Topological Aspects of Polyhedral Isomerizations.
GEORGIA UNIV ATHENS DEPT OF CHEMISTRY
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Some topological properties of polyhedra are summarized with particular reference to properties relevant to polyhedral isomerizations such as inherent rigidity or fluxionality. Among the chemically significant deltahedra, the tetrahedron, octahedron, 4,4-bicapped square antiprism, and icosahedron are inherently rigid whereas the normally encountered 5, 8, 9, and 11 vertex deltahedra are inherently fluxional. Topological representations are graphs in which the vertices correspond to different polyhedral isomers and the edges represent single polyhedral isomerization steps. The Petersens and Desargues-Levy graphs are topological representations of 5 vertex systems and the double group pentagonal dodecahedron with K5 complete graphs on each of the 12 faces is a topological representation of a 6 vertex system. Additional ideas including hyperoctahedral restriction are necessary for tractable topological representations of systems having more than 6 vertices. Thus a hyperoctahedrally restricted topological representation of an 8 vertex system is a K4,4 bipartite graph of hexagons with cubes at the vertices of the K4,4 graph, hexagonal bipyramids at the edge midpoints of the K4,4 graph, square antiprisms at the vertices of the 8 hexagons, and bisdisphenoids at the midpoints of the edges of the 8 hexagons. In the study of 5 and 6 vertex polyhedra, Gale transformations can be used to reduce their effective dimensionalities to 1 and 2, respectively, so that all possible types of their non-planar isomerizations can be found.
- Theoretical Mathematics
- Atomic and Molecular Physics and Spectroscopy