= Solution
Write $p_i$ for the orthonormal $2p_z$ <atomic orbital> on numbered site $i$, with every positive lobe pointing along $+z$. A normalized set of <symmetry-adapted linear combinations> is
$$
\begin{array}{c|c|l}
\text{symmetry}&\text{name}&\text{symmetry orbital}\\\hline
B_{1u}&O_+&(p_1+p_8)/\sqrt2\\
B_{1u}&C_+&(p_2+p_7)/\sqrt2\\
B_{1u}&R_+&(p_3+p_4+p_5+p_6)/2\\
B_{3g}&O_-&(p_1-p_8)/\sqrt2\\
B_{3g}&C_-&(p_2-p_7)/\sqrt2\\
B_{3g}&R_-&(p_3+p_4-p_5-p_6)/2\\
B_{2g}&X&(p_3-p_4+p_5-p_6)/2\\
A_u&Y&(p_3-p_4-p_5+p_6)/2
\end{array}
$$
Each has unit squared coefficient sum, and every distinct pair has zero <inner product>. The orbital representation therefore reduces to $\boxed{3B_{1u}+3B_{3g}+B_{2g}+A_u}$. For example, the fully symmetric site sum still has $B_{1u}$ orbital symmetry because a $p_z$ orbital itself changes sign under operations that reverse $z$. Ignoring that sign would assign the wrong representations.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2025/paper-1-quinone-symmetry-orbitals.png]
{title=Original phase maps of the eight normalized <para-quinone> <symmetry orbitals>}
{description=Filled circles label the phases of the $+z$ lobes; the opposite lobe has the opposite phase. White circles mean zero coefficient. Areas scale with coefficient magnitude. Site numbers follow the source paper.}
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