This chapter geometrically investigated the structure of clusters, the core of which represent the metal chains (linear or curved) of both identical and different elements. It was shown that the dimension of the structures of these clusters is more than three. To create a model of these chains in a higher dimension space, a new geometric approach has been developed that allows us to construct convex, closed polytopes of these chains. It consists of removing part of the octahedron edges necessary for constructing the octahedron and adding the same number of new edges necessary to build a closed polytope chain while maintaining the number of metal atoms and ligands and their valence bonds. As a result, it was found that metal chain polytopes consist of polytopes of higher dimension, adjacent to each other along flat sections.
TopAn example of a linear homo - element metal chain is shown in Figure 1 (Smart, Cook, & Woodward, 1977; Evans, Okrasinski, Pribula, & Norton, 1976; Gochin & Moss, 1980).
Figure 1. Schematic structure of cluster connection Os3(CO)12R2
In this example, we illustrate the sequence of actions to calculate the dimension of the metal chain. When determining the dimension of metal chains, atoms that are linearly located and do not have branching of the chain at their location do not matter (Zhizhin, 2018). At the same time, not only metal atoms, but also ligand atoms are of importance for determining the dimension of a compound. Therefore, Figure 1 can be represented as Figure 2.
Figure 2. The geometric structure of cluster connection Os3(CO)12R2
In Figure 2, the red edges correspond to chemical bonds, and the black edges serve only to create a convex closed figure. It can be seen that each link of the metal chain creates an octahedron with the center as a convex figure. Moreover, the center of each octahedron is simultaneously the vertex of another neighboring octahedron or two neighboring octahedrons. In Figure 2 at the centers of octahedrons are disposed Os atoms (vertices 2, 3, 4). At the vertices1, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17are disposed the functional group CO, at the vertices 5, 16 are disposed the functional group R (Zhizhin, 2016). It is easy to prove that an octahedron with a center has a dimension of 4. To do this, refer to Figure 3, which shows separately the first of the chain octahedron.
Polytope on Figure 3 has 7 vertices (f0=7), 18 edges (f1=18)
1 - 2, 2 - 3, 2 - 5, 2 - 9, 1 - 5, 5 - 3, 3 - 9, 1 - 9, 2 - 4, 2 - 11, 1 - 7, 5 - 7, 3 - 7, 7 - 9, 1 - 11, 5 - 11, 9 - 11, 3 - 11.
Polytope composed of eight tetrahedrons
1 - 2 - 9 - 11, 2 - 9 - 3 - 11, 2 - 5 - 3 - 11, 1 - 2 - 5 - 11, 1 - 2 - 5 - 7, 2 - 5 - 3 - 7, 2 - 9 - 3 - 7, 1 - 2 - 9 - 7.
To these three - dimensional figures you need to add an octahedron without a center
1 - 5 - 3 - 7 - 9 - 11.