Coordinated polyhedra

KinR81 King, R. B. Chemical applications of topology and group theory. 9. The symmetries of coordination polyhedra. Inorg. Chem. 20 (1981) 363-372. 

It is furthermore to be anticipated that the cation-cation repulsion will operate in some cases to displace the cations from the centers of their coordinated polyhedra. This action will be large only in case the radius ratio approaches the lower limit for stability, so that the size of the polyhedron is partially determined by the characteristic anion-anion repulsive 21 Linus Pauling, Z. Krist., 67, 377 (1928). 

The configuration of the twelve ligands about the small MnlV atom is that of an approximately regular icosahedron. The thirteen-cornered and sixteen-cornered coordination polyhedra about Mnl, Mnll, and Mnlll are appropriate to axial ratios slightly greater than unity. 

The Fe—O distances in hematite are 1.99 and 2.06 A. The (Mn,Fe)—O distances in bixbyite are expected to be the same in case that (Mn, Fe) has the coordination number 6, and slightly smaller, perhaps 1.90 A, for coordination number 4. The radius of 0= is 1.40 A, and the average O—O distance in oxide crystals has about twice this value. When coordinated polyhedra share edges the O—O distance is decreased to a minimum value of 2.50 A, shown by shared edges in rutile, anatase, brookite, corundum, hydrargillite, mica, chlorite, and other crystals. Our experience with complex ionic crystals leads us to believe that we may... 

Fig. 3. The four different coordination polyhedra in the Mg32(Al, Zn)49 structure.

The coordination polyhedron results when the centers of mutually adjacent coordinated atoms are connected with one another. For every coordination number typical coordination polyhedra exist (Fig. 2.2). In some cases, several coordination polyhedra for a given coordination number differ only slightly, even though this may not be obvious at first glance by minor displacements of atoms one polyhedron may be converted into another. For example, a trigonal bipyramid can be converted into a tetragonal pyramid by displacements of four of the coordinated atoms (Fig. 8.2, p. 71). 

The most important coordination polyhedra and their symbols for explanation of the symbols see page 6... 

When coordination polyhedra are connected to chains, layers or a three-dimensional network, this can be expressed by the preceding symbols 2, or 2, respectively. Examples ... 

Important structural principles for ionic crystals, which had already been recognized in part by V. Goldschmidt, were summarized by L. Pauling in the following rules. First rule Coordination polyhedra... 

According to the preceding statements certain coordination polyhedra occur preferentially for compounds of transition metals, depending on the central atom, the oxidation state, and the kind of ligand. The general tendencies can be summarized as follows ... 

Table 9.2 Most common coordination polyhedra for coordination numbers 2 to 6 for transition metal compounds...

Table 9.3 Number of possible geometric isomers depending on the number of different ligands (designated by A, B, C,. ..) for some coordination polyhedra (excluding chelate complexes). Of every pair of enantiomers only one representative was counted...

The structures of antimony and bismuth correspond to that of gray arsenic. With increasing atomic weight the distances between adjacent atoms within a layer and between layers become less different, i.e. the coordination polyhedra deviate less from a regular octahedron. This effect is enhanced under pressure (cf. next section). 

The composition of a compound is intimately related to the way of linking the polyhedra. An atom X with coordination number c.n.(X) that acts as a common vertex to this number of polyhedra makes a contribution of l/c.n.(X) to every polyhedron. If a polyhedron has n such atoms, this amounts to n/c.n.(X) for this polyhedron. This can be expressed with Niggli formulae, as shown in the following sections. To specify the coordination polyhedra, the formalism presented at the end of Section 2.1 and in Fig. 2.2 (p. 5) is useful. 

W. H. Baur, Bond length variation and distorted coordination polyhedra in inorganic crystals. Trans. Am. Crystallogr. Assoc. 6 (1970) 129. 

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