Localized bonds in clusters

The electronic structures of borane clusters were first successfully described using localized three-center- and two-center-two-electron bonds. These treatments have been replaced by the cluster electron-counting rule based on MO methods hence, why bother with the three-center bond model in a book about clusters Let s consider why there is value in a more localized approach. 

It should be no surprise that in a full MO treatment, the HOMO of B5H9 is the degenerate ring-cap tt MO set and the LUMO is the non-bonding ring 8 orbital. For the binary boron hydrides, the frontier orbitals and, by implication, the reactivity 

Exercise 2.4. Use the localized-bond model to rationalize the dipole moment exhibited by B5H9 (1.7 D). 

Exercise 2.5. Develop a three-center bond description of one resonance structure for [B6H6]2 and show that each B atom satisfies the eight-electron rule (review Section 1.3.5). 

Answer. There are a total of 30 orbitals and 26 electrons to be utilized in bonding. The external B-H bonds utihze 12 orbitals and 12 electrons leaving 18 orbitals and 14 electrons for cluster bonding. Four three-center B-B-B and three two-center B-B bonds utilize 12 + 6=18 orbitals and 8 + 6 = 14 electrons. They may be placed on the framework as shown in the diagram above where the top and the bottom of the octahedron are shown separately. By counting you can find that B(l) and B(2) are associated with three three-center-two-electron bonds (and a B-H bond), B(3) and B(4) with two three-center and one two-center bonds (and a B-H), and B(5) and B(6) with one three-center and two two-center bonds (and a B-H), i.e., eight electrons around each B. Notice that one would need to draw a considerable number of resonance structures to give all the boron atoms the same electronic environment. 

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