Centre-of-Mass Separation in Quantum Mechanics: Implications for the Many-Body Treatment in Quantum Chemistry and Solid State Physics

  • Michal Svrček
Part of the Progress in Theoretical Chemistry and Physics book series (PTCP, volume 22)


We address the question to what extent the centre-of-mass (COM) separation can change our view of the many-body problem in quantum chemistry and solid-state physics. We show that the many-body treatment based on the electron-vibrational Hamiltonian is fundamentally inconsistent with the Born-Handy ansatz so that such a treatment can never fully account for the COM problem. The Born-Oppenheimer (B-O) approximation reveals a secret: it is the limiting case where the degrees of freedom can be treated classically. Beyond the B-O approximation they are in principle inseparable. The (unique) covariant description of all the equations, with respect to the individual degrees of freedom, leads to new types of interactions: in addition to the known vibronic (electron-phonon) ones the rotonic (electron-roton) and translonic (electron-translon) interactions arise. We have proved that as a result of the COM problem only the hypervibrations (hyperphonons, i.e. phonons + rotons + translons) have a general physical meaning in molecules and crystals; nevertheless, the use of pure vibrations (phonons) is a justified procedure only for so-called adiabatic systems. This state of affairs calls for a total revision of our contemporary view of general non-adiabatic effects, especially in connection with the Jahn-Teller effect and in formulating better approaches to superconductivity. Although the vibronic coupling is primarily responsible for the removal of the electron (quasi-) degeneracies the explanation of symmetry breaking and the formation of molecular and crystallic structures, rotonic and translonic couplings are necessary.


Adiabatic Approximation Adiabatic Limit Vibronic Coupling Degenerate Ground State Adiabatic Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The author wishes to express his gratitude to E. Brändas for his valuable advice during compilation of this paper, to O. Šipr for critical reading of the manuscript and useful suggestions and to V. Žárský for constant help and encouragement.


  1. 1.
    Born M, Oppenheimer R (1927) Ann Phys (Leipzig) 84:457Google Scholar
  2. 2.
    Primas H, Müller-Herold U (1984) Elementare Quantenchemie. Teubner, Stuttgart, p 147 ffGoogle Scholar
  3. 3.
    Monkhorst HJ (1999) Int J Quant Chem 72:281CrossRefGoogle Scholar
  4. 4.
    Cafiero M, Adamowicz L (2004) Chem Phys Letters 387:136–141CrossRefGoogle Scholar
  5. 5.
    Kutzelnigg W (1997) Mol Phys 90:909CrossRefGoogle Scholar
  6. 6.
    Handy NC, Lee AM (1996) Chem Phys Lett 252:425CrossRefGoogle Scholar
  7. 7.
    Jahn HA, Teller E (1937) Proc R Soc Lond A 161:220CrossRefGoogle Scholar
  8. 8.
    Bersuker IB (2006) The Jahn-Teller effect. Cambridge University Press, Cambridge, EnglandCrossRefGoogle Scholar
  9. 9.
    Fröhlich H (1950) Phys Rev 79:845CrossRefGoogle Scholar
  10. 10.
    Fröhlich H (1952) Proc R Soc Lond A215:291Google Scholar
  11. 11.
    Bardeen J, Cooper LN, Schrieffer JR (1957) Phys Rev 108:1175CrossRefGoogle Scholar
  12. 12.
    Svrček M (1986) Faculty of mathematics and physics. PhD thesis, Comenius University, BratislavaGoogle Scholar
  13. 13.
    Svrček M (1988) The break down of Born-Oppenheimer approximation, the unifying formalism for quantum chemistry and solid-state theory, unpublishedGoogle Scholar
  14. 14.
    Hubač I, Svrček M (1988) Int J Quant Chem 23:403Google Scholar
  15. 15.
    Hubač I, Svrček M, Salter EA, Sosa C, Bartlett RJ (1988) Lecture notes in chemistry, vol 52. Springer, Berlin, pp 95–124Google Scholar
  16. 16.
    Svrček M, Hubač I (1991) Czech J Phys 41:556CrossRefGoogle Scholar
  17. 17.
    Svrček M (1992) Methods in computational chemistry. In: Molecular vibrations, vol 4. Plenum Press, New York, pp 145–230Google Scholar
  18. 18.
    Svrček M, Baňacký P, Zajac A (1992) Int J Quant Chem 43:393CrossRefGoogle Scholar
  19. 19.
    Svrček M, Banacký P, Zajac A (1992) Int J Quant Chem 43:415CrossRefGoogle Scholar
  20. 20.
    Svrček M, Baňacký P, Zajac A (1992) Int J Quant Chem 43:425CrossRefGoogle Scholar
  21. 21.
    Svrček M, Baňacký P, Zajac A (1992) Int J Quant Chem 43:551CrossRefGoogle Scholar
  22. 22.
    Svrček M, Baňacký P, Biskupič S, Noga J, Pelikán P, Zajac A (1999) Chem Phys Lett 299:151CrossRefGoogle Scholar
  23. 23.
    Gerratt J, Mills JM (1968) J Chem Phys 49:1719–1730CrossRefGoogle Scholar
  24. 24.
    Pople JA, Raghavachari K, Schlegel HB, Binkley JS (1979) Int J Quant Chem Symp 13:225Google Scholar
  25. 25.
    Kołos W, Wolniewicz W (1964) J Chem Phys 41:3663CrossRefGoogle Scholar
  26. 26.
    Wolniewicz W (1993) J Chem Phys 99:1851CrossRefGoogle Scholar
  27. 27.
    Kleinman LI, Wolfsberg M (1974) J Chem Phys 60:4749CrossRefGoogle Scholar
  28. 28.
    Moller C, Plesset MS (1934) Phys Rev 46:618, SosaGoogle Scholar
  29. 29.
    Köppel H, Domcke W, Cederbaum LS (1984) Adv Chem Phys 57:59CrossRefGoogle Scholar
  30. 30.
    Bersuker IB, Polinger BZ (1983) Vibronic interactions in molecules and crystals (in Russian). Nauka, MoscowGoogle Scholar
  31. 31.
    Van Vleck JH (1939) J Chem Phys 7:61CrossRefGoogle Scholar
  32. 32.
    Low W (1960) Paramagnetic resonance in solids. Academic Press, New YorkGoogle Scholar
  33. 33.
    Renner R (1934) Z Phys 92:172CrossRefGoogle Scholar
  34. 34.
    von Neumann J, Wigner E (1929) Phys Z 30:467Google Scholar
  35. 35.
    Lee TD, Low FE, Pines D (1953) Phys Rev 90Google Scholar
  36. 36.
    Wagner M (1981) Phys Stat Sol B107:617Google Scholar
  37. 37.
    Lenz P, Wegner F (1996) Nucl Phys B 482:693–712CrossRefGoogle Scholar
  38. 38.
    Hanic F, Baňacký P, Svrček M, Jergel M, Smrčok L, Koppelhuber B, unpublishedGoogle Scholar
  39. 39.
    Yang CN (1962) Rev Mod Phys 34:694CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Michal Svrček
    • 1
  1. 1.Centre de Mechanique Ondulatoire AppliquéeCMOA Czech BranchCarlsbadCzech Republic

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