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Aluminium (III) chloride forms a dimer because
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Higher coordination number can be achieved by aluminium
$\mathrm{AlCl}_3$ can form dimer and exists as $\mathrm{Al}_2 \mathrm{Cl}_6$, aluminium has vacant d-orbitals which can accommodate electron from chlorine atom. $\mathrm{AlCl}_3$ is an electron deficient compound in $\mathrm{Al}$ (octet incomplete) thus behaves as Lewis acid and $\mathrm{Al}$ completes it by taking electron pair from $\mathrm{Cl}$-atom.
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