The Geometry of Higher-Dimensional Multi-Shell Clusters With Common Center and Different Centers: The Geometry of Metal Clusters With Ligands

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Introduction

In a recent work of the author (Zhizhin, 2019a) higher dimensions of clusters of intermetallic compounds were considered. In particular, multi-shell intermetallic clusters were considered, each shell of which is a convex regular three-dimensional polyhedron (Plato bodies). It was assumed that all shells in a certain arbitrary cluster have a common center and are of the same type. The statement is proved.

• Theorem: The dimension d of a cluster of N shells with a common center is N + 2, if there is no atom in the common center, and is equal to N + 3, if there is an atom in the common center.

The proved statement allows us to calculate the dimension of many well-known intermetallic clusters: Mackay, Bergman, Samson, and others (Mackay, 1962; Pauling, 1960; Nyman & Anderson,1979; Bergman et al., 1952, 1957; Komura et al., 1960; Audier et al., 1998; Samson, 1972).

Continuing the mathematical descriptions begun in this work of real clusters of chemical compounds with the determination of their dimension, this article discusses multi-shell metal clusters with ligands. Moreover, the shells in one cluster may have a different shape, as well as a common center or several centers at the same time. The study is based on the classification of various real clusters of chemical compounds with ligands (Gubin, 2019). This line of research is fundamentally different from abstract cluster studies that are not associated with specific chemical compounds and do not have technological significance (McMullen, & Schulte, 2002; Diudea & Nagy, 2007; Ashrafi, Cataldo, Iraumanesh, & Ori, 2013; Diudea, 2018).

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The Dimension Of A Metal Skeleton Of Giant Palladium Cluster

It was shown (Vargaftik, et al., 1985) that upon the reduction of palladium acetate with hydrogen in the presence of nitrogen-containing ligands (L), polynuclear complexes are formed, which are easily converted into cluster compounds. Based on the study of their structure by electron microscopy, they were assigned the composition Pd₅₆₁L₆₀(O₂)₁₈₀(OAc)₁₈₀. The palladium atoms in this compound form a dense packing in the form of five icosahedrons with a common center in which the palladium atom is located. Ligands are located on the surface of a metal skeleton. The number of atoms in the metal core of this cluster is determined by the formula:

is the number of layers around the central metal atom (Lord et al., 2006).

From Theorem in introduction (Zhizhin, 2019 a) at once it follows that the dimension of a metal skeleton of giant palladium cluster, containing 561 atoms in five shells, is 8. The claims of some authors that the palladium cluster in this case is an lattice (Shevchenko, 2011) are groundless. The lattice , as you know (Conway, & Sloane, 1988), is a collection of points in an eight-dimensional space with coordinates , where units can stand anywhere on the line with arbitrary signs, as well as points with coordinates: