= Solution
For the doubly bridged tetracarbonyl, each <iron> has one terminal <carbonyl ligand>, two <bridging carbonyl ligands> and one <cyclopentadienyl ligand>. Neutral <electron counting> gives $8+5+2+2=17$ before direct bonding, so a single Fe–Fe bond gives 18 at each <iron>. For the triply bridged tricarbonyl, each <iron> instead has $8+5+3=16$ before direct bonding; formal completion to 18 requires a double Fe–Fe bond. Equivalently, the total fragment counts before <metal>–<metal> bonding are 34 and 32 respectively.
$$
\boxed{\begin{aligned}
\mathrm{Cp_2Fe_2(CO)_2(\mu\! -CO)_2}:&\quad d_{\mathrm{Fe-Fe}}=2.539\ \text{Å},\quad\text{formal single bond},\\
\mathrm{Cp^{*}_2Fe_2(\mu\! -CO)_3}:&\quad d_{\mathrm{Fe-Fe}}=2.265\ \text{Å},\quad\text{formal double bond}.
\end{aligned}}
$$
The increased direct <bond order> explains the substantial contraction. The <pentamethylcyclopentadienyl ligands> also alter steric and electronic effects, and bridge geometry changes, so this is not a controlled measurement of <bond order> alone. Nevertheless, the markedly shorter distance in the less carbonyl-saturated dimer is consistent with stronger direct Fe–Fe bonding. The isolated triply bridged compound was characterized structurally and magnetically in the primary work.<Dinuclear, 18-electron species having a triplet ground state: isolation, characterization, and crystal structure of photogenerated (η5-C5Me5)2Fe2(μ-CO)3 by Josephine Paw Blaha and others>[1985, pp. 4561–4562, isolation, crystal structure and triplet assignment]{ref}
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