chemistry

Polyhedral skeletal electron pair theory

Polyhedral skeletal electron pair theory is a chemistry topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Polyhedral skeletal electron pair theory rather than just read about it. In short: In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of clusters such as borane and carborane clusters. The electron counting rules were originally formulated by Kenneth Wade, and were further developed by others including Michael Mingos; they are sometimes known as Wade's rules or the Wade–Mingos rules.

Key takeaways

Polyhedral skeletal electron pair theory belongs to chemistry; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Polyhedral skeletal electron pair theory to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Polyhedral skeletal electron pair theory from memory before moving on to harder problems.

Reference excerpt

In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of clusters such as borane and carborane clusters. The electron counting rules were originally formulated by Kenneth Wade, and were further developed by others including Michael Mingos; they are sometimes known as Wade's rules or the Wade–Mingos rules. The rules are based on a molecular orbital treatment of the bonding. These rules have been extended and unified in the form of the Jemmis mno rules.

Predicting structures of cluster compounds

Different rules (4n, 5n, or 6n) are invoked depending on the number of electrons per vertex. The 4n rules are reasonably accurate in predicting the structures of clusters having about 4 electrons per vertex, as is the case for many boranes and carboranes. For such clusters, the structures are based on deltahedra, which are polyhedra in which every face is triangular. The 4n clusters 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. However, hypho clusters are relatively uncommon due to the fact that the electron count is high enough to start to fill antibonding orbitals and destabilize the 4n structure. If the electron count is close to 5 electrons per vertex, the structure often changes to one governed by the 5n rules, which are based on 3-connected polyhedra. As the electron count increases further, the structures of clusters with 5n electron counts become unstable, so the 6n rules can be implemented. The 6n clusters have structures that are based on rings. A molecular orbital treatment can be used to rationalize the bonding of cluster compounds of the 4n, 5n, and 6n types.

4n rules

The following polyhedra are closo polyhedra, and are the basis for the 4n rules; each of these have triangular faces. The number of vertices in the cluster determines what polyhedron the structure is based on.

Using the electron count, the predicted structure can be found. n is the number of vertices in the cluster. The 4n rules are enumerated in the following table.

When counting electrons for each cluster, the number of valence electrons is enumerated. For each transition metal present, 10 electrons are subtracted from the total electron count. For example, in Rh6(CO)16 the total number of electrons would be 6 × 9 + 16 × 2 − 6 × 10 = 86 – 60 = 26. Therefore, the cluster is a closo polyhedron because n = 6, with 4n + 2 = 26.

Other rules may be considered when predicting the structure of clusters:

For clusters consisting mostly of transition metals, any main group elements present are often best counted as ligands or interstitial atoms, rather than vertices. Larger and more electropositive atoms tend to occupy vertices of high connectivity and smaller more electronegative atoms tend to occupy vertices of low connectivity. In the special case of boron hydride clusters, each boron atom connected to 3 or more vertices has one terminal hydride, while a boron atom connected to two other vertices has two terminal hydrogen atoms. If more hydrogen atoms are present, they are placed in open face positions to even out the coordination number of the vertices. For the special case of transition metal clusters, ligands are added to the metal centers to give the metals reasonable coordination numbers, and if any hydrogen atoms are present they are placed in bridging positions to even out the coordination numbers of the vertices. In general, closo structures with n vertices are n-vertex polyhedra. To predict the structure of a nido cluster, the closo cluster with n + 1 vertices is used as a starting point; if the cluster is composed of small atoms a high connectivity vertex is removed, while if the cluster is composed of large atoms a low connectivity vertex is removed. To predict the structure of an arachno cluster, the closo polyhedron with n + 2 vertices is used as the starting point, and the n + 1 vertex nido complex is generated by following the rule above; a second vertex adjacent to the first is removed if the cluster is composed of mostly small atoms, a second vertex not adjacent to the first is removed if the cluster is composed mostly of large atoms.

Example: Pb2−10

Electron count: 10 × Pb + 2 (for the negative charge) = 10 × 4 + 2 = 42 electrons. Since n = 10, 4n + 2 = 42, so the cluster is a closo bicapped square antiprism. Example: S2+4

Electron count: 4 × S – 2 (for the positive charge) = 4 × 6 – 2 = 22 electrons. Since n = 4, 4n + 6 = 22, so the cluster is arachno. Starting from an octahedron, a vertex of high connectivity is removed, and then a non-adjacent vertex is removed. Example: Os6(CO)18

Electron count: 6 × Os + 18 × CO – 60 (for 6 osmium atoms) = 6 × 8 + 18 × 2 – 60 = 24 Since n = 6, 4n = 24, so the cluster is capped closo. Starting from a trigonal bipyramid, a face is capped. The carbonyls have been omitted for clarity.

Example: B5H4−5

Electron count: 5 × B + 5 × H + 4 (for the negative charge) = 5 × 3 + 5 × 1 + 4 = 24 Since n = 5, 4n + 4 = 24, so the cluster is nido. Starting from an octahedron, one of the vertices is removed. The rules are useful in also predicting the structure of carboranes. Example: C2B7H13

Electron count = 2 × C + 7 × B + 13 × H = 2 × 4 + 7 × 3 + 13 × 1 = 42 Since n in this case is 9, 4n + 6 = 42, the cluster is arachno. The bookkeeping for deltahedral clusters is sometimes carried out by counting skeletal electrons instead of the total number of electrons. The skeletal orbital (electron pair) and skeletal electron counts for the four types of deltahedral clusters are:

n-vertex closo: n + 1 skeletal orbitals, 2n + 2 skeletal electrons n-vertex nido: n + 2 skeletal orbitals, 2n + 4 skeletal electrons n-vertex arachno: n + 3 skeletal orbitals, 2n + 6 skeletal electrons n-vertex hypho: n + 4 skeletal orbitals, 2n + 8 skeletal electrons The skeletal electron counts are determined by summing the total of the following number of electrons:

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

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Polyhedral skeletal electron pair theory

Start with the simplest possible case. Write down what Polyhedral skeletal electron pair theory claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In chemistry, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Polyhedral skeletal electron pair theory before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Polyhedral skeletal electron pair theory ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Polyhedral skeletal electron pair theory

In research: Polyhedral skeletal electron pair theory appears in chemistry research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Polyhedral skeletal electron pair theory in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Polyhedral skeletal electron pair theory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Chemical bonding, Cluster chemistry, Inorganic chemistry, so understanding it makes those chapters shorter.
In everyday life: Look for Polyhedral skeletal electron pair theory outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Polyhedral skeletal electron pair theory” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Polyhedral skeletal electron pair theory in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Polyhedral skeletal electron pair theory means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Polyhedral skeletal electron pair theory out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Polyhedral skeletal electron pair theory in simple terms?

Why does Polyhedral skeletal electron pair theory matter?

Because it connects several chemistry ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Polyhedral skeletal electron pair theory?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Polyhedral skeletal electron pair theory.