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

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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 ….