= Solution
Use a common Coulomb integral $\alpha$ on each site, a resonance integral $\beta$ on each adjacent pair, zero interaction across a diagonal, and the identity overlap <matrix>. In site order $(1,2,3,4)$ the <Hückel method> gives
$$
H=\begin{pmatrix}\alpha&\beta&0&\beta\\\beta&\alpha&\beta&0\\0&\beta&\alpha&\beta\\\beta&0&\beta&\alpha\end{pmatrix}.
$$
Applying it to the four <symmetry-adapted linear combinations> gives
$$
\boxed{E_{A_{1g}}=\alpha+2\beta,\qquad E_{E_u}=\alpha\ \text{(twice)},\qquad E_{B_{1g}}=\alpha-2\beta.}
$$
For the conventional bonding sign $\beta<0$, the all-in-phase combination is lowest and the alternating combination is highest. The $E_u$ combinations have canceling nearest-neighbour interactions and are nonbonding within this model.
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