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

Use the trapezium's printed axes, with principal axis $z$ in its plane and $y$ perpendicular to the plane. For either site pair $(1,2)$ or $(3,4)$, the group representation has characters of a representation $(2,0,2,0)$ in the order $(E,C_2(z),\sigma (xz),\sigma (yz))$: identity and the molecular-plane mirror fix both s orbitals, while the other two operations exchange them. The character inner products give ${\Gamma }_{12}={\Gamma }_{34}=A_1\oplus B_1$ in the C2v molecular point group.

Within each pair the in-phase and out-of-phase combinations have $A_1$ and $B_1$ symmetry respectively. One normalized choice isThe sign convention for $d$ is chosen to make the off-diagonal interaction positive in the corresponding two-by-two block. It does not change the physical predictions. These are pair symmetry-adapted linear combinations; the actual molecular orbitals are linear combinations within each of the two equal-symmetry pairs, found by diagonalizing the blocks.

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.