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ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2022/paper-1/a4/b/i/solution

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Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 1 / A4 / b / i / Solution by

Codex  0 2026-10-20

Write

p_{j} = p_{z j}

and use the orthonormal basis of the Hückel method. The four normalized one-dimensional symmetry-adapted linear combinations are

Φ_{A_{1 u}} Φ_{A_{2 u}} Φ_{B_{1 u}} Φ_{B_{2 u}} = \frac{p _{1} - p _{2} + p _{3} - p _{4} + p _{5} - p _{6} + p _{7} - p _{8}}{8}, = \frac{p _{1} + p _{2} + p _{3} + p _{4} + p _{5} + p _{6} + p _{7} + p _{8}}{8}, = \frac{p _{1} - p _{2} - p _{3} + p _{4} + p _{5} - p _{6} - p _{7} + p _{8}}{8}, = \frac{p _{1} + p _{2} - p _{3} - p _{4} + p _{5} + p _{6} - p _{7} - p _{8}}{8} .

For example, the

x z

mirror plane exchanges sites

1 \leftrightarrow 6

,

2 \leftrightarrow 5

,

3 \leftrightarrow 4

,

8 \leftrightarrow 7

, without changing the sign of an individual

p_{z}

orbital. The all-positive combination is unchanged by every vertical mirror but changes sign under

σ_{h}

A_{2 u}

. The alternating combination changes sign under both sorts of vertical mirror, as required for

A_{1 u}

. The two remaining sign patterns distinguish

B_{1 u}

from

B_{2 u}

in the printed axes. The signs refer to the upper

p_{z}

lobes; the lower lobes have opposite quantum phases, and neither colour denotes an electric charge.

For a normalized coefficient vector

c

, the energy is

E = α + 2 \sum_{⟨ j, k ⟩} β_{jk} c_{j} c_{k} .

Every site has one short-bond and one long-bond neighbour. Equivalently, apply the Hückel method matrix to any coefficient and read off its two neighbouring signs. The results are

symmetry A_{1 u} A_{2 u} B_{1 u} B_{2 u} E α - β_{S} - β_{L} α + β_{S} + β_{L} α - β_{S} + β_{L} α + β_{S} - β_{L}

With negative resonance integrals and

∣ β_{S} ∣ > ∣ β_{L} ∣

, the

A_{2 u}

and

B_{2 u}

levels are occupied, while

B_{1 u}

and

A_{1 u}

are unoccupied in the closed-shell alternating-bond state.

Figure 1.
Original phase patterns for the four one-dimensional carbon-ring symmetry orbitals in the printed numbering and axes
. Details.

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