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Lower Bounds to Energy Eigenvalues

  • Charles E. Reid

Abstract

Since the publication of Schrödinger’s equation in 1926, a vast amount of human effort and, more recently, of computer time, has been expended on the calculation of approximate energy levels of atoms and molecules. Much, perhaps most, of this work has utilized methods based on the variation theorem. This includes the treatment of the helium atom by Hylleraas and that of the hydrogen molecules by James and Coolidge, as well as all calculations by the Hartree-Fock method and its many variants, or by configuration interaction. It is well known that these methods yield upper bounds to the true eigenvalues. From the early days of quantum mechanics there has been some interest in the much more difficult problem of calculating lower bounds also, so that the eigenvalues could be rigorously delimited within a known range. Temple’s /1/ formula was derived in a non-quantum mechanical context, but it and the related formula of D.H. Weinstein /2/ were soon applied to energy levels. These methods were improved by Stevenson and Crawford /3,4/ and, despite severe limitations, are still occasionally used. Starting about 1960, there was an upsurge of interest in lower bounds, as a result of the adaptation of A. Weinstein’s /5/ intermediate problem method to quantum mechanics by Bazley and Fox /6/ and later of the bracketing function by Löwdin /7/. A flurry of activity during the next few years showed that these methods, at least as usually understood, were restricted to rather trivial systems such as the helium atom, and the amount of attention given to lower bounds has again dwindled. Whether recent breakthroughs /7,8/ will revive interest still remains to be seen.

Keywords

Lower Bound Helium Atom Trial Function Essential Spectrum Positive Semidefinite 
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.

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Copyright information

© Springer Science+Business Media New York 1976

Authors and Affiliations

  • Charles E. Reid
    • 1
  1. 1.Quantum Theory ProjectUniversity of FloridaGainesvilleUSA

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