Chains of Metallic Clusters With Ligands

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.

Incident Conservation Law

This chapter first establishes the existence of integral equality in relation to the issue of the transmission of information by elements of lower and higher dimensions in the polytopes of higher dimension that describe natural objects. This integral equality is called the law of conservation of incidents. There is the incidence interpreted as the transfer of information from one material body to another. The fulfillment of the law of conservation of incidents for the n-simplex of the n-cube and the n-cross-polytope is proved in general terms. It is shown that the law of conservation of incidents is valid for both regular bodies and irregular bodies, which can be clusters of chemical compounds. The incident conservation law can serve as a mathematical basis for the recently discovered epigenetic principle of the transmission of hereditary information without changing the sequence of genes in DNA and RNA molecules.

Higher-Dimensional Space of Nanoworld

In this chapter, a geometrical model to accurately describe the distribution of light points in diffraction patterns of quasicrystals is proposed. It is shown that the proposed system of parallel lines has axes of the fifth order and periodically repeating the fundamental domain of the quasicrystals. This fundamental domain is 4D-polytope, called the golden hyper-rhombohedron. It consists of eight rhombohedrons densely filling the 4D space. Faces of the hyper-rhombohedron are connected by the golden section; they can be scaled as needed. On this universal lattice of the vertices of the golden hyper-rhombohedrons, famous crystallographic lattices—Bravais, Delone, Voronoi, etc.—can be embedded. On the lattice of the vertices of the golden hyper-rhombohedrons, projections of all regular three-dimensional convex bodies—Plato's bodies—can be constructed.

Closed Metal Cycles in Clusters With Ligands

This chapter considers closed three-membered metal cycles of one or several chemical elements surrounded by ligands connected to them. It has been proven that the widespread opinion in the literature about the formation of ligands by atoms in some cases of the semi-correct polyhedron of the anti-cube-octahedron is wrong. Geometrical analysis of the interpenetration of the coordinates of ligand atoms around each of the metal atoms of a closed chain showed that this leads to a different class of special three-dimensional irregular polyhedrons for different clusters. In all cases of homo-element and hetero-element closed metal chains, the cycle itself, located in a certain plane, creates a cross section of the cluster, dividing the cluster into two parts. Each of the parts of a cluster has dimension 4.

Mathematical Design of Nanomaterials From Clusters of Higher Dimension

Using concrete examples of clusters of chemical compounds of various types (intermetallic clusters, metal chains with ligands, polyhedral metal clusters with ligands), it is shown how nanomaterials are formed from individual clusters by multiplying their geometric structure by other geometric elements of different dimensions. The considered examples correspond to nanomaterials with a structure of limited complexity. However, the mathematical apparatus developed on the basis of the geometry of high-dimensional polytopes allows, in principle, to describe and study and design nanomaterials of this type of any complexity and any dimension. In particular, nanomaterials with the simultaneous use of elements with different metric characteristics can be attributed to such nanomaterials.

Dimension of Chemical Compounds of Atoms Metals With Atoms No Metals

The structures of compounds of a metal atom with ligands were studied by sequentially changing the groups and subgroups of the periodic system of elements in which the metal atom is located. It is shown that all metals from the first to the eighth groups form chemical compounds of a higher dimension. The formation of molecules of higher dimension occurs due to the chemical bonds of the metal atom with ligands both due to the influence of electron pairs and due to the attraction of ions. Moreover, the apparent valence of the metal atom, as a rule, exceeds the value of the valence determined by the location of the metal in the periodic table of chemical elements.

Nanostructures as Tillings of Higher Dimension Spaces

It is proved that clusters in the form of the polytopic prismahedrons have the necessary properties for partitioning the n-dimensional spaces of a face into a face, that is, they satisfy the conditions for solving the eighteenth Hilbert problem of the construction of n-dimensional spaces from congruent figures. Moreover, they create extended nanomaterials, in principle, of any size. General principles and an analytical method for constructing n-dimensional spaces with the help of polytopic prismahedrons are developed. On the example of specific types of the polytopic prismahedrons (tetrahedral prism, triangular prismahedron), the possibility of such constructions is analytically proved. It was found that neighboring polytopic prismahedrons in these constructions can have common geometric elements of any dimension less than n or do not have common elements.

Metallic Clusters With Ligands and Polyhedral Core

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.

Higher Dimensions of Clusters of Intermetallic Compounds

The author has previously proved that diffraction patterns of intermetallic compounds (quasicrystals) have translational symmetry in the space of higher dimension. In this chapter, it is proved that the metallic nanoclusters also have a higher dimension. The internal geometry of clusters was investigated. General expressions for calculating the dimension of clusters are obtained from which it follows that the dimension of metallic nanoclusters increases linearly with increasing number of cluster shells. The dimensions of many experimentally known metallic nanoclusters are determined. It is shown that these clusters, which are usually considered to be three-dimensional, have a higher dimension. The Euler-Poincaré equation was used, and the internal geometry of clusters was investigated.