$($c$)$ An octahedral cluster molecule $E_6$ in $O_h$ point symmetry can be formed without a central

atom and is bonded together using the six s and eighteen p orbitals on the six atoms. Each

vertex comprises a sp hybrid orbital $($radial$)$ pointing into the cluster, and two p orbitals

$($tangential$)$ at right angles to each other and to the centre of the cluster $($see figure below$).$

These are the same as the ligand orbitals used in $-$ and $-$bonding in octahedral transition

metal complexes. The bonding overlaps between these orbitals are shown below in A$,$ B

and $C,$ and two of these are each part of a triply degenerate set and the other is singly

degenerate $-$ remember in $O_{\text{d}\text{a}\text{r}\text{r}}$ symmetry $x,y$ and $z$ are degenerate.

two tangential

p orbitals

radial

sp orbital

The reducible representation for these bonding orbitals is:

$($i$)$ Determine the three irreducible representations for the orbitals that contribute to cluster

bonding to form the seven MOs.

$($ii$)$ Match the symmetry labels of the irreducible representations with MOs A to C$.$ Hint: use

the inversion symmetry to do this.

$($iii$)$ Deduce the order in increasing energy of $A,B$ and $C$ by considering the number of nodal

planes that slice through the middle of the octahedral cage.