= Solution
The long edge is $3$–$4$. In the orthonormal pair <basis> $(a,b)$, its weaker resonance integral $n\beta$ changes the second diagonal element. The two short side edges contribute an inter-pair <matrix element> $\beta$, giving
$$
H_{A_1}=\begin{pmatrix}\alpha+\beta&\beta\\\beta&\alpha+n\beta\end{pmatrix}.
$$
With $\lambda=(E-\alpha)/\beta$, its secular equation is
$$
(1-\lambda)(n-\lambda)-1=0,
\quad\lambda^2-(n+1)\lambda+(n-1)=0.
$$
The <eigenvalues> are therefore
$$
\boxed{E_{A_1}=\alpha+\frac\beta2\left[n+1\pm\sqrt{n^2-2n+5}\right].}
$$
At $n=1$ they become $\alpha+2\beta$ and $\alpha$, agreeing with the square limit. <Symmetry> permits mixing of the two $A_1$ pair functions; treating each pair as an isolated bond would miss the off-diagonal term.
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