In
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
the polyhedral skeletal electron pair theory (PSEPT) provides
electron counting
In chemistry, electron counting is a formalism for assigning a number of valence electrons to individual atoms in a molecule. It is used for classifying compounds and for explaining or predicting their electronic structure and chemical bond, bondi ...
rules useful for predicting the structures of
clusters
may refer to:
Science and technology Astronomy
* Cluster (spacecraft), constellation of four European Space Agency spacecraft
* Cluster II (spacecraft), a European Space Agency mission to study the magnetosphere
* Asteroid cluster, a small ...
such as
borane
Borane is an inorganic compound with the chemical formula . Because it tends to dimerize or form adducts, borane is very rarely observed. It normally dimerizes to diborane in the absence of other chemicals. It can be observed directly as a c ...
and
carborane
Carboranes (or carbaboranes) are electron-delocalized (non-classically bonded) clusters composed of boron, carbon and hydrogen atoms.Grimes, R. N., ''Carboranes 3rd Ed.'', Elsevier, Amsterdam and New York (2016), . Like many of the related boron ...
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
In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding ...
treatment of the bonding.
[ These notes contained original material that served as the basis of the sections on the 4''n'', 5''n'', and 6''n'' rules.] These rules have been extended and unified in the form of the
Jemmis ''mno'' rules.
Predicting structures of cluster compounds

Different rules (4''n'', 5''n'', or 6''n'') are invoked depending on the number of electrons per vertex.
The 4''n'' rules are reasonably accurate in predicting the structures of clusters having about 4 electrons per vertex, as is the case for many
boranes
A borane is a compound with the formula although examples include multi-boron derivatives. A large family of boron hydride clusters is also known. In addition to some applications in organic chemistry, the boranes have attracted much attention ...
and
carborane
Carboranes (or carbaboranes) are electron-delocalized (non-classically bonded) clusters composed of boron, carbon and hydrogen atoms.Grimes, R. N., ''Carboranes 3rd Ed.'', Elsevier, Amsterdam and New York (2016), . Like many of the related boron ...
s. For such clusters, the structures are based on
deltahedra
A deltahedron is a polyhedron whose faces are all equilateral triangles. The deltahedron was named by Martyn Cundy, after the Greek capital letter Delta (letter), delta resembling a triangular shape Δ.
Deltahedra can be categorized by the prope ...
, which are
polyhedra
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
in which every face is triangular. The 4''n'' clusters are classified as ''closo-'', ''nido-'', ''arachno-'' or ''hypho-'', based on whether they represent a complete (''closo-'')
deltahedron
A deltahedron is a polyhedron whose faces are all equilateral triangles. The deltahedron was named by Martyn Cundy, after the Greek capital letter delta resembling a triangular shape Δ.
Deltahedra can be categorized by the property of convexi ...
, 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 4''n'' 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 6''n'' rules can be implemented. The 6''n'' clusters have structures that are based on rings.
A molecular orbital treatment can be used to rationalize the bonding of cluster compounds of the 4''n'', 5''n'', and 6''n'' types.
4''n'' rules

The following
polyhedra
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
are ''closo'' polyhedra, and are the basis for the 4''n'' 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 4''n'' rules are enumerated in the following table.

When counting electrons for each cluster, the number of
valence electrons
In chemistry and physics, valence electrons are electrons in the outermost shell of an atom, and that can participate in the formation of a chemical bond if the outermost shell is not closed. In a single covalent bond, a shared pair forms with b ...
is enumerated. For each
transition metal
In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. The lanthanide and actinid ...
present, 10 electrons are subtracted from the total electron count. For example, in Rh
6(CO)
16 the total number of electrons would be = = 26. Therefore, the cluster is a ''closo'' polyhedron because , with .

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
Diborane(6), commonly known as diborane, is the chemical compound with the formula . It is a highly toxic, colorless, and pyrophoric gas with a repulsively sweet odor. Given its simple formula, borane is a fundamental boron compound. It has attr ...
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
In coordination chemistry, a ligand is an ion or molecule with a functional group that binds to a central metal atom to form a coordination complex. The bonding with the metal generally involves formal donation of one or more of the ligand's ...
are added to the metal centers to give the metals reasonable coordination numbers, and if any
hydrogen
Hydrogen is a chemical element; it has chemical symbol, symbol H and atomic number 1. It is the lightest and abundance of the chemical elements, most abundant chemical element in the universe, constituting about 75% of all baryon, normal matter ...
atoms
Atoms are the basic particles of the chemical elements. An atom consists of a nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other ...
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:
:Electron count: 10 × Pb + 2 (for the negative charge) = 10 × 4 + 2 = 42 electrons.
:Since ''n'' = 10, 4''n'' + 2 = 42, so the cluster is a ''closo'' bicapped square antiprism.
Example:
:Electron count: 4 × S – 2 (for the positive charge) = 4 × 6 – 2 = 22 electrons.
:Since ''n'' = 4, 4''n'' + 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: Os
6(CO)
18
:Electron count: 6 × Os + 18 × CO – 60 (for 6 osmium atoms) = 6 × 8 + 18 × 2 – 60 = 24
:Since ''n'' = 6, 4''n'' = 24, so the cluster is capped ''closo''.
:Starting from a trigonal bipyramid, a face is capped. The carbonyls have been omitted for clarity.

