Abstract
The geometry of clusters with ligands and a polyhedral frame is considered by the methods of studying the geometry of higher-dimensional polytopes, developed in the author's monograph. It is shown that these methods allow us to establish important details of cluster geometry, which elude analysis based on the representations of three-dimensional geometry. It is established that the well-known Kuban cluster is a 4-cross-polytope, which allows different variants of the Kuban cluster. A cluster of gold with a tetrahedral backbone is a 5-cross-polytope. The cluster tetra anion of cobalt is a polytope of dimension 5 of a new type. Different types of ligands limit the cobalt skeleton from above and below.
TopAmong a wide variety of clusters, a special place is occupied by compounds based on the skeleton in the form of the simplest metal - polyhedron — the tetrahedron. Tetrahedral cluster compounds are widely distributed (Cubin, 2019; Garner, 1980). The special significance of tetrahedral clusters lies in the fact that compounds based on them are found in nature in the form of multi - core redox homo - metallic and hetero - metal enzymes (Lichtensins, 1979; Neuton & Otsuka, 1980). Let us consider in more detail the structural features of cluster compounds with tetrahedral metal cores. A classic example of compounds of this type is the tetrahedral cluster Ir4(CO)12 (Gubin, 2019). The scheme of this compound is shown in Figure 1.
Figure 1. Homo - metallic tetrahedral iridium cluster Ir4(CO)12
Theorem 1. The homo - metallic tetrahedral iridium cluster Ir4(CO)12 has dimension 4.
Proof. The spatial structure of the cluster Ir4(CO)12 is shown in Figure 2.
Figure 2. The spatial structure of the clusterIr4(CO)12
As follows from Figures 1, 2 the spatial structure of the cluster Ir4(CO)12 contains 16 vertices. In vertices 1÷2 is arrangement of the functional groups CO, in vertices 13÷6 is arrangement of the atoms Ir.
For further it is necessary to use the Euler - Poincaré equation (Poincaré, 1895)
(1) Key Terms in this Chapter
Dimension of the Space: The member of independent parameters needed to describe the change in position of an object in space.
Binding (or Bridge) Ligands: Non-metal atoms or functional groups bonded by a chemical bond to several atoms of the metal backbone of the cluster simultaneously.
Terminal Ligands: Non-metal atoms or functional groups bonded by chemical bonding to only one of the atoms in the metal core of the cluster.
Polytope: Polyhedron in the space of higher dimension.
Nanocluster: A nanometric set of connected atoms, stable either in isolation state or in building unit of condensed matter.