Octahedron notations

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. ... 

The structure of AIF3 can be described by the same notation as VF3, 2 2POi/3, but there is more distortion (Dg, R32 R.W.G. Wycoff, Crystal Structures, vol. 2. 2nd ed., p. 47, Wiley, New York, 1964). The octahedron around each Al3+ ion is distorted with three F ions at greater distance (1.89 A) than the other three F (1.70A). 

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

Define notations, b, c, and d for a truncated cube, g and s for truncated octahedron, as shown in Figure 5.20, and t for a growth time, then v(lll), v(lOO), and the a-parameter are expressed as shown in Table 5.1. This method has a great advantage that no comparison of diamond crystal shapes in an interval of growth time t is... 

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. 

As a complement to this section on crystallography, some remarks on the old notations are given. In the nineteenth-century treatises, the crystallographic notations are varied and difficult to understand. However C. Friedel (1893) recorded all notations used from Haiy onward. For all the simple forms, he compared five different modes (Levy, Weiss, Naumann, Dana, Miller) which could be reduced substantially to two different schemes. Some crystallographers assigned a symbol to a simple form (e.g., p is cube in Levy, O is octahedron in Naumann), whereas others referred to the intercepts of the faces on axes and assigned some numbers to each face in comparison with a reference face, with direct (Weiss) or inverse (Miller) proportionality. The following table, which reports the various symbols for some simple forms of the cubic system, clarifies these differences. 

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],...

Figure 1. Representative polyoxometalates in polyhedral notation. A. the isopolyanion decatungstate (Wio032 ) (C>4h point group symmetry) and B. the heteropolyanion family, (TM)PWii039 , where TM is a first row divalent transition metal ion, P is the heteroatom (Cs point group symmetry). In the latter class of complexes, which constitute functional oxidatively resistant inorganic metalloporphyrin analogs, P is one of many elements that can function as the heteroatom. The darker octahedron on the surface and the veiy dark internal tetrahedron of B represent the TM ion and the heteroatom, respectively. In polyhedral notation a complementary notation to ball-and-stick or bond representations, the vertices of the polyhedra, principally W06 octahedra, are the nuclei of the oxygen atoms. The metal atoms lie inside each polyhedron.

To indicate the representation of a simple form by indices, braces instead of parentheses are used for the notation of simple faces. For example 111 octahedron indicates the face (111) of octahedron of the positive octant, while 110 rhomboidal dodecahedron indicates the face (110) of ihomboidal dodecahedron of positive octant. 

---