Unsöld's Theorem

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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 ….


Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA



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

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.