Polyhedral Isomerization

A polyhedral isomerization may be defined as a deformation of a specific polyhedron until its vertices define a new polyhedron t 2. Of particular interest are sequences of two polyhedral isomerization steps in which the polyhedron 

Consider a polyhedral isomerization sequence P, rp2 S3 in which Pi and are combinatorially equivalent. Such a polyhedral isomerization sequence may be called a degenerate polyhedral isomerization with Pj as the intermediate polyhedron. Structures undergoing such degenerate isomerization processes are often called fluxional. A degenerate polyhedral isomerization with a planar intermediate polyhedron (actually a polygon) may be called a planar polyhedral isomerization. The simplest 

Now consider the symmetry point group G (or, more precisely, the framework group ) of the above ML coordination compound. This group has IGI operations of which lf l are proper rotations so that IGI/I/ I = 2if the compound is achiral and IGI/I I = 1 if the compound is chiral (i.e., has no improper rotations). The n distinct permutations of the n sites in the coordination compound or cluster are divided into nM R right cosets which represent the permutational isomers since the permutations corresponding to the IWI proper rotations of a given isomer do not change the isomer but merely rotate it in space. This leads naturally to the concept of isomer count, I, namely, 

The concept of a topological representation is conveniently illustrated by the topological representations for rearrangements of coordination polyhedra having four and five vertices. More complicated topological representations for coordination polyhedra having six and eight vertices are discussed elsewhere.  

AetStet = sqSsq = 6 this is an example of the closure condition 48a = 485 required for a topological representation with vertices representing more than one type of polyhedron. 

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The role of polyhedra in the static description of chemical structures, including those of coordination compounds, makes the dynamic properties of polyhedra also of considerable interest. The central concept in the study of dynamic properties of polyhedra is that of a polyhedral isomerization, which may be defined as the deformation of a specific polyhedron lIi until the vertices define a new polyhedron II2. Of particular interest are sequences of two polyhedral isomerization steps Hi 112 113 in which the polyhedron II3 is equivalent to the polyhedron Hi but with some permutation of the vertices. Such a polyhedral isomerization process is called a degenerate polyhedral isomerization with II2 as the intermediate polyhedron. A degenerate polyhedral isomerization in which the intermediate II2 is a planar polygon may be called a planar polyhedral isomerization. 

Now consider some macroscopic aspects of polyhedral isomerizations as depicted by topological representations, which are graphs in which the vertices correspond to isomers and the edges correspond to isomerization steps. For isomerizations of a polyhedron having n vertices, the number of vertices in the topological representation is the isomer count I defined by the following... 

Polyhedral isomerizations may be studied using either a microscopic or macroscopic approach. The microscopic approach uses details of polyhedral topology to elucidate possible single polyhedral isomerization steps, namely which types of isomerization steps are possible. Such isomerization steps consist most commonly of so-called diamond square-diamond processes or portions thereof. The microscopic approach to polyhedral isomerizations is relevant to understanding fluxional processes in borane and metallaborane polyhedra. 

A polyhedral isomerization can be defined as a deformation of a specific polyhedron Vi until its vertices define a new polyhedron V2- Of particular interest are sequences of two polyhedral isomerization steps V Vi V3 in which the final polyhedron V3 is combinatorially (i.e., topologically) equivalent to the initial polyhedron although with some permutation of its vertices, generally not the identity permutation. In this sense two polyhedra V and V3 may be considered to be combinatorially equivalent [36] when there are three one-to-one mappings V, , and from the vertex, edge, and face sets, respectively, of Pi to... 

Pi and V3) through a square planar intermediate V2. Except for this simplest example, planar polyhedral isomerizations are unfavorable owing to excessive intervertex repulsion. 

Now consider polyhedra in the ordinary three-dimensional space of interest in chemical structures (i.e., d = 3). Gale diagrams of five- and six-vertex polyhedra can be embedded into one- or two-dimensional space, respectively, thereby simplifying analysis of their possible vertex motions leading to non-planar polyhedral isomerizations of these polyhedra of possible interest in a chemical context. 

A brief review of graph-theoretical ideas is to be found in Mathematical methods in coordination chemistry topological and graph-theoretical ideas in the study of metal clusters and polyhedral isomerizations by R. B. King, Coord. Chem. Rev. (1993) 122, 91. 

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