= Solution
Write $p_j=p_{zj}$ and use the orthonormal basis of the <Hückel method>. The four normalized one-dimensional <symmetry-adapted linear combinations> are
$$
\begin{aligned}
\Phi_{A_{1u}}&=\frac{p_1-p_2+p_3-p_4+p_5-p_6+p_7-p_8}{\sqrt8},\\
\Phi_{A_{2u}}&=\frac{p_1+p_2+p_3+p_4+p_5+p_6+p_7+p_8}{\sqrt8},\\
\Phi_{B_{1u}}&=\frac{p_1-p_2-p_3+p_4+p_5-p_6-p_7+p_8}{\sqrt8},\\
\Phi_{B_{2u}}&=\frac{p_1+p_2-p_3-p_4+p_5+p_6-p_7-p_8}{\sqrt8}.
\end{aligned}
$$
For example, the $xz$ mirror plane exchanges sites $1\leftrightarrow6$, $2\leftrightarrow5$, $3\leftrightarrow4$, $8\leftrightarrow7$, without changing the sign of an individual $p_z$ orbital. The all-positive combination is unchanged by every vertical mirror but changes sign under $\sigma_h$, giving $A_{2u}$. The alternating combination changes sign under both sorts of vertical mirror, as required for $A_{1u}$. The two remaining sign patterns distinguish $B_{1u}$ from $B_{2u}$ 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 $\mathbf c$, the <energy> is
$$
E=\alpha+2\sum_{\langle j,k\rangle}\beta_{jk}c_jc_k.
$$
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
$$
\boxed{\begin{array}{c|c}
\text{symmetry}&E\\\hline
A_{1u}&\alpha-\beta_S-\beta_L\\
A_{2u}&\alpha+\beta_S+\beta_L\\
B_{1u}&\alpha-\beta_S+\beta_L\\
B_{2u}&\alpha+\beta_S-\beta_L
\end{array}}
$$
With negative resonance integrals and $|\beta_S|>|\beta_L|$, the $A_{2u}$ and $B_{2u}$ levels are occupied, while $B_{1u}$ and $A_{1u}$ are unoccupied in the closed-shell alternating-bond state.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2022/paper-1-ring-symmetry.png]
{title=Original phase patterns for the four one-dimensional carbon-ring symmetry orbitals in the printed numbering and axes}
Back to article page
