Abstract
General analytical expressions are obtained for calculating the dimension of multi-shell clusters with a common center of shells in those cases when there is a metal atom in the center of the shells and when it is not. The shells can be in the shape of any body of Plato. It has been established that the gamma-copper cluster has the form of a cross-polytope of high dimension. The forms of clusters with ligands of the core of which is a chain of metal atoms or a metal polyhedron are geometrically investigated. It is shown that if the skeleton is a chain of metal atoms, then the cluster is polytope composed of two polytopes of higher dimension adjacent to each other along a flat section containing a chain. If the skeleton is a metal polyhedron, then a cluster of higher dimension has several ligand shells.
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