Polyhedral skeletal electron pair theory: Difference between revisions

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{{Short description|Electron counting rules}}

In [[chemistry]] the '''polyhedral skeletal electron pair theory''' (PSEPT) provides [[electron counting]] rules useful for predicting the structures of [[cluster compound|clusters]] such as [[Boranes|borane]] and [[carborane]] clusters. The electron counting rules were originally formulated by [[Kenneth Wade]],<ref name=wade71>{{cite journal|title=The structural significance of the number of skeletal bonding electron-pairs in carboranes, the higher boranes and borane anions, and various transition-metal carbonyl cluster compounds|author-link=Kenneth Wade |first=K. |last=Wade |journal=J. Chem. Soc. D |date=1971 |volume=1971|pageissue=15 |pages=792–793 |DOIdoi=10.1039/C29710000792}}</ref> and were further developed by others including [[Michael Mingos]];<ref name=mingos72>{{cite journal|title=A General Theory for Cluster and Ring Compounds of the Main Group and Transition Elements |author-link=Michael Mingos|first=D. M. P. |last=Mingos |year = 1972|journal=Nature Phys.Physical Sci.Science |volume=236 |pageissue=68 |pages=99–102 |doi=10.1038/physci236099a0|bibcode=1972NPhS..236...99M }}</ref> and others; they are sometimes known as '''Wade's rules''' or the '''Wade–Mingos rules'''.<ref name=welch13>{{cite journal|title=The significance and impact of Wade's rules |first=Alan J. |last=Welch |journal=Chem. Commun. |date=2013|volume=49 |pageissue=35 |pages=3615–3616 |DOIdoi=10.1039/C3CC00069A|pmid=23535980 }}</ref> The rules are based on a [[molecular orbital]] treatment of the bonding.<ref name=Wade>{{cite journal|title=Structural and Bonding Patterns in Cluster Chemistry|last=Wade |first=K.|authorlink=Kenneth Wade|journal=Adv. Inorg. Chem. Radiochem. |series=Advances in Inorganic Chemistry and Radiochemistry |year=1976|volume=18|pages=1–66|doi=10.1016/S0065-2792(08)60027-8|isbn=9780120236183 }}</ref><ref name=lecture>{{cite journal|title= Lecture notes distributed at the University of Illinois, Urbana-Champaign|last=Girolami |first=G.|date=Fall 2008}} These notes contained original material that served as the basis of the sections on the 4''n'', 5''n'', and 6''n'' rules.</ref><ref name=Nyholm>{{cite journal|title=Nyholm Memorial Lectures|last=Gilespie |first=R. J.|journal=[[Chemical Society Reviews|Chem. Soc. Rev.]]|year=1979|volume=8|issue=3|pages=315–352|doi=10.1039/CS9790800315}}</ref><ref name=Mingosmingos84>{{cite journal|title=Polyhedral Skeletal Electron Pair Approach|last=Mingos |first=D. M. P.|authorlink=D. M. P. Mingos|journal=[[Acc. Chem. Res.]]|year=1984|volume=17|issue=9|pages=311–319|doi=10.1021/ar00105a003}}</ref> These rules have been extended and unified in the form of the [[Jemmis mno rules|Jemmis ''mno'' rules]].<ref name=mnorulesjemmis01>{{cite journal|title=A Unifying Electron-counting rule for Macropolyhedral Boranes, Metallaboranes, and Metallocenes|journal=[[J. Am. Chem. Soc.]]|year=2001|volume=123|issue=18|pmid=11457198|pages=4313–4323|doi=10.1021/ja003233z|last1=Jemmis|first1=Eluvathingal D.|last2=Balakrishnarajan|first2=Musiri M.|last3=Pancharatna|first3=Pattath D.}}</ref><ref name=mnoreviewjemmis02>{{cite journal|title=Electronic Requirements for Macropolyhedral Boranes|journal=[[Chem. Rev.]]|year=2002|volume=102|issue=1|pages=93–144|doi=10.1021/cr990356x|last1=Jemmis|first1=Eluvathingal D.|last2=Balakrishnarajan|first2=Musiri M.|last3=Pancharatna|first3=Pattath D.|pmid=11782130}}</ref>

==Predicting structures of cluster compounds==

[[File:Re4CO122-.svg|thumb|The structure of the [[butterfly cluster compound]] [Re₄(CO)₁₂]^2-2− conforms to the predictions of PSEPT.]]

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 [[boraneboranes]]s and [[carborane]]s. For such clusters, the structures are based on [[deltahedra]], which are [[polyhedra]] 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]], 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.

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A molecular orbital treatment can be used to rationalize the bonding of cluster compounds of the 4''n'', 5''n'', and 6''n'' types.

[[File:Re4CO122-.svg|thumb|The structure of the [[butterfly cluster]] [Re₄(CO)₁₂]^2- conforms to the predictions of PSEPT.]]

