Symmetry polyhedra

D-penicillamine, HSC(Me)2CH(C02)NH3] (239), also contain a mixture of high-symmetry polyhedra. The latter cage is made by mixing D-penicillamine and copper chloride in water, followed by layering with a solution containing a suitable cation—[ColNHsle], [Ru(NH3)6] +, or T1+ can be used. The purple crystals contain a central chloride anion, surrounded by the S atoms from twelve pen ligands, which form an... 

Table 1 summarises the calculated moments of inertia for some clusters with cubic symmetry and edge lengths of 1.6A. This bond length corresponds to a typical B-B distance in a boron hydride cluster. The dimensions a=b=c for the corresponding sphere are also summarised in the Table. It is noteworthy that the moments of inertia do not distinguish these high symmetry polyhedra from a sphere. Higher order moments are required to make this distinction. 

The symmetry of the structure we are looking for is imposed on the field 0(r) by building up the field inside a unit cubic cell of a smaller polyhedron, replicating it by reflections, translations, and rotations. Such a procedure not only guarantees that the field has the required symmetry but also enables substantial reduction of independent variables 0/ the function F (f)ij k )- For example, structures having the symmetry of the simple cubic phase are built of quadrirectangular tetrahedron replicated by reflection. The faces of the tetrahedron lie in the planes of mirror symmetry. The volume of the tetrahedron is 1 /48 of the unit cell volume. 

Figure 6.1 The icosahedron and some of its symmetry elements, (a) An icosahedron has 12 vertices and 20 triangular faces defined by 30 edges, (b) The preferred pentagonal pyramidal coordination polyhedron for 6-coordinate boron in icosahedral structures as it is not possible to generate an infinite three-dimensional lattice on the basis of fivefold symmetry, various distortions, translations and voids occur in the actual crystal structures, (c) The distortion angle 0, which varies from 0° to 25°, for various boron atoms in crystalline boron and metal borides.

The compounds of the MMe205F type, where Me = Nb or Ta M = Rb, Cs, Tl, crystallize in cubic symmetry and correspond to a pyrochlore-type structure [235-237]. This structure can be obtained from a fluorite structure by replacing half of the calcium-containing cubic polyhedrons with oxyfluoride octahedrons. 

You say that your nonlinear molecule has the high symmetiy of a regular polyhedron, such as a tetrahedron, cube, octahedron, dodecahedron, icosahedron,... sphere. If it is a sphere, it is monatomic. On the other hand, if it is not monatomic, it has the symmetry of one of the Platonic solids (see the introduction to Chapter 8). 

You have replied that your molecule, that is not a regular polyhedron, does not have a proper rotation axis of order greater than one. If its only symmetry element is a plane, it belongs to the group 6Jih a... 

It is noteworthy that the to-bonded structure for ArF6 differs from that predicted by VSEPR theory. ArF6 is predicted to be of octahedral (Oh) symmetry, with three mutually perpendicular F i- Ar -h F triads and an s-type lone pair. In contrast, VSEPR predicts a pentagonal bipyramid (or other seven-vertex polyhedron) with some or all F-Ar-F angles less than 90°. The calculated equilibrium structure is in agreement with the co-bonding model. 

The effect of ligand-field symmetries is less easily understood. In Fe3(CO)12, two Fe-Fe distances of 2.56 A and 2.68 A are observed. It may be argued that these distances reflect the packing of a triangle within the icosahedron, each polyhedron undergoing distortion until a common C2v symmetry is reached. 

The structure of the heptaosmium carbonyl Os7(CO)2i has been determined by X-ray studies, and the 13C-NMR is consistent with the same structure in solution (231). The stereochemistry of the metal polyhedron follows Wade theory and is a monocapped octahedron (see Fig. 56). As anticipated for this structure, the 13C-NMR spectrum of an isotopically enriched sample (13CO 40%) shows three different chemical shifts, with an intensity ratio of 3 5 1. This spectrum is related to that of [HOs6(CO)18], which also has an idealized C3v symmetry, and may be viewed as [Os7(CO)21] with the Os(CO)3 cap replaced by the hy-drido group. The NMR spectrum of Os7(CO)2j only begins to show evidence of structure to the three bonds at —120°, but the resolution of the 2 1 component for each group, expected on the basis of the X-ray struc-... 

The relationship of the wheel-rim-type structure of 62 to the metal can be demonstrated by a 30° rotation of the two centered Alg rings followed by a shift of the six rings towards each other (cf. Figure 2.3-13) [92], The other possibility of the formation of an Al14 polyhedron with point symmetry by displacement of the two naked central atoms in the direction of a polyhedral entity has been shown to be energetically unfavorable by quantum chemical calculations i.e., the observed metalloid structure (Figure 2.3-13) is favored over the anticipated polyhedral structure as described by Wade-Mingos [5, 96] (see Chapter 1.1.2). 

A cluster may be considered, in a sense, the antithesis of a complex even if there are many similarities between the two groups of compounds due to common symmetry properties. In both cases a set of atoms defines the vertices of a polyhedron in the complex, however, these atoms may be considered as bound to another, central, atom and not to each other. In the cluster, on the other hand, there is not necessarily a central atom and the atoms at the vertices can be described as directly connected to each other. 

As shown by ab initio studies with the anionic model species 49, the energy difference between the trigonal-bipyramidal (C2 symmetry) and the alternative square-pyramidal structure (C2v symmetry) is rather small (6.9 kJ mor1).40 From this result one may conclude that related zwitterionic A5S7-silicates, such as 47 and 48, should also have a chance to exist with a square-pyramidal /-coordination polyhedron in the solid state. The square-... 

Structural analysis of several octahedral Sc(III) complexes have recently been reported. The structure 44) of anhydrous scandium-tris-acetylacetonate, Sc(acac)3, consists of discrete Sc(CH3COCHCOCH3)3 entity, with slight deviation from Dz symmetry. The coordination polyhedron around Sc (III) is a distorted... 

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