= Solution
At $\beta_L=0$, the preceding <matrix> separates into identical $(\phi_1,\phi_2)$ and $(\phi_3,\phi_4)$ blocks. For $\beta_S<0$, a normalized pair at the lower <energy> $\alpha+\beta_S$ is
$$
\boxed{\begin{aligned}
\psi_{-,1}&=\frac{\phi_1+\phi_2}{\sqrt2}
=\frac{p_1+p_2-p_5-p_6}{2},\\
\psi_{-,2}&=\frac{\phi_3+\phi_4}{\sqrt2}
=\frac{p_3+p_4-p_7-p_8}{2}.
\end{aligned}}
$$
A pair at the upper <energy> $\alpha-\beta_S$ is
$$
\boxed{\begin{aligned}
\psi_{+,1}&=\frac{\phi_1-\phi_2}{\sqrt2}
=\frac{p_1-p_2-p_5+p_6}{2},\\
\psi_{+,2}&=\frac{\phi_3-\phi_4}{\sqrt2}
=\frac{p_3-p_4-p_7+p_8}{2}.
\end{aligned}}
$$
The four coefficient vectors are mutually orthonormal. Any orthogonal mixing within either equal-energy pair is an equally valid choice of <molecular orbitals>. At this limiting parameter value, additional coincidences with the one-dimensional levels occur: the full eight-orbital system is four isolated dimers, with four bonding and four antibonding levels.
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