Icosahedral structure, numbering

In order to describe derivatives of B12 or B12H122 having icosahedral structures, it is necessary to have a way to designate positions of atoms or substituent groups. In order to do that, the positions are identified by a numbering system that is illustrated as shown in Figure 13.5. 

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. 

From the above discussion, it is evident that mixed cluster ions of the type Ar M exhibit strong magic numbers at values of (n + m) = 13, 19, 55, 71, and 147 in a variety of different studies. These values correspond to the completion of the first, second, and third icosahedral shells occurring at 13, 55, and 147 whereas 19 and 71 correspond to especially stable subshells formed by interpenetrating double icosahedron structures. The size and symmetry of the dopant moiety appear to be the most important factors in observing magic numbers that can be rationalized on the basis of icosahedral-like structures. The inability to observe magic numbers has been attributed to the distortion of the icosahedral structure due to size and steric factors associated with the dopant ion which destroys the delicate balance between the monomer interactions. One of the issues that has been interpreted differently involves the location of the dopant atomic/molecular... 

A derivative of B12H122 is the carborane, Bi0C2H12. Note that this species is neutral because each of the carbon atoms has one more electron than does the boron atom and can, therefore, replace a B-H unit in the structure. Because all the positions in an icosahedron are identical, there will be three isomers of Bi0C2H12 that differ in the location of the two carbon atoms in the structure. The positions in the icosahedral structure are identified using the numbering system shown in Figure 8.8. Therefore, the structures for the three isomers of... 

We may also employ eqn. (29) to estimate the bond numbers in tetragonal boron. Prom Table 3, we have D(l) = 1.672 A, leading to a normalized icosahedral bond number of 0.4776 and an interstitial bond number of 1.0170. Therefore, the average bond number per atom for the icosahedral B atoms, n, 0j, is 0.5068, and the average number of unsynchronized resonance structures for the icosahedral B atoms, is 6.0702. A check of the internal consistency of these numbers is provided by eqn. (28) D(n) = 1.7932 A. This compares with the experimental average icosahedral bond distance of 1.7874 A, which differs from the calculated bond distance by only 0.3 %. 

The first indications that certain systems might violate the phase rule came from computer simulations of small clusters of atoms. A number of studies revealed clearly defined solid-like and liquid-like forms [5-14]. These embraced both molecular dynamics and Monte Carlo simulations, and explored a variety of clusters. These included several based on atomic models with interparticle Lennard-Jones forces, which mimic rare gas clusters rather well. There were also models of alkali halide clusters. Hence, the existence of solid and liquid forms for such small systems seemed not only plausible but general, not restricted to any one kind of system. Shortly after these studies appeared, another, of a 55-atom cluster with Lennard-Jones interparticle forces, showed not only solid and liquid forms but also a form in which the surface of the cluster (with icosahedral structure) is liquid... 

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. 

Many other kinds of clusters differ sharply from this simple relationship. Clusters of rare-gas atoms such as argon tend to have structures based on icosahedral geometry. This geometry cannot be the basis for a lattice it simply does not have the translational symmetry necessary to build a lattice. The spacings between neighbors in shells distant from the central atom differ from those near the core. The cluster sizes for which complete, filled icosahedra can be made, namely 13, 55, 137,..., are called magic numbers and the structures are called Mackay icosahedra. Not all the most stable structures of such clusters have icosahedral structures, and the specific structure of the most stable form depends on the forces that bind the cluster together. ° 4i por example, a number of clusters bound by Lennard-Jones or Morse potentials, potentials V(R) that depend only on the distance R between pairs of particles. The first of these has the form... 

Icosahedral structures with triangulation numbers of T = 7 and 3 are indicated by three-dimensional image reconstruction of human wart virus and tomato bushy stunt virus, and a three-dimensional X-ray analysis at 30 A resolution of the bushy stunt virus is consistent with this interpretation. ... 

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