Icosahedral structures, truncated

Fig 4. Frontal view uf truncated icosahedral structure of Cm cluster... 

Since the discovery of the soccer-ball-like, 60-carbon buckminsterfullerene, scientists have been fascinated by this molecule. The molecule possesses a truncated icosahedral structure, of Ih symmetry, and is depicted in Fig. 4-37. 

The other important symmetry-related discovery was the quasicrystals. Both the truncated icosahedral structure of buckminsterfullerene and the regular but nonperiodic network of the quasicrystals are related to fivefold symmetry. In spite of this intimate connection between them at an intellectual level, their stories did not cross. The conceptual linkage between them is provided by Fuller s physical geometry and this is also what relates them to the icosahedral structure of viruses (see, Section 9.5.2 on Icosahedral Packing). 

Figure 2.21 Pairwise coalescence of the decorated decagons of the regular orbit of 111 from Figure 2.19 leading to the truncated icosahedron [column a] and the small rhombicosidodecahedron [column b]. Note that the truncated icosahedral structure is 3-valent and is the archetypal C q cage of fullerene chemistry.

Fig. 1 A football (in the United States, a soccerball) on Texas grass. The Qo molecule featured in this letter is suggested to have the truncated icosahedral structure formed by replacing each vertex on the seams of such a ball by a carbon atom.

Carbon clusters have been the focus of many experimental studies using a variety of techniques [1-4]. Recently, Kroto, Heath, O Brien, Curl and Smalley [5] have obtained a remarkably stable cluster containing 60 carbon atoms and have proposed a highly symmetric truncated icosahedral structure ( football or soccerball like) for this molecule [5,6]. These authors have proposed the name Buckminster-fuUerene for this unusual structure of Cgg-... 

A LONG search for molecular allotropic forms of carbon culminated in the discovery of a method for preparing large quantities of the C o molecule and the subsequent confirmation of its cage-like truncated-icosahedral structure. The 70 molecule prepared by the same method was later also isolated and found to have the predicted cylindrical structure. Incomplete... 

The structures of organic polynuclear aromatic compounds are not limited to planar systems of carbon and hydrogen atoms. A classification of three-dimensional aromatic compounds is proposed on the basis of the number of recognizable edges (boundaries) in the molecular structure. Aromatic structures with no edges are included in this classification an example is the recently proposed truncated icosahedral structure for C6o (Buckminsterfullerene). The current literature and activity in the subfield of nonplanar aromatic compounds is reviewed. Three-dimensional aromatic compounds are possible tools for use in studies of polynuclear aromatic chemistry, and some possible applications to the particular chemical topics presented in this book are outlined. 

Notably in the context of our advocated description the curvature-related ideas marshaled in Section 9.3 elaborate on an important stereochemical aspect, and these ideas lead in later sections to insight concerning strain and its effects. For example, the global shape of large (hypothetical) fullerene shapes is better understood, and the truncated-icosahedral structure of C is verified to be uniquely elegant. ... 

Figure 3-18. Examples with symmetry (a) The regular icosahedral boron skeleton of the ion (b) truncated icosahedral structure of buckminsterfullerene, C, and of a climber in Sapporo, Japan. Photograph by the authors.

Figure 3.9 The truncated icosahedral structure of a Ceo buckybalT a carbon atom is situated at each vertex, (b) The hexagonal structure of a single sheet of carbon atoms arranged as in graphite, (c) Carbon nanotubes, which consist of similar sheets roUed up into a variety of configurations...

There is at present much interest in large cage (cluster) molecules based on carbon [62]. The best known of these is the established Cgo molecule (fullerene) which has exceptional stability and has the highly symmetrical truncated icosahedral structure depicted in Figure 4.7. Derivatives of C o, and carbon nanotubes, currently attract much interest because of their exceptional electrical, mechanical and structural properties. Phosphorus analogues may exist (P is used as a co-catalyst for growing of carbon nanotubes [63,64]). 

Baraldi and Vanossi, also following up on earlier formal work, use (in a fairly conventional manner) the general Polya-theoretic machinery to enumerate substitutional isomers for several cychc or polyhedral skeletons. They conclude with enumerations for icosahedral-symmetry skeletons, both for an icosahedron and for a truncated icosahedron, such as have become of some degree of popularity (as in ref. 86) over the last decade or so because of the elegantly beautiful truncated-icosahedral structure of buckminsterfullerene. 

Fig. 4.20. Schema of the truncated icosahedral structure expected for the carbon backbone in the cluster C6o...

For N < 1,000, Lennard-Jones clusters follow an icosahedral pattern growth with magic numbers corresponding to Mackay icosahedra (Mackay 1962) for N = 13, 55,147, 309, etc. In between these magic numbers, most of the structures are Mackay-like with incomplete outer layers. Exceptions occur when there are alternative structures with complete shells. These are mostly Marks decahedra (Doye 2003) but there are instances of an fee truncated octahedron and a Leary tetrahedron (Noya and Doye 2006). The preference for icosahedral structures of Lennard-Jones clusters at small sizes is thought to be due to a trade-off between optimal bond distance and strain (Hartke 2002 Krainyukova 2006). 

In work of Nava and coworkers (2003), Pd clusters were studied using the spin-polarized DFT method in the range N = 2-309. The N = 13 cluster was foimd to have an icosahedral structure with a high spin state. It is seen to undergo a very slight Jahn-Teller distortion, which increases the cohesive energy only by 0.01 eV. The truncated decahedron and the cuboctahedron are found to be less stable. 

Ascendo, J.A., Perez, M. and Jose-Yacaman, M. (2000) A truncated icosahedral structure observed in gold nanopartides. Surface Science, 447(1-3), 73-80. 

Fullerenes are described in detail in Chapter 2 and therefore only a brief outline of their structure is presented here to provide a comparison with the other forms of carbon. The C o molecule, Buckminsterfullerene, was discovered in the mass spectrum of laser-ablated graphite in 1985 [37] and crystals of C o were fust isolated from soot formed from graphite arc electrodes in 1990 [38]. Although these events are relatively recent, the C o molecule has become one of the most widely-recognised molecular structures in science and in 1996 the codiscoverers Curl, Kroto and Smalley were awarded the Nobel prize for chemistry. Part of the appeal of this molecule lies in its beautiful icosahedral symmetry - a truncated icosahedron, or a molecular soccer ball, Fig. 4A. 

Clathrins, first observed by Pearse, are structures with pentagonal and hexagonal faces that can form three dimensional structures with the same truncated icosahedral morphology as a C6o molecule or a soccer ball [1], The vertices of the clathrin coating are formed by three intertwined proteins composed of heavy and light chains that emanate from a central hub in a structure known as a triskelion, shown in Fig. 3.1. 

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