= Solution
Each $\mathrm{Cp^*Al}$ fragment leaves two electrons after allocating one aluminium valence electron to its monovalent cyclopentadienyl ligand. Four fragments therefore give \b[eight cluster electrons]. In the tetrahedral arrangement, fragment lone-pair/donor and vacant acceptor orbitals combine into delocalized cluster orbitals. The occupied low-energy set has $a_1^2t_2^6$ symmetry, analogous to a filled 1S and 1P shell; higher combinations remain empty. This is a multicentre description, not six independent two-electron Al–Al bonds, which would require twelve electrons.
A schematic molecular-orbital diagram should show the occupied nondegenerate $a_1$ level below the occupied triply degenerate $t_2$ set, separated from the empty higher levels. Detailed energies and ordering of the empty levels require an orbital calculation. On descending Group 13, stabilization of the $ns^2$ pair and reduced favourable $s$–$p$ mixing can weaken the tendency towards the same compact Al-like bonding pattern; metal–metal distances and preferred aggregation change. Ligand size, donor/acceptor properties and hapticity also tune fragment orbital energies and steric access, so neither substitution of the metal nor of Cp\* is electronically innocent.
\Image[/past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2-cluster-counting.png]
{title=Original cluster orbital filling, octahedral electron counting and capped-cube connectivity}
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