Cuboctahedron symmetry

Many of the magic number combinations observed in the CMS of inert gas atoms have been identified with stable structures having an icosahedral symmetry (Echt et al. 1981). The Mackay icosahedra series (Hoare 1979 Mackay 1962) exhibits completion of the first three solvation shells as = 13, 55, and 147, respectively, such that the completion of solvation shells at n = 13, 55, etc., can arise from structures with a cuboctahedron symmetry (Hoare 1979). However, theoretical studies indicate that the icosahedral structures are more stable than those with cuboctahedral symmetry (Hoare 1979). The theoretical studies of Farges et al. (1986) and Northby (1987) provide insight into the growth of icosahedral structures. 

Lu Q, Hu J, Tang K, Qian Y, Zhou G, Liu X (2000) Synthesis of nanocrystaUine CuMS2 (M = In or Ga) through a solvothermal process. Inorg Chem 39 1606-1607 Wu C, Yu S-H, Antoniette M (2006) Complex concaved cuboctahedrons of copper sulfide crystals with highly geometrical symmetry created by a solution process. Chem Mater 18 3599-3601... 

M14 and M15 in Fig. 9 differ from Mu (hep) in Fig. 8 by the addition of one or two metal atoms to the cuboctahedron. Consequently, this leads to a decrease of symmetry. The cluster nucleus of [Rh CO ]4- results from the stacking of 4 closest-packed layers with C3v-symmetry. Only one of the 4 layers is completed (7 atoms). The sequence ABAC corresponds to a mixture of hexagonal (hep) and cubic (ccp) packing. 

Figure 2.9 shows how contractions of local sets of 8, 6 and 4 vertices onto the poles of C4, C3 and C2 axes, respectively, recover the octahedron, cube and cuboctahedron and thereby identify the Oe, Og and O12 orbits of Oh symmetry. 

Figure 2.9 The results of applying contractions to the coloured sets of vertices of the regular orbit cage of symmetry to give the octahedron, the cube and the cuboctahedron. The colour codings refer to those in Figure 2.7 and so the first row displays the results for contractions onto the Cq poles, the second row the results for contractions onto the C3 poles and in the third row, contractions onto the C2 axes lead to the cuboctahedron.

The sequence ABC ABC... possesses cubic symmetry, that is, 3-fold axes of symmetry in four directions parallel to the body-diagonals of a cube, and is therefore described as cubic closest packing (c.c.p.). A unit cell is shown in Fig. 4.1(e) (p. 120) and the packing is also illustrated in Fig. 4.14. The close-packed layers seen in plan in Fig. 4.12 are perpendicular to any body-diagonal of the cube. Since the atoms in c.c.p. are situated at the corners and mid-points of the faces of the cubic unit cell the alternative name face-centred cubic (f.c.c.) is also applied to this packing. All atoms have their twelve nearest neighbours at the vertices of a cuboctahedron. (Fig. 4.5(b)). 

Twelve Vertices. Twelve-atom cages are not widespread but play a dominant role in boron chemistry. The most highly symmetrical arrangement is the icosahedron [Fig. l-8(b)], which has 12 equivalent vertices and Ih symmetry. Icosahedra of boron atoms occur in all forms of elemental boron, in B12Hi 2 and in the numerous carboranes of the B10C2H12 type. A related polyhedron, the cuboctahedron (1-LI) is found in several borides of stoichiometry MB12. As shown on page 249, the icosahedron and cuboctahedron have a rather close relationship. 

Cuboctahedron 5.58 space filling viewed along C3-symmetry axis. The four units of tris(Ld -cycltn) are shown in light green, green, yellow and blue (reprinted with permission from reference ). 

Fig. 3.24 There are several modes of 12-coordination, all of high symmetry (although no complex containing the twelve monodentate ligands needed to give these symmetries is known). That shown is the cuboctahedron (0),.

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