-dependent quantum-mechanical Hamiltonian

  • Francisco M. Fernández
Original Article


The non-crossing rule for the energy levels of a parameter-dependent Hamiltonian is revisited and a flaw in a commonly accepted proof is revealed. Some aspects of avoided crossings are illustrated by means of simple models. One of them shows the close relationship between avoided crossings and exceptional points.


Avoided crossings Non-crossing rule Parameter-dependent operators Exceptional points PT symmetry 



This report has been financially supported by PIP No. 11420110100062 (Consejo Nacional de Investigaciones Cientificas y Tecnicas, República Argentina).


  1. 1.
    A. Devaquet, Avoided crossings in photochemistry, Presented at the Fifth International Symposium on Photochemistry (Enschede, The Netherlands, 1974)Google Scholar
  2. 2.
    G. Herzberg, Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules (Van Nostrand Reinhold, New York, 1950)Google Scholar
  3. 3.
    M. Teller, J. Phys. Chem. 41, 109 (1937)CrossRefGoogle Scholar
  4. 4.
    H. Eyring, J. Walter, G.E. Kimball, Quantum Chemistry (Wiley, New York, 1944)Google Scholar
  5. 5.
    F.L. Pilar, Elementary Quantum Chemistry (McGraw-Hill, New York, 1968)Google Scholar
  6. 6.
    K. Razi Naqvi, W. Byers Brown, Int. J. Qunatum Chem. 6, 271 (1972)CrossRefGoogle Scholar
  7. 7.
    R.P. Feynman, Phys. Rev. 56, 340 (1939)CrossRefGoogle Scholar
  8. 8.
    D.J. Klein, P.L. DeVries, J. Chem. Phys. 68, 160 (1978)CrossRefGoogle Scholar
  9. 9.
    E.A. Castro, F.M. Fernández, J. Chem. Phys. 73, 4711 (1980)CrossRefGoogle Scholar
  10. 10.
    F.M. Fernández, Phys. Rev. B 69, 037101 (2004)CrossRefGoogle Scholar
  11. 11.
    R. Balawender, A. Holas, N.H. March, Phys. Rev. B 69, 037103 (2004)CrossRefGoogle Scholar
  12. 12.
    S.R. Vatsya, Phys. Rev. B 69, 037102 (2004)CrossRefGoogle Scholar
  13. 13.
    F.A. Cotton, Chemical Applications of Group Theory (Wiley, New York, 1990)Google Scholar
  14. 14.
    M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, New York, 1964)Google Scholar
  15. 15.
    K. Razi Naqvi, Chem. Phys. Lett. 15, 634 (1972)CrossRefGoogle Scholar
  16. 16.
    W.D. Heiss, A.L. Sannino, J. Phys. A 23, 1167 (1990)CrossRefGoogle Scholar
  17. 17.
    W.D. Heiss, Phys. Rev. E 61, 929 (2000)CrossRefGoogle Scholar
  18. 18.
    W.D. Heiss, H.L. Harney, Eur. Phys. J. D 17, 149 (2001)CrossRefGoogle Scholar
  19. 19.
    W.D. Heiss, Czech. J. Phys. 54, 1091 (2004)CrossRefGoogle Scholar
  20. 20.
    C.M. Bender, Rep. Prog. Phys. 70, 947 (2007)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.División Química TeóricaINIFTA (UNLP, CCT La Plata-CONICET)La PlataArgentina

Personalised recommendations