= Solution
Number the four sites of the <cyclobutadienyl dianion> cyclically and apply the <Hückel method>. The <orthonormal> pi basis gives
$$
H=\alpha I+\beta\begin{pmatrix}0&1&0&1\\1&0&1&0\\0&1&0&1\\1&0&1&0\end{pmatrix},\qquad \beta<0.
$$
A convenient real <orthonormal> eigenbasis is
$$
\begin{array}{c|c|c}
E& (c_1,c_2,c_3,c_4)&\text{occupation}\\ \hline
\alpha+2\beta&\tfrac12(1,1,1,1)&2\\
\alpha&\tfrac1{\sqrt2}(1,0,-1,0)&2\\
\alpha&\tfrac1{\sqrt2}(0,1,0,-1)&2\\
\alpha-2\beta&\tfrac12(1,-1,1,-1)&0
\end{array}
$$
Thus \b[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>.
\Image[/past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2a-orbitals.png]
{title=Original orbital levels and occupations for the four-site dianion and valence O2}
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