Electron density near the nucleus of a large atom

  • Ole J. Heilmann
  • Elliott H. Lieb

Abstract

The density of electrons on a distance scale 1/Z near the nucleus of a large atom with nuclear charge Ze is given (asymptotically as Z→ ∞) by the sum of the squares of all the hydrogenic bound-state functions (with nuclear charge Ze). This density function, which is an important limiting function in quantum chemistry, is investigated here in detail. Several analytic results are found: In particular, the asymptotic expansion for large r is derived and it is shown that the function falls off as r −3/2 for large r; this behavior coincides with the Thomas-Fermi density for small r. “Shell structure” is visible, but barely so.

Keywords

Seco sinO 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  1. [1]
    G. Zhislin, Tr. Mosk. Mat. Obshch. 9, 81 (1960).Google Scholar
  2. [2]
    E. H. Lieb, Phys. Rev. A 29, 3018 (1984).Google Scholar
  3. [3]
    E. H. Lieb, I. M. Sigal, B. Simon, and W. Thirring, Phys. Rev. Lett. 52, 994 (1980); Commun. Math. Phys. 116, 635 (1988).MathSciNetCrossRefGoogle Scholar
  4. [4]
    C. L. Fefferman and L. A. Seco, Commun. Math. Phys. 128, 109 (1990).MathSciNetADSMATHCrossRefGoogle Scholar
  5. [5]
    E. H. Lieb and B. Simon, Adv. Math. 23, 22 (1977).MathSciNetCrossRefGoogle Scholar
  6. [6]
    E. H. Lieb, Rev. Mod. Phys. 53, 603 (1981); 54, 311 (E) (1982).Google Scholar
  7. [7]
    J. M. C. Scott, Philos. Mag. 43, 859 (1952).Google Scholar
  8. [8]
    H. Siedentop and R. Weikard, Invent. Math. 97, 213 (1989).MathSciNetCrossRefGoogle Scholar
  9. [9]
    H. Siedentop and R. Weikard, Commun. Math. Phys. 112, 471 (1987).MathSciNetADSMATHCrossRefGoogle Scholar
  10. [10]
    W. Hughes, Adv. Math. 79, 213 (1990).MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    A. Iantchenko, E. H. Lieb, and H. Siedentop, J. Reine Angew. Math. (to be published).Google Scholar
  12. [12]
    A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions ( McGraw-Hill, New York, 1953 ).Google Scholar
  13. [13]
    F. W. J. Giver, SIAM J. Math. 5, 19 (1974).Google Scholar
  14. [14]
    G. N. Watson, A Treatise on the Theory of Besse! Functions ( Cambridge University Press, Cambridge, 1962 ).MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Ole J. Heilmann
    • 1
  • Elliott H. Lieb
    • 2
  1. 1.Chemistry Laboratory 3, H.C. Orsted InstituteUniversity of CopenhagenCopenhagenDenmark
  2. 2.Departments of Physics and MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations