Advertisement

The Born-Oppenheimer Approximation

  • J. M. Combes
  • P. Duclos
  • R. Seiler
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 74)

Abstract

In physics and chemistry the Born-Oppenheimer approximation is a very important method for analyzing the spectrum of molecules1). It is based on the important fact that the molecular Schrödinger operator contains one small parameter, the ratio k4 of the electronic to the nuclear mass2). Perturbation theory in this parameter is however very singular.

Keywords

Asymptotic Expansion Dirichlet Boundary Condition Discrete Spectrum Energy Eigenvalue Absolute Minimum 
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. Aguilar J. and Combes J.M., 1971; A class of Analytic Perturbations for One-body Schrödinger Hamiltonians, Com.Math.Phys. 22, 269–279.MathSciNetADSMATHCrossRefGoogle Scholar
  2. Born M. and Oppenheimer R., 1927; Zur Quantentheorie der Molekeln, Annalen der Physik 84, 457.ADSCrossRefGoogle Scholar
  3. Brattsev V.F., 1965; Dokl.Akad.Nauk. SSSR 16o, 570.Google Scholar
  4. Brezin E. and Zinn-Justin J., 1979; Expansion of the \(H_2^ + \) ground state energy in inverse powers of the distance between the two protons; Le journal de physique, lettres, 40L-511.Google Scholar
  5. Combes J.M. and Seiler R., 1978; Regularity and Asymptotic Properties of the Discrete Spectrum of Electronic Hamiltonians; Int.J. of Quantum Chemistry 14, 213.CrossRefGoogle Scholar
  6. Combes J.M. and Seiler R., 1980; Spectral Properties of Atomic and Molecular Systems, Ed.G.Wooley, Plenum.Google Scholar
  7. Combes J.M., Duclos P. and Seiler R.,1981; Decoupling and Krein’s Formula I, to appear.Google Scholar
  8. Combes J.M. and Thomas L., 1973; Asymptotic Behaviour of Eigenfunctions for Multiparticle Schrödinger Operators; Commun.math.Phys. 34, 251.MathSciNetADSMATHCrossRefGoogle Scholar
  9. Deift P., Hunziker W., Simon B., Vock E.,1979; Pointwise Bounds of Eigenfunctions and Wave Packets in N-Body Quantum Systems IV; Com. Math. Phys. 64, 1–34.Google Scholar
  10. Dieudonné J., 1968; Calcul Infinitesimal, Chapter VIII. 7, Herman, Paris.Google Scholar
  11. Duclos P., Thèse, Dept.de Mathématique Université de Toulon.Google Scholar
  12. Epstein S., 1966; Ground-State Energy of a Molecule in the Adiabatic Approximation, J.Chem.44, 863 and 44, 4062.Google Scholar
  13. Herring C., 1962; Critique of the Heitler-Landau method of calculating spin coupling at large distances, Rev.Mod.Phys. 34, 631–645.MathSciNetADSCrossRefGoogle Scholar
  14. Hoffmann-Osterhoff T., 1980; A Compairison Theorem for Differential Inequalities with Applications in Quantum Mechanics; Journal of Physics AB, 417.Google Scholar
  15. Hoffmann-Osterhoff M. and Seiler R., 1981; to appear in Phys. Rev.A.Google Scholar
  16. Kato T., 1966; Perturbation theory for linear Operators, Springer-Verlag, Berlin.MATHGoogle Scholar
  17. Kato T., 1957; Commun.Pure and Appl.Math. X, 151.CrossRefGoogle Scholar
  18. Komarov J.N., Ponomarev L.I., Slavianov S.I., 1976; Spherical and Coulomb Spheroidal functions. Nauka, Moscow (in Russian).Google Scholar
  19. Krein M., 1946; Über Resolventen hermitescher Operatoren mit Defektindex (m,m) Doklady Akad.Nauk. SSSR 52 No. 8, 657–660.MathSciNetGoogle Scholar
  20. Lieb E. and Simon B., 1978; Monotonicity of the Electronic Contribution to the Born-Oppenheimer Energy, J.Phys.B. 5 37.Google Scholar
  21. Morgan J.D. Ill and Simon B., 1980; Behavior of Molecular Potential Energy Curves for Large Nuclear Seperations; Int.J.Quant. Chemistry, 17, 1143.CrossRefGoogle Scholar
  22. Pack R.T. and Hirschfelder J.O., 1968; Separation of Rotational Coordinates from the N-electrons Diatomic Schrödinger Equation, Journal of Chemical Physics 49, 4009–4020.MathSciNetADSCrossRefGoogle Scholar
  23. Protter M.H. and Weinberger H.F.,1967; Maximum Principles in Differential Equations, Prentice-Hall, Juc.Google Scholar

Copyright information

© Plenum Press, New York 1981

Authors and Affiliations

  • J. M. Combes
    • 1
  • P. Duclos
    • 1
  • R. Seiler
    • 2
  1. 1.Département de mathématiqueUniversité de ToulonFrance
  2. 2.Institut f. theor.PhysikFreie Universität BerlinGermany

Personalised recommendations