Polyhedra classification

Vertices in Parent Polyhedron Classification Boron Atoms in Cluster Valence Electrons Framework Electron Pairs Examples Formally Derived From... 

Vertices in parent polyhedron Classification Boron atoms in cluster Valence electrons Framework electron pairs Examples Formally derived from... 

The structures of boranes can be grouped into several classifications. If the structure contains a complete polyhedron of boron atoms, it is referred to as a closo borane (closo comes from a Greek word meaning closed ). If the structure has one boron atom missing from a comer of the polyhedron, the structure is referred to as a nido borane (nido comes from a Latin word for nest ). In this type of structure, a polyhedron having n comers has (n - 1) comers that are occupied by boron atoms. A borane in which two comers are unoccupied is referred to as an arachno structure (arachno comes from a Greek word for web ). Other types of boranes have structures that are classified in different ways, but they are less numerous and will not be described. 

In addition, it is sometimes useful to relate the total valence electron count in boranes to the structural type. In closo boranes, the total number of valence electron pairs is equal to the sum of the number of vertices in the polyhedron (each vertex has a boron-hydrogen bonding pair) and the number of framework bond pairs. For example, in there are 26 valence electrons, or 13 pairs (= 2n + 1, as mentioned previously). Six of these pairs are involved in bonding to the hydrogens (one per boron), and seven pairs are involved in framework bonding. The polyhedron of the closo structure is the parent polyhedron for the other structural types. Table 15-8 summarizes electron counts and classifications for several examples of boranes. 

The purpose of this section is 1) to characterize and symbolize each polyhedron of Table 1 with the help of an appropriate notation that allows the classification of the various homopolyhedra arrangements I [i] I and their representation in the network built from them, 2) to illustrate these arrangements from a few examples to demonstrate the generality and the usefulness of our systematic and, in the process, to reveal the internal coherence of structures apparently very different. Unknown I [i] I arrangements will be suggested. 

Valence electron counts provide a convenient basis to classify these structural types. Various schemes for relating electron counts to structures have been proposed most are based on rules formulated by Wade. Wade s classification scheme is summarized in Table 15.7. It is remarkable that the pairs of framework bonding electrons solely depend on the number of corners of the parent polyhedron. The challenge is identifying the parent polyhedron on the basis of the borane formula. An example of a counting scheme to conveniently deduce the number of framework bonding pairs (and therefore the parent polyhedron) is presented later in this section. 

Symmetry point groups specify the symmetry properties of a general three-dimensional object of finite extension and apply to all such objects. However, if the object consists of only a finite number of points, such as the vertices of a polyhedron or the atoms in a molecule, then further classification of... 

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