Polyhedral skeletal electron pair theory
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 (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 and carboranes. 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. 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. 4n" rules. The following polyhedra 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 is enumerated. For each transition metal present, 10 electrons are subtracted from the total electron count. For example, in Rh(CO) 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: 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: Example: Example: Os(CO) Example: The rules are useful in also predicting the structure of carboranes. Example: CBH 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: The skeletal electron counts are determined by summing the total of the following number of electrons: 5"n" rules. As discussed previously, the 4"n" rule mainly deals with clusters with electron counts of , 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 duals of the deltahedra. The common types of 3-connected polyhedra are listed below. The 5"n" rules are as follows. Example: P Example: PS Example: PO 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 Hexane (CH ) 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) provides 2 electrons. The derivation of this is briefly as follows: Bonding in cluster compounds. 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. 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 [FeC(CO)]. In such cases, the rules of electron counting assume that the interstitial atom contributes all valence electrons to cluster bonding. In this way, [FeC(CO)] is equivalent to [Fe(CO)] or [Fe(CO)].