Past exam of the chemistry A course of the University of Cambridge / 2025 / Paper 1 / A5 / c

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 areThe one-dimensional $B_{2g}$ and $A_u$ blocks have $\lambda =1$ and $-1$, respectively. The characteristic polynomial of the $B_{1u}$ block factors asso its three molecular orbital energies areFor 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 $-(1.86-2.12)\simeq 0.26$. 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 $\beta <0$, areThe 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 $D_{2h}$.