Chemical topology polyhedra

In this chapter we have reviewed the basic tenets of a chemical topology scheme, one that can be applied to classify and effectively map the innumerable polyhedra, tessellations and networks, based upon a simple computation of their Schlafli symibols (n, p), from translation of their corresponding Wells point symbols. A restriction pointed up by this work, is that all structures in such a chemical topology scheme must, indeed, be simply connected. The phrase simply connected means that all edges, E, in a structure, be it a polyhedron, tessellation or network, must terminate... 

The choice of favored coordination polyhedra, or even the assignment of a coordination polyhedron to a given chemical structure, would appear to be very complicated in view of the large number of topologically distinct polyhedra with as few as seven vertices. However, the properties of atomic orbitals coupled with an assumption of maximum symmetry for a given hybrid of atomic orbitals makes this problem both tractable and interesting. 

Fig. 4.17 Unit cells of PAFs, a PAF-301, b PAF-302, c PAF-303, and d PAF-304, derived from topology design and geometry optimization with the force field method. Here, gray and pink spheres represent carbon and hydrogen atoms, respectively, while the blue polyhedron represents the tetrahedrally bonded carbon atoms. In addition, the yellow sphere denotes the pores in 3D PAFs. Reproduced from Ref. [103] with permission from the American Chemical Society...

The simplest and most important polyhedra are the so-called simple polyhedra, which can be continuously deformed into spheres (i.e., are topologically homeomorphic to a sphere). The simple polyhedra of chemical interest generally have the additional property of convexity. All vertices of a convex polyhedron lie entirely on one side of each of its faces so that a convex polyhedron can be laid on a flat tabletop with any face down. Convexity is not a topological property, for convex polyhedra in general can be continuously transformed into nonconvex polyhedra. 

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