= Solution
The nitrogen pair and the four-carbon set are separately invariant under the <D2h point group>. Construct <symmetry-adapted linear combinations> within these two sets; mixing equivalent representations across the sets is left for the <Hückel method> calculation.
For the nitrogen $p_z$ <atomic orbitals>, normalized combinations are
$$
\boxed{N_+=\frac{\chi_1+\chi_4}{\sqrt2}\sim B_{1u},\qquad
N_-=\frac{\chi_1-\chi_4}{\sqrt2}\sim B_{3g}.}
$$
For the carbon $p_z$ <atomic orbitals>, take
$$
\boxed{\begin{aligned}
C_0&=\tfrac12(\chi_2+\chi_3+\chi_5+\chi_6)\sim B_{1u},\\
C_y&=\tfrac12(\chi_2-\chi_3-\chi_5+\chi_6)\sim B_{3g},\\
C_x&=\tfrac12(\chi_2+\chi_3-\chi_5-\chi_6)\sim B_{2g},\\
C_{xy}&=\tfrac12(\chi_2-\chi_3+\chi_5-\chi_6)\sim A_u.
\end{aligned}}
$$
These six functions are <orthonormal> under the zero-overlap assumption. The suffixes $x,y,xy$ describe coefficient patterns across the carbon sites; the sign acquired by each underlying $p_z$ function must also be included when assigning an <irreducible representation>. In particular, reflection through the molecular plane changes the sign of every function, while inversion sends $\chi_1$ to $-\chi_4$. Ignoring that orbital sign would give incorrect $g/u$ labels.
The complete $\pi$ <group representation> is therefore
$$
\boxed{\Gamma_\pi=2B_{1u}+2B_{3g}+B_{2g}+A_u.}
$$
The diagram shows all six ring <symmetry-adapted linear combinations> and the resulting <energy> ordering used in part (iii).
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2023/paper-1-pyrazine-symmetry.png]
{title=Original normalized pyrazine symmetry-orbital phase patterns and Hückel energy diagram, with nitrogen at sites 1 and 4}
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