Electron density near the nucleus of a large atom

  • Ole J. Heilmann
  • Elliott H. Lieb


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.


Asymptotic Expansion Shell Structure Nuclear Charge Distance Scale Large Atom 
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  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

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