Projective fullerenes

The projective plane arises as a quotient space of the sphere, the required group being C,-. It is obtained by identifying antipodal points of the spherical surface in other words, it is the antipodal quotient of the sphere (see Section 1.2.2). P2 is the simplest compact non-orientable surface in the sense that it can be obtained from the sphere by adding just one cross-cap. 

In this terminology, our definition of projective fullerenes amounts to selection of cell-complex projective-planar 3-valent maps with only 5- and 6-gonal feces. As noted above, P5 — 6 for these maps. Thus, the Petersen graph is die smallest projective fullerene. In general, the projective fullerenes are exactly the antipodal quotients of the centrally symmetric spherical fullerenes. 

the problem of enumeration and construction of projective fullerenes reduces simply to that for centrally symmetric conventional spherical fullerenes. The point symmetry groups that contain the inversion operation are Q, C, h, (m even), Dmh (m even), Dmd (m odd), 7, Oh, and 7. A spherical fullerene may belong to one of 28 point groups ([FoMa95]) of which eight appear in the previous list C,-, C2h, Dm, Da, D3d, Du, 7, and /. Clearly, a fullerene with v vertices can be centrally symmetric only if v is divisible by four as p6 must be even. After the minimal case v = 20, the first centrally symmetric fullerenes are at v = 32 (Dm) and v = 36 (Dm). 

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Figure 3.1 Smallest spherical, toroidal, Klein bottle and projective fullerenes. The first column lists the graphs drawn m the plane, the second the map on the appropriate surface and the third the dual on the same surface. The examples are (a) Dodecahedron (dual Icosahedron), (b) the Heawood graph (dual Ky), (c) a smallest Klein bottle polyhex (dual 3,3,3), and (d) the Petersen graph (dual Ke).

Today, the laboratory employs 3,5GG scientists and has an annual budget of about 1.6 billion. In addition to projects such as Manaa s fullerene research, Lawrence Livermore National Laboratory operates the world s largest laser facility—National Ignition Facility—as well as a number of powerful supercomputers. This equipment provides researchers with the tools to study atoms and molecules on an experimental basis as weii as supplying computational power to simulate their activity on a theoretical level. 

Figure 7. A78 isomers. The unfolded surface lattice nets at the left are drawn with boundaries along the vectors between nearest neighbour V5s which are marked by the black circular sectors, whereas the boundaries for the nets at the right are along the edges of deltahedral facets. The projected views of the fullerene polyhedra and deltahedra duals in the centre column are all oriented with a corresponding two-fold axis horizontal. For the four mirror-symmetric isomers, there is one mirror plane in the plane of projection and an orthogonal horizontal one. Marking the symmetry elements for each isomer on the deltahedral surface lattice net defines the asymmetric unit.

Figure 17 Mono (50) and regio-isomeric bis-adducts (51, ( )-52 and ( )-53) formed by Bingel cyclopropanation of [70]fullerene. Also shown are Newman-type projections looking down the C s-symmetry axis of the C70 core on to the two polar pentagons, which show the relative orientations of the addends.

Figure 1.26. Double Bingel addition to C70 leads to an achiral top) and two inherently chiral (center and bottom) addition patterns. Combination of each of the latter with chiral ester moieties affords two diastereoisomeric pairs of enantiomers. The enantiomers of each pair were prepared separately by addition of either (R,R) or (S, -configured malonates to C70, and all stereoisomers were isolated in pure state. The black dots mark intersections of the C2-symmetry axis with the [70]fullerene spheroid. Next to the three-dimensional representations, constitution and configuration of the addition patterns are shown schematically in a Newman type projection along the Cs-axis of C70. Of the two concentric five-membered rings, the inner one corresponds to the polar pentagon closest to the viewer, and the attached vertical line represents the bond C(l)-C(2) where the first addition occurred. The functionalized bonds at the distal pole depart radially from the outer pentagon.

Solution of the Kohn-Sham equations as outlined above are done within the static limit, i.e. use of the Born-Oppenheimer approximation, which implies that the motions of the nuclei and electrons are solved separately. It should however in many cases be of interest to include the dynamics of, for example, the reaction of molecules with clusters or surfaces. A combined ab initio method for solving both the geometric and electronic problem simultaneously is the Car-Parrinello method, which is a DFT dynamics method [52]. This method uses a plane wave expansion for the density, and the inner ions are replaced by pseudo-potentials [53]. Today this method has been extensively used for studies of dynamic problems in solids, clusters, fullerenes etc [54-61]. We have recently in a co-operation project with Andreoni at IBM used this technique for studying the existence of different isomers of transition metal clusters [62,63]. 

Acknowledgements TS gratefully acknowledges Prof. Arnout Ceulemans and Prof. Liviu F. Chibotaru for valuable discussions on the dynamic Jahn-Teller problem and vibronic couplings in fullerene ions. Numerical calculation was partly performed in the Supercomputer Laboratory of Kyoto University and Research Center for Computational Science, Okazaki, Japan. This work was supported by Grant-in-Aid for Scientific Research (C) (20550163), Priority Areas Molecular theory for real system (20038028) from Japan Society for the Promotion of Science (JSPS), and the JSPS-FWO (Fonds voor Wetenschappelijk Onderzoek-Vlaanderen) Joint Research Project. 

The question remains on the size of the innermost shell. An estimate can be made by measuring the diameter in projections of entirely or partially filled carbon onions. The values obtained this way frequently correspond to the diameter of Ca). It is not possible, however, to exclude fullerenes of similar size hke C50, with certainty aU the more as the low stability observed for these species in their isolated... 

The purpose of this article is to investigate the dielectric response of fullerenes under static uniform electric fields in order to predict their effects in composite materials used for coating of electronic circuits. This study is a part of the larger project devoted to electromagnetic wave absorbing media. 

Acknowledgements Author thanks Prof. Mircea Diudea from Babes-Bolyai University of Cluj-Napoca for courtesy in providing the Hyper file for the Fullerene structure and to Romanian Ministry of Education and Research for supporting the present work through the CNCS-UEFISCDI (former CNCSIS-UEFISCSU) project Chemical Bond Within Orthogonal Spaces of Reactivity. Applications on Molecules of Bio-, Eco- and Pharmaco- Logical Interest>, Code PN II-RU-TE-2009-1 grant no. TE-16/2010-2011. 

Feynman s vision of miniaturization and the Drexler-versus-Smalley debate on feasibility of mechanosynthetic reactions using molecular assemblers were discussed. Fullerenes are the third allotropic form of carbon. Soccer-ball-structured Cgo with a surface filled with hexagons and pentagons satisfies Euler s law. Howard patented the first generation combustion synthesis method for fullerene production. The projected price of the fullerenes has decreased from 165,000 per kg to 200 per kg in the second-generation process. Fullerenes can also be synthesized using chemical methods, a supercritical extraction method, and the electric arc process. Applications of fullerenes include high temperature superconductors, bucky onion catalysts, advanced composites and electromechanical systems, synthetic diamonds. 

FIGURE 10.3 Two different projections of a carbon onion consisting of four fullerene molecules, each of which contains 60n carbon atoms, where n is the number of the fullerene shell in the carbon onion. The carbon onion has evident icosahedral shape due to the presence of 12 pentagons in each of the constituent fullerene shells. 

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