Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2B / Section B / 36 / b / iii / 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 $\beta$, givingWith $\lambda =(E-\alpha )/\beta$, its secular equation isThe eigenvalues are thereforeAt $n=1$ 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.