Advertisement

Communications in Mathematical Physics

, Volume 143, Issue 3, pp 607–639 | Cite as

On the Born-Oppenheimer expansion for polyatomic molecules

  • M. Klein
  • A. Martinez
  • R. Seiler
  • X. P. Wang
Article

Abstract

We consider the Schrödinger operatorP(h) for a polyatomic molecule in the semiclassical limit where the mass ratioh2 of electronic to nuclear mass tends to zero. We obtain WKB-type expansions of eigenvalues and eigenfunctions ofP(h) to all orders inh. This allows to treat the splitting of the ground state energy of a non-planar molecule. Our class of potentials covers the physical case of the Coulomb interaction. We use methods ofh-pseudodifferential operators with operator valued symbols, which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials.

Keywords

Neural Network Statistical Physic Complex System State Energy Nonlinear Dynamics 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A] Agmon, S.: Lectures on exponential decay of solutions of second-order elliptic equations. Princeton, NJ: Princeton University Press 1982Google Scholar
  2. [AS] Aventini, P., Seiler, R.: On the electronic spectrum of the diatomic molecule. Commun. Math. Phys.22, 269–279 (1971)CrossRefGoogle Scholar
  3. [BK] Balazard-Konlein, A.: Calcul fonctionel pour des opérateurh-admissible à symbole opérateurs at applications. Thése de 3ème cycle, Université de Nantes (1985)Google Scholar
  4. [BO] Born, M., Oppenheimer, R.: Zur Quantentheorie der Molekeln. Annal. Phys.84, 457 (1927)Google Scholar
  5. [BT] Bott, R., Tu, L.W.: Differential forms in algebraic topology. Berlin, Heidelberg, New York: Springer 1982Google Scholar
  6. [CDS] Combes, J.M., Duclos, P., Seiler, R.: The Born-Oppenheimer approximation. In: Rigorous atomic and molecular physics. Velo, G., Wightman, A. (eds.). pp. 185–212. New York: Plenum Press 1981Google Scholar
  7. [CS] Combes, J.M., Seiler, R.: Regularity and asymptotic properties of the discrete spectrum of electronic hamiltonians. Int. J. Quant. Chem.XIV, 213–229 (1978)CrossRefGoogle Scholar
  8. [GMS] Gerard, C., Martinez, A., Sjöstrand, J.: A mathematical approach to the effective Hamiltonian in perturbed periodic problems. Preprint Orsay 1990, submitted to Commun. Math. Phys.Google Scholar
  9. [HS1] Helffer, B., Sjöstrand, J.: Multiple wells in the semiclassical limit I. Commun. Partial Differ. Equations9, (4), 337–408 (1984)Google Scholar
  10. [HS2] Helffer, B., Sjöstrand, J.: Puits multiples en mécanique semi-classique VI (Cas des puits sous-variétés). Ann. Inst. Henri Poincaré46, 353–372 (1987)Google Scholar
  11. [Ha1] Hagedorn, G.A.: High order corrections to the time-independent Born-Oppenheimer approximation. I. Smooth potentials. Ann. Inst. Henri Poincaré47, 1–16 (1987)Google Scholar
  12. [Ha2] Hagedorn, G.A.: High order corrections to the time-independent Born-Oppenheimer approximation. II. Diatomic Coulomb systems. Commun. Math. Phys.116, 23–44 (1988)CrossRefGoogle Scholar
  13. [Hu] Hunziker, W.: Distortion analycity and molecular resonance curves. Ann. Inst. H. Poincaré45, 339–358 (1986)Google Scholar
  14. [Hus] Husemoller, D.: Fiber bundles. Berlin, Heidelberg, New York: Springer 1975Google Scholar
  15. [K] Klein, M.: On the mathematical theory of predissociation. Ann. Phys.178, 48–73 (1987)Google Scholar
  16. [Ma1] Martinez, A.: Estimations de l'effet tunnel pour le double puits I. J. Math. Pures Appl.66, 195–215 (1987)Google Scholar
  17. [Ma2] Martinez, A.: Développements asymptotiques et effet tunnel dans l'approximation de Born-Oppenheimer: Ann. Inst. Henri Poincaré49, 239–257 (1989)Google Scholar
  18. [Ma3] Martinez, A.: Resonances dans l'approximation de Born-Oppenheimer I.J. Diff. Eq. (to appear)Google Scholar
  19. [Ma4] Martinez, A.: Resonances dans l'approximation de Born-Oppenheimer II — Largeur des resonances. Commun. Math. Phys.135, 517–530 (1991)Google Scholar
  20. [S] Seiler, R.: Does the Born-Oppenheimer approximation work? Helv. Phys. Acta46, 230–234 (1973)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • M. Klein
    • 1
  • A. Martinez
    • 2
  • R. Seiler
    • 1
  • X. P. Wang
    • 1
  1. 1.Fachbereich Mathematik MA 7-2Technische Universität BerlinBerlin 12Federal Republic of Germany
  2. 2.Département de Mathématiques et d'InformatiqueUniversité Paris-Nord C.S.P.VilletaneuseFrance

Personalised recommendations