= Solution
Set $E=\alpha+\lambda\beta$, with the usual bonding convention $\beta<0$. In the <Hückel method>, the dimensionless matrix $M=(H-\alpha I)/\beta$ has diagonal entries one on oxygen sites 1 and 8, zero on carbon, and unit entries between bonded neighbours. In the bases $(O_+,C_+,R_+)$ and $(O_-,C_-,R_-)$ above, its blocks are
$$
M_{B_{1u}}=\begin{pmatrix}1&1&0\\1&0&\sqrt2\\0&\sqrt2&1\end{pmatrix},\qquad
M_{B_{3g}}=\begin{pmatrix}1&1&0\\1&0&\sqrt2\\0&\sqrt2&-1\end{pmatrix}.
$$
The one-dimensional $B_{2g}$ and $A_u$ blocks have $\lambda=1$ and $-1$, respectively. The <characteristic polynomial> of the $B_{1u}$ block factors as
$$
\det(\lambda I-M_{B_{1u}})=(\lambda-1)(\lambda^2-\lambda-3),
$$
so its three <molecular orbital> energies are
$$
\boxed{E_{B_{1u}}=\alpha+\beta,\qquad\alpha+\frac{1+\sqrt{13}}2\beta,\qquad\alpha+\frac{1-\sqrt{13}}2\beta.}
$$
For the $B_{3g}$ block the <matrix trace> is zero. Thus its three $\lambda$ values sum to zero, and the pair supplied in the paper gives the remaining one as $-(1.86-2.12)\simeq0.26$. Directly, its secular equation is $\lambda^3-4\lambda+1=0$, whose roots are approximately $1.860806$, $0.254102$ and $-2.114908$. The supplied pair is approximate, so $0.26$ from its trace and $0.2541$ from the exact block should not be mistaken for an additional physical splitting.
All eight energies, ordered from low to high for $\beta<0$, are
$$
\begin{array}{c|c|c}
\lambda&\text{symmetry}&\text{neutral occupancy}\\\hline
2.302776&B_{1u}&2\\
1.860806&B_{3g}&2\\
1&B_{1u}&2\\
1&B_{2g}&2\\
0.254102&B_{3g}&0\\
-1&A_u&0\\
-1.302776&B_{1u}&0\\
-2.114908&B_{3g}&0
\end{array}
$$
The eight $\pi$ electrons fill the four lowest <molecular orbitals>. The equality at $\lambda=1$ is an accidental degeneracy of this parameterized model, not a multidimensional representation of $D_{2h}$.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2025/paper-1-quinone-energy-levels.png]
{title=Original <Hückel method> energy-level diagram for neutral <para-quinone>, with eight $\pi$ electrons and its unoccupied $B_{3g}$ level}
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