===4''n'' rules===

[[File:Deltahedral-borane-cluster-array-numbered-3D-bs-17.png|thumb|[[:en:Ball-and-stick model|Ball-and-stick model]]s showing the structures of the [[:en:Boron|boron]] skeletons of [[:en:Boranes|borane]] [[:en:Atom cluster|clusters]].]]

The following [[polyhedra]] are ''closo'' polyhedra, and are the basis for the 4''n'' rules; each of these have triangular faces.<ref name="Cotton&Wilkinson6th">{{Cotton&Wilkinson6th}}</ref> The number of vertices in the cluster determines what polyhedron the structure is based on.

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|-

| 10

| [[Gyroelongated square bipyramid|Bicapped square antiprismantiprismatic molecular geometry]]

|-

| 11

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| ''hypho''

| ''n'' + 3 vertex ''closo'' polyhedron with 3 missing vertices

|-

| 4n + 10

| ''klado''

| ''n'' + 4 vertex ''closo'' polyhedron with 4 missing vertices

|}

[[File:Pb9 Cluster.png|thumb|150px|Rightright|{{chem|Pb|10|2−}}]]

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 {{nowrap|6 × 9 + 16 × 2 − 6 × 10}} = {{nowrap|86 – 6 × 1060}} = 26. Therefore, the cluster is a ''closo'' polyhedron because {{nowrap|1=''n'' = 6}}, with {{nowrap|1=4''n'' + 2 = 26}}.

[[File:S4 Cluster.png|thumb|150px|Rightright|{{chem|S|4|2+}}]]

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 [[Boranes|boron hydride]] clusters, each boron atom connected to 3 or more vertices has one terminal hydride, while a boron atom connected to 2two other vertices has 2two terminal hydrogenshydrogen atoms. If more hydrogenshydrogen 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.

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:Starting from a trigonal bipyramid, a face is capped. The carbonyls have been omitted for clarity.

[[File:B5H5 cluster.png|thumb|150px|{{chem|B|5|H|5|4−}}, hydrogenshydrogen atoms omitted]]

Example:<ref name=Cotton3>{{cite book| last = Cotton| first = Albert| title = Chemical Applications of Group Theory| year = 1990| publisher = John Wiley & Sons| pages = [https://archive.org/details/isbn_9780471510949/page/205 205–251]| isbn = 0-471-51094-7| url-access = registration| url = https://archive.org/details/isbn_9780471510949/page/205}}</ref> {{chem|B|5|H|5|4−}}

Example:<ref name=Cotton3>{{cite book

| last= Cotton| first = Albert| title = Chemical Applications of Group Theory

| year = 1990|publisher = John Wiley & Sons

| pages = 205–251| isbn = 0-471-51094-7}}</ref> {{chem|B|5|H|5|4−}}

:Electron count: 5 × B + 5 × H + 4 (for the negative charge) = 5 × 3 + 5 × 1 + 4 = 24

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The rules are useful in also predicting the structure of [[carborane]]s.

Example: C₂B₇H₁₃

:Electron count = 2 × C + 7 × B + 13 × H = 2 × 4 + 37 × 73 + 13 × 1 = 42

:Since n in this case is 9, 4''n'' + 6 = 42, the cluster is ''arachno''.

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As discussed previously, the 4''n'' rule mainly deals with clusters with electron counts of {{nowrap|4''n'' + ''k''}}, in which approximately 4 [[electrons]] 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 which each vertex is connected to 3 other vertices. The 3-connected polyhedra are the [[dual polyhedron|duals]] of the deltahedra. The common types of 3-connected polyhedra are listed below.

[[File:P4 diagram.gif|thumb|150px|right|5''n'' cluster: P₄]]

[[File:P4S3 diagram.gifpng|thumb|150px|right|5''n'' + 3 cluster: P₄S₃]]

[[File:Phosphorus trioxide.svg|thumb|100px|right|5''n'' + 6 cluster: P₄O₆]]

{|class = "wikitable" style="text-align:center"

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|14||Dual of triaugmented triangular prism (K₅ [[associahedron]])

|-

|16||Square [[Square truncated trapezohedron]]

|-

|18||Dual of edge-contracted icosahedron

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===Isolobal vertex units===

Provided a vertex unit is [[Isolobal principle|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.<ref name="Cotton&Wilkinson6th"/> 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 [[sigma bond|σ]]- and [[pi bond|π-bonding]] are taken into account making 14 electrons.

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==Bonding in cluster compounds==

===Polyhedron===

;B₂H₆

[[File:B2H6MOdiagram.JPG|thumb|Molecular-orbital (MO) diagram of B₂H₆. Atoms and their corresponding orbitals are colored the same. Green MOs symbolize bonding, while red symbolize anti-bonding.]]

:The bonding in diborane is best described by treating each B as sp³-[[Orbital hybridisation|hybridized]]. Two sp³-hybrid orbitals on each boron form the bonds to the terminal hydrogens. The remaining sp³-orbitals create the bonds with the bridging hydrogens. Because the angles in the diborane structure are not tetrahedral the orbitals also likely contain some sp² character.

