Number the four sites of the cyclobutadienyl dianion cyclically and apply the Hückel method. The orthonormal pi basis gives
(1)
A convenient real orthonormal eigenbasis is
(2)
Thus one bonding and two degenerate nonbonding orbitals are filled, and the antibonding orbital is empty. There are six pi electrons, including the two added by the dianion charge. Rotating the two occupied nonbonding orbitals within their degenerate subspace changes their individual sketches but not the total density matrix.
Figure 1.
Original orbital levels and occupations for the four-site dianion and valence O2
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