9722

Unsöld's Theorem

A theorem due to A. Unsöld, Ann. Physik, 82, 1927 pp. 355-365 states that a filled or half-filled subshell of atomic orbitals with is spherically symmetrical and thus contributes an orbital angular momentum of zero. This can be illustrated by evaluating the sum of atomic orbital densities , or equivalently , giving a spherically symmetrical function (independent of and ). The nitrogen atom in its ground state has the configuration …, with three electrons of parallel spins singly occupying the three degenerate -orbitals. Neon has a completely filled subshell with configuration …. Likewise, the half-filled subshells in Cr and Mn lead to spherically symmetrical ground states. The mathematical proof of Unsöld's theorem follows from the spherical-harmonic identity
.
In this Demonstration, you can add sums of , , or atomic orbital densities to approach a spherical distribution. A filled or half-filled shell missing one orbital behaves like a positive hole with the same angular momentum as the missing electron. Thus the ground state of C … is a state, constructed, in concept, by removing a electron from N ….

THINGS TO TRY

SNAPSHOTS

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

DETAILS

Snapshot 1: a configuration, showing a orbital hole
Snapshot 2: spherically symmetrical or configuration
Snapshot 3: configuration, which accounts for square-planar complexes of Ni, Pd and Pt
    • 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.









 
RELATED RESOURCES
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+