Tetrahedron notations

The coordination conditions can be expressed in a chemical formula using a notation suggested by F. Machatschki (and extended by several other authors for recommendations see [35]). The coordination number and polyhedron of an atom are given in brackets in a right superscript next to the element symbol. The polyhedron is designated with a symbol as listed in Fig. 2.2. Short forms can be used for the symbols, namely the coordination number alone or, for simple polyhedra, the letter alone, e.g. t for tetrahedron, and in this case the brackets can also be dropped. For example ... 

Because there are numerous silicates whose structures are made up of repeating patterns based on the Si04 tetrahedron, a type of shorthand notation has been developed for drawing the structures. For example, the Si04 unit can be shown as follows. 

Most of the stereochemical arrangements found in these complexes are based oil the octahedron or on the tetrahedron, and may be derived from these configurations by a suitable distortion. Both basic systems have cubic symmetry and d-orbital energies are split into a triplet and a doublet. The group-theoretical notation will be used for their description. ... 

If we call f the number of factors, a simplex is a convex and closed figure formed by/— 1 vertices in a space of/dimensions. This is a triangle for the case when we consider two factors (/= 2) or a tetrahedron if/= 3 for greater values of / the design cannot be drawn and we need to resort to matrix notation see Table 2.20. 

Ca atoms forming the base of a tetrahedron and three Si atoms form a trigonal prism with the Si3 face capped by the axial Ca atom. The notation is 3 3PTT(h). 

Assuming to simplify the notations that for all the incident bonds the atom A is the right-end atom A = Rm) we obtain the overall quasitorque acting upon the hybridization tetrahedron centered on the atom A and the corresponding energy minimum conditions with respect to orientations of all hybridization tetrahedra in the molecule ... 

The variations of the one-electron densities 6Pria with a = a, ,v,( and the polarity (/ — P[l) of the bond with m = 1 deserve some discussion. As it is seen from eqs. (3.86), (3.105) each bond incident to an atom contributes an increment to the quasitorque and to the pseudotorque acting upon its hybridization tetrahedron. In the equilibrium these increments separately sum up to zero. We can think that the equilibrium shape and orientation of the hybridization tetrahedron is obtained within a TATO DMM model applied to the entire system. Then, within such a model, there exists an atom corresponding to the left end of the bond with m = 1 having number Li according to our previous notation. The HOs obtained in this approximation provide an initial guess for HOs in the system including those of the atom Pi, which... 

Figure 9-33. Shorthand notation for some common structural units after Wells [55] (a) Notations for tetrahedron (b) Notations for octahedron.

The tetragonal (P4j = C ) TIBO2 with the notation 2 oo(A + T) consists of ohe boron-oxygen triangle and one boron-oxygen tetrahedron linked together to form an infinite (3204) " ion... 

The ion [86011] has been found in-monoclinic (P2i/b = Cfh) 3 B2O3 -2 CaO = Ca2[B60iiI with the notation 6 003(6 2A + 3T, IT), A = B, network a penta-borate unit and a single BO4 tetrahedron are connected to build up a three-dimensional network... 

Q notation specifies the number of oxygens of the SiO/ unit bridging to other Si (or Al) atoms (e.g., a Q silicate has three bridging and one nonbridging oxygen per silicate tetrahedron). 

The four bonds with which carbon attaches to other atoms are equally distributed in a singly bonded carbon atom. Picture, then, that the bond sites of carbon are like the comers of a tetrahedron. Organic molecules, therefore, are three dimensional. Because it is difficult to draw complex, three-dimensional figures, we represent organic molecules by convention with a two-dimensional system of notation. 

By representing each atom of quadrivalent carbon in the form of a r ular tetrahedron, stereochemical notation attributes to these two acids two symbols as distinct as the crystals giving them. The S3rmbol of the right-handed acid is not superposable upon itself in a mirror, but on reflecting it in a mirror it reproduces the formula of the left-handed acid. 

We now apply our model to investigate the formation of four C-H bonds in the methane molecule CH4 of symmetry Tc (Magnasco, 2004a). For this molecule, tetrahedral sp3 hybridization is completely determined by molecular symmetry. We use the usual notation for the eight valence AOs, callings, x, y, zthe2s and 2p orbitals on C, and h, h2,h3, h4 the Is orbitals on the H atoms at the vertices of the tetrahedron (Figure 2.14). Molecular... 

Flexure 1. Drawing (in polyhedral notation) of a-SiVWn04o (1). The central tetrahedron represents the Si 04 unit and the shaded octahedron represents V O. The unshaded octahedra represent W 06 units. 

There are five regular (Platonic) polyhedra with equivalent faces and equivalent vertices. Of these, the tetrahedron, octahedron, cube (hexahedron) and icosahedron are widely represented in cluster polyhedra. The structure descriptions in Table 1 illustrate the occurrence of regular polyhedra in cages and demonstrate the descriptive notation based on expanding concentric... 

Figure 9-33. Shorthand notations for the tetrahedron (a) and the octahedron (b). After Wells [9-26],...

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