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 …

.