{{clear}}

;''closo''-{{chem|B|6|H|6|2−}}

[[File:B6H6MOdiagram.jpg|thumb|200px|MO diagram of {{chem|B|6|H|6|2−}} showing the orbitals responsible for forming the cluster. Pictorial representations of the orbitals are shown; the MO sets of T and E symmetry will each have two or one additional pictorial representation, respectively, that are not shown here.]]

:The boron atoms lie on each vertex of the octahedron and are sp hybridized.<ref name=Cotton3 /> 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.<ref name="mnorulesjemmis02"/> 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. {{nowrap|''n'' + 1}}.

{{clear}}

;Main group atom clusters

:The bonding in other main group cluster compounds follow similar rules as those described for the boron cluster bonding. The atoms at the vertex [[Orbital hybridisation|hybridize]] in a way which allows the lowest energy structure to form.

: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. {{nowrap|''n'' + 1}}.

===Transition metal clusters===

Transition metal clusters use the d orbitals for [[Chemical bond|bonding]]. Thus, sothey have up to nine bonding orbitals, instead of only the four present in boron and main group clusters.<ref name=king>{{cite journal|title=Chemical Applications of Group Theory and Topology.7. A Graph-Theoretical Interpretation of the Bonding Topology in Polyhedral Boranes, Carboranes, and Metal Clusters|last1=King |first1=R. B. |last2=Rouvray |first2=D. H.|journal=[[J. Am. Chem. Soc.]]|year=1977|volume=99|issue=24|pages=7834–7840|doi=10.1021/ja00466a014}}</ref><ref Therename=RCR>{{cite isjournal|last1=Kostikova also|first1=G. moreP. bonding|last2=Korolkov flexibility|first2=D. inV.|title=Electronic transitionStructure metalof clustersTransition dependingMetal onCluster whetherComplexes vertexwith metalWeak- electronand pairsStrong-field areLigands|journal=Russ. involvedChem. inRev.|year=1985|volume=54|issue=4|pages=591–619|doi=10.1070/RC1985v054n04ABEH003040|bibcode cluster= bonding1985RuCRv..54..344K or|s2cid=250797537 appear}}</ref> as lonePSEPT pairs.also applies to [[metallaborane]]s

The cluster chlorides and carbonyls of transition metals will be briefly discussed here as they represent opposite ends of the [[spectrochemical series]] and show important features of the differences between transition metal clusters with different ligands.<ref name=RCR>{{cite journal|last1=Kostikova |first1=G. P. |last2=Korolkov |first2=D. V.|title=Electronic Structure of Transition Metal Cluster Complexes with Weak- and Strong-field Ligands|journal=Russ. Chem. Rev.|year=1985|volume=54|issue=4|pages=591–619|doi=10.1070/RC1985v054n04ABEH003040|bibcode = 1985RuCRv..54..344K }}</ref> In chloride clusters the energy splitting of the valence d orbitals increases upon formation of the [[Cluster chemistry|cluster]]. The number and symmetry of these orbitals are dependent upon the type and structure of each individual cluster complex.<ref name=RCR /> Conversely in the carbonyl clusters the energy splitting of the valence d orbitals is greater before the formation of the cluster.<ref name=RCR />

===Clusters with interstitial atoms===

<gallery caption="MO diagram clusters metal chlorides and metal carbonyls" widths="250px" heights="300px" perrow="2">

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 [Fe₆C(CO)₁₆]^2-. In such cases, the rules of electron counting assume that the interstitial atom contributes all valence electrons to cluster bonding. In this way, [Fe₆C(CO)₁₆]^2- is equivalent to [Fe₆(CO)₁₆]^6- or [Fe₆(CO)₁₈]^2-.<ref>{{cite book |doi=10.1002/0470862106.ia097|chapter=Cluster Compounds: Inorganometallic Compounds Containing Transition Metal & Main Group Elements|title=Encyclopedia of Inorganic Chemistry|year=2006|last1=Fehlner|first1=Thomas P.|isbn=0470860782}}</ref>

Image:TMclusterCl.JPG| General MO diagram of metal chloride structures. Green MOs represent bonding orbitals while red represent anti-bonding orbitals. The labeling on the MOs is as follows: s = σ, p = π, d = δ bonding, with * denoting anti-bonding interactions.

Image:TMclusterCO.JPG| General MO diagram for metal carbonyl clusters. Green MOs represent bonding orbitals while red represent anti-bonding orbitals. The labeling on the MOs is as follows: s = σ, p = π, d = δ bonding with * denoting anti-bonding interactions.

==See also==

</gallery>

* [[Styx rule]]

==References==

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{{Organometallics}}

{{Chemical bonds}}

{{DEFAULTSORT:Polyhedral Skeletal Electron Pair Theory}}

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[[Category:Inorganic chemistry]]

[[Category:Organometallic chemistry]]

[[Category:Cluster chemistry]]