Polyhedral skeletal electron pair theory

In chemistry the polyhedral skeletal electron pair theory provides electron counting rules used to predict the structure of electron deficient clusters. They were originally formulated by K. Wade and further developed by Mingos and others, and are sometimes known as Wade's Rules or Wade / Mingos rules. The rules were originally developed to aid the prediction of the structures of borane and carborane cluster compounds and are based on a molecular orbital treatment of the bonding.
The rules only apply to clusters that have the same deltahedral geometry as the boranes and carboranes. That is to say their structures must be based on polyhedra where every face is triangular. The deltahedral boranes and carboranes are classified as closo-, nido-, arachno- or hypho-, based on whether they represent a complete (closo-) Deltahedron, or a Deltahedron that is missing one (nido-), two (arachno-) or three (hypho) vertices.

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The rules for boranes and carboranes

The skeletal electron counts for the four types of deltahedral cluster are:-

• n vertex closo (n+1)
• n vertex nido (n+2)
• n vertex arachno (n+3)
• n vertex hypho (n+4)

The skeletal electron counts are determined by totalling the following:-

• 2 from each BH unit
• 3 from each CH unit
• 1 from each additional hydrogen atom (over and above the ones on the BH and CH units)
• the anionic charge electrons

Example of C₂B₇H₁₃

Framework electron count: 14(7 X BH) + 6 (2 X CH)+ 4 (additional H) = 24

As the number of vertices is 9 the prediction is arachno (24 = 2(9+3))

Example of B₆H₆^2-

Framework electron count: 12(6 X BH) + 2 (anionic charge) = 14

As the number of vertices is 6 the prediction is closo (14 = 2(6+1))

Total electron count formulation for boranes and carboranes

There is an alternative and more straightforward version of the rules that uses a total valence electron count. For boranes and carboranes the total valence electrons are counted plus any charge electrons,the rules are simply:-

• n vertex closo (4n+2) electrons
• n vertex nido (4n+4) electrons
• n vertex arachno (4n+6) electrons
• n vertex hypho (4n+8) electrons

Example of C₂B₇H₁₃

Total valence electrons = (2 X 4) + (3 X 7) + (13X1)= 42

As 42 = (4 X 9) + 6 the structure is predicted to be arachno

Example of B₆H₆^2-

Total valence electrons = (6 X 3) + (6 X 1) + 2 = 26

As 26 = (4 X 6) + 2 the structure is predicted to be closo

Theoretical basis for the rules for boranes

A brief insight into the theory behind the rules through a consideration of the anionic closo cluster B₆H₆^2-. The boron atoms sit at the vertices of an octahedron, each using one sp hybrid, that points radially out of the polyhedron, to form a σ-bond to a hydrogen atom. Each boron atom therefore has 3 orbitals remaining to contribute to framework molecular orbitals i.e.:

• 1 sp hybrid orbital pointing radially into the center of the cluster
• 2 p orbitals tangential to the surface

The 18 framework molecular orbitals, (MOs), derived from the 18 boron atomic orbitals are :-

• 1 bonding MO at the center of the cluster and 5 antibonding MOs from the 6 sp radial hybrid orbitals
• 6 bonding MOs and 6 antibonding MOs from the 12 tangential p orbitals.

The total skeletal bonding orbitals is therefore 7 i.e (n+1).

Isolobal vertex units

Provided a vertex unit is isolobal with BH then it can, in principle at least, be substituted for a BH unit. The CH⁺ unit is isolobal, hence the reason why the rules are applicable to carboranes.
Additionally there are isolobal transition metal units. For example Fe(CO)₃ provides 2 electrons. The derivation of this is briefly as follows:-

• Fe has 8 valence electrons.
• Each carbonyl group is a net 2 electron donor after the internal σ and π bonding are taken into account making 14 electrons.
• 3 pairs are considered to be involved in Fe - CO σ-bonding and 3 pairs are involved in π back bonding from Fe to CO reducing the 14 to 2.

Transition metal clusters

As for boranes and carboranes there is an alternative and more straightforward version of the rules that uses a total valence electron count. The total valence electrons are counted plus any charge electrons. The rules are simply:-

• n vertex closo (14n+2) electrons
• n vertex nido (14n+4) electrons
• n vertex arachno (14n+6) electrons
• n vertex hypho (14n+8) electrons

Example Rh₆CO₁₆

Rhodium is group 9 with 9 valence electrons

Total electron count = (6 X 9) + (16 X 2)= 86

86 = (14 x 6) + 2

Therefore the prediction is closo

General References

• WebElements
• Greenwood, N. N.; Earnshaw, A. (1997). Chemistry of the Elements, 2nd Edition, Oxford:Butterworth-Heinemann. ISBN 0-7506-3365-4. 
• Cotton, F. Albert; Wilkinson, Geoffrey; Murillo, Carlos A.; Bochmann, Manfred (1999). Advanced Inorganic Chemistry (6th Edn.) New York:Wiley-Interscience. ISBN 0-471-19957-5.