Abstract
Polytopes of the highest dimension of molecules of compounds of inorganic and organic chemistry are considered on specific examples. It is shown that they all have dimensions greater than three. They may include polytopes of a known shape, discussed in previous chapters, but in general they differ significantly from the standard shapes. These include linear and nonlinear chains of metal atoms with ligands, closed chains of metal atoms with ligands, clusters with ligands and a metal polyhedral backbone. A class of polytopic prismahedrons (a special type of polytopes of higher dimension) is considered, from which parallelotopes of higher dimension are formed, which are necessary for constructing n-dimensional spaces, using them to create extended nanomaterials based on clusters of chemical compounds.
TopIntroduction
The studies carried out in the monograph of G. Zhizhin “Chemical Compound Structures and Higher Dimension of Molecules: Emerging Research and Opportunities” (Zhizhin, 2018) convincingly showed that many elements of the periodic system of Mendeleev elements form molecules whose spatial structure can be geometrically represented as a convex polytope. Moreover, the dimension of this polytope, determined by the Euler-Poincaré relation between the geometric elements forming the polytope (Poincare, 1895), is usually higher than three. In addition, in this monograph it is noted that most elements of the periodic system exhibit valence in compounds higher than the valence determined by the location of the element in the periodic table. It is known that the calculation of the valence of a chemical element based on its location in the periodic table is based on the electronic structure of atoms (s -, p -, d -, f – orbitals), determined by solving the Schrodinger equation (Schrodinger, 1927) under the assumption of the three - dimensionality of the space surrounding the nucleus of the atom. To explain the observed differences in the valence of atoms from valence, determined from the location of the element in the periodic table, researchers resorted to the construction of various models of the interaction of electrons in the vicinity of atomic nuclei. These models are quite diverse and still remain at the level of hypotheses.
Nevertheless, the concept of the electronic structure of atoms, based on the solution of the Schrödinger equation in a three - dimensional industrial space, is currently prevailing in the description of the structures of chemical compounds in both inorganic and organic chemistry. Moreover, these representations are supported by the opinion that there are no solutions to the Schrödinger equations in a space of dimension more than three (Büchel, 1963; Freeman, 1969; Gurevich & Mostepanenko, 1971). However, in the works of the author (Zhizhin, 2018, 2019 a), it was found that the proof of the impossibility of solving the Schrödinger equation in a space with a dimension greater than three contains an error associated with an incorrect writing of Coulomb's law in a space of higher dimension. When correcting this error, the author first obtained a solution to the Schrödinger equation in four - dimensional space (Zhizhin, 2018), and then obtained a solution to the Schrödinger equation in an n - dimensional space (Zhizhin, 2019 a). It is proved that an increase in the dimension of space in the vicinity of an atom leads to an increase in the number of quantum numbers that determine the structure of the electron cloud around the nucleus of the atom, as a consequence, an increase in the number of quantum cells in the electron orbitals of the atom. This may be the real reason for the experimentally observed increase in the valence of elements. As an illustration of the higher dimensionality of molecules, this chapter presents studies of molecules involving anomalous elements, binary compounds, and metal clusters with ligands. In addition, it is considered how spaces of higher dimension are constructed from polytopes of higher dimension.
Key Terms in this Chapter
Linear Metal Chains: Metal chains in which metal atoms are located in a straight line.
N-Cross-Polytope: The convex polytope of dimension n in which opposite related of centrum edges not have connection of edge.
Undivided Electron Pair: A non-bonding electron pair belonging to one atom in a molecule.
Metal Chains: Cluster compounds that have a skeleton in the form of metal chains, i.e., polymetallic chains formed by metal-metal localized covalent bonds.
S- and P-Elements: The chemical elements in which is filling with electrons s- and p-orbitals of atoms.
Tetrahedral Coordination of Electron Pairs: The location of the electronic pairs of the outer and the pre-outer electron layer at the vertices of the tetrahedron.
Homo-Element Metal Chains: Clusters in which the core contains atoms of the same element.
Divided Electron Pair: The binding electron pair, which simultaneously belongs to two atoms in the molecule.
N–Simplex: The convex polytope of dimension n in which each vertex is joined by edges with all remain vertices of polytope.