Example:
:Electron count: 5 × B + 5 × H + 4 (for the negative charge) = 5 × 3 + 5 × 1 + 4 = 24
:Since ''n'' = 5, 4''n'' + 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
carborane
Carboranes (or carbaboranes) are electron-delocalized (non-classically bonded) clusters composed of boron, carbon and hydrogen atoms.Grimes, R. N., ''Carboranes 3rd Ed.'', Elsevier, Amsterdam and New York (2016), . Like many of the related boron ...
s.
Example: C
2B
7H
13
:Electron count = 2 × C + 7 × B + 13 × H = 2 × 4 + 7 × 3 + 13 × 1 = 42
:Since n in this case is 9, 4''n'' + 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, 2''n'' + 2 skeletal electrons
*''n''-vertex ''nido'': ''n'' + 2 skeletal orbitals, 2''n'' + 4 skeletal electrons
*''n''-vertex ''arachno'': ''n'' + 3 skeletal orbitals, 2''n'' + 6 skeletal electrons
*''n''-vertex ''hypho'': ''n'' + 4 skeletal orbitals, 2''n'' + 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
5''n'' rules
As discussed previously, the 4''n'' rule mainly deals with clusters with electron counts of , in which approximately 4
electrons
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
are on each vertex. As more electrons are added per vertex, the number of the electrons per vertex approaches 5. Rather than adopting structures based on deltahedra, the 5n-type clusters have structures based on a different series of polyhedra known as the 3-connected
polyhedra
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
, in which each vertex is connected to 3 other vertices. The 3-connected polyhedra are the
duals
''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers.
Track listing
:* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, ...
of the deltahedra. The common types of 3-connected polyhedra are listed below.

The 5''n'' rules are as follows.
Example: P
4
:Electron count: 4 × P = 4 × 5 = 20
:It is a 5''n'' structure with ''n'' = 4, so it is tetrahedral
Example: P
4S
3
:Electron count 4 × P + 3 × S = 4 × 5 + 3 × 6 = 38
:It is a 5''n'' + 3 structure with ''n'' = 7. Three vertices are inserted into edges
Example: P
4O
6
:Electron count 4 × P + 6 × O = 4 × 5 + 6 × 6 = 56
:It is a 5''n'' + 6 structure with ''n'' = 10. Six vertices are inserted into edges
6''n'' rules
As more electrons are added to a 5''n'' cluster, the number of electrons per vertex approaches 6. Instead of adopting structures based on 4''n'' or 5''n'' rules, the clusters tend to have structures governed by the 6''n'' rules, which are based on rings. The rules for the 6''n'' structures are as follows.

Example: S
8
:Electron count = 8 × S = 8 × 6 = 48 electrons.
:Since ''n'' = 8, 6''n'' = 48, so the cluster is an 8-membered ring.
Hexane (C
6H
14)
:Electron count = 6 × C + 14 × H = 6 × 4 + 14 × 1 = 38
:Since ''n'' = 6, 6''n'' = 36 and 6''n'' + 2 = 38, so the cluster is a 6-membered chain.
Isolobal vertex units
Provided a vertex unit is
isolobal with BH then it can, in principle at least, be substituted for a BH unit, even though BH and CH are not isoelectronic. The CH
+ unit is isolobal, hence the rules are applicable to carboranes. This can be explained due to a
frontier orbital treatment.
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-backbonding from Fe to CO reducing the 14 to 2.
Bonding in cluster compounds
;''closo''-

:The boron atoms lie on each vertex of the octahedron and are sp hybridized.
One sp-hybrid radiates away from the structure forming the bond with the hydrogen atom. The other sp-hybrid radiates into the center of the structure forming a large bonding molecular orbital at the center of the cluster. The remaining two unhybridized orbitals lie along the tangent of the sphere like structure creating more bonding and antibonding orbitals between the boron vertices.
The orbital diagram breaks down as follows:
::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. .
Transition metal clusters
Transition metal clusters use the d orbitals for
bonding. Thus, they have up to nine bonding orbitals, instead of only the four present in boron and main group clusters.
PSEPT also applies to
metallaboranes
Clusters with interstitial atoms
Owing their large radii, transition metals generally form clusters that are larger than main group elements. One consequence of their increased size, these clusters often contain atoms at their centers. A prominent example is
6C(CO)16">e6C(CO)16sup>2-. In such cases, the rules of electron counting assume that the interstitial atom contributes all valence electrons to cluster bonding. In this way,
6C(CO)16">e6C(CO)16sup>2- is equivalent to
6(CO)16">e6(CO)16sup>6- or
6(CO)18">e6(CO)18sup>2-.
See Also
*
Styx rule
The styx rule, also known as Lipscomb's styx rule, can be used to calculate the structures of boranes. It was developed by William Lipscomb in 1954. The rule defines boranes to have four types of bonds besides the terminal Boron, B-Hydrogen, H b ...
References
General references
*
*
{{DEFAULTSORT:Polyhedral Skeletal Electron Pair Theory
Chemical bonding
Inorganic chemistry
Organometallic chemistry
Cluster chemistry