The Polytopes of the Higher Dimension in Physics and Chemistry and Construction of Spaces of the Higher Dimension

Introduction

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.