Shell Structure of Noble Gas Atoms

Accurate Hartree–Fock computations were carried out long ago on all the atoms in the periodic table [1]. From these results, Wang and Parr [2] have plotted radial distribution functions (RDFs) , where is the total electron density, for the noble gas atoms He (), Ne (), Ar (), Kr (), and Xe (). These are shown as insets in the graphics. The shell structure of the electron distributions is evident, and can be identified with the K, L, M, N, and O shells, corresponding to principal quantum numbers 1 to 5, respectively.
This Demonstration proposes a simple analytic approximation for the RDFs, based on a superposition of simple Slater-type atomic orbitals (STOs). An STO has the form , where is a constant, is the nuclear charge, is an empirical shielding constant, and is an effective principal quantum number equal to 1, 2, 3, 3.7, and 4 for the principal quantum numbers , respectively. The optimal shielding constants can be determined from Slater's rules [3] but, in this application, we treat them as adjustable constants. We treat the spherically symmetric complete shells in noble gas atoms containing 2, 8, or 18 electrons as if they were single -orbitals, normalized to , the number of electrons in a shell. Thus, the contribution to the RDF from a single shell with quantum number has the form . The total RDF is the sum of up to five shells, with the appropriate values of , , , and (for Xe, we replace in the fifth term by .25).
The object is to find values of , , , , and for each of the noble gas atoms that best reproduces the appearance of the exact RDFs. Quantitative accuracy is not attainable, particularly for Ar, Kr, and Xe, unless more adjustable parameters are introduced, say modified values for and .


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


[1] E. Clementi, "Tables of Atomic Functions," IBM Journal of Research and Development, 9(1), 1965 pp. 87–89. doi:10.1147/JRD.1965.5392159.
[2] W.-P. Wang and R. G. Parr, "Statistical Atomic Models with Piecewise Exponentially Decaying Electron Densities," Physical Review A, 16(3), 1977 pp. 891–902. doi:10.1103/PhysRevA.16.891.
[3] Wikipedia. "Slater's Rules." (Apr 15, 2014) en.wikipedia.org/wiki/Slater's_rules.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+