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

Write $p_i$ for the orthonormal $2p_z$ atomic orbital on numbered site $i$, with every positive lobe pointing along $+z$. A normalized set of symmetry-adapted linear combinations isEach has unit squared coefficient sum, and every distinct pair has zero inner product. The orbital representation therefore reduces to $\boxed{3B_{1u}+3B_{3g}+B_{2g}+A_u}$. For example, the fully symmetric site sum still has $B_{1u}$ orbital symmetry because a $p_z$ orbital itself changes sign under operations that reverse $z$. Ignoring that sign would assign the wrong representations.