Hypothetical toroidal, cylindrical, and helical analogs of C 60 (original) (raw)
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Playing with Hexagons and Pentagons: Topological and Graph Theoretical Aspects of Fullerenes
Regular fullerenes are cubic planar connected graphs consisting of pentagons and hexagons only and come in many different shapes. There has been great progress over the last two decades describing the topological and graph theoretical properties of fullerenes, but leaving still many unsolved and interesting mathematical and chemical problems open in this field. For example, i) how to generate all possible fullerene isomers for a fixed atom count (where an efficient algorithm was introduced only very recently) ii) are fullerenes Hamiltonian (Barnette's conjecture) and what is the number of distinct Hamiltonian cycles (longest carbon chains), iii) the Pauling bond order and the number of different Kekulé structures (perfect matchings), iv) the search for suitable topological (chemical) indices to find the most stable fullerene structure out of the many (∼ N 9 ) possibilities, or how to pack fullerenes in 3D space (the Hilbert problem) to name but a few. Here we present a general overview on recent topological and graph theoretical developments in fullerene research over the past two decades.
Constructing Molecules with Beads: The Geometry of Topologically Nontrivial Fullerenes
Three-dimensional molecular structures of topologically nontrivial fullerenes (consisting of either finite or extended structures) are usually aesthetically pleasing. In this article, we demonstrate that beads such as the ones commonly used in decorative art and ornament making can also be used to construct arbitrary fullerene structures. Based on the spiral codes of fullerenes, we developed a systematic strategy for making physical models of cage-like fullerenes use common beads. The resulting beaded model structure is similar to the true three-dimensional molecular structure of corresponding fullerene due to an interesting analogy between the hard-sphere repulsion among neighboring beads and the microscopic valence shell electron pair repulsion for the sp 2-hybridized carbon atoms. More complicated fullerenes models that have nontrivial topology (e.g. toroidal carbon nanotubes, helically coiled carbon nanotubes, and high-genus fullerenes) can also be faithfully constructed using beads. Beaded models of extended graphitic structures such as those that correspond to tiling of graphene sheet on a Schwartz P-and Dsurfaces, Shoen I-WP, and Nervious surfaces, can also been created.
Distributed curvature and stability of fullerenes
Physical chemistry chemical physics : PCCP, 2015
Energies of non-planar conjugated π systems are typically described qualitatively in terms of the balance of π stabilisation and the steric strain associated with geometric curvature. Curvature also has a purely graph-theoretical description: combinatorial curvature at a vertex of a polyhedral graph is defined as one minus half the vertex degree plus the sum of reciprocal sizes of the faces meeting at that vertex. Prisms and antiprisms have positive combinatorial vertex curvature at every vertex. Excluding these two infinite families, we call any other polyhedron with everywhere positive combinatorial curvature a PCC polyhedron. Cubic PCC polyhedra are initially common, but must eventually die out with increasing vertex count; the largest example constructed so far has 132 vertices. The fullerenes Cn have cubic polyhedral molecular graphs with n vertices, 12 pentagonal and (n/2 - 10) hexagonal faces. We show that there are exactly 39 PCC fullerenes, all in the range 20 ≤n≤ 60. In th...
A Spherical Molecule with a Carbon-Free I h -C 80 Topological Framework
Angewandte Chemie International Edition, 2009
The discovery of novel fullerene carbon allotropes and the exploration of their reactivity and chemical properties is a success story of the last two decades. Within the rapidly growing class of fullerenes, the most stable and therefore accessible molecules are those that fulfill both the isolated pentagon rule (IPR), which describes the stability of the s core, and the criterion of spherical aromaticity of the p system of these molecules. In contrast, those fullerenes that only follow the IPR are less stable and, therefore, less accessible. The non-aromatic C 80 molecule is one of the derivatives which has long been of interest in this chemistry. Of the seven structural isomers (D 5d , C 2v (I), D 2 , C 2v (II), D 5h , D 3 , and I h ) satisfying the IPR, the D 2 isomer was isolated in small quantities in 1996 by Kappes and co-workers. In 2000, Shinohara and co-workers reported on the synthesis of the D 5d isomer, in which they also obtained the D 2 isomer. According to quantum chemical calculations the first six isomers are all stable within an energy range of 30 kJ mol À1 . The icosahedral isomer, however, is calculated to be 72 kJ mol À1 less stable than the energetically most favored D 5d isomer and has never been reported. Lately, endohedral C 80 has been synthesized using a four-atom trimetallic Angewandte Chemie 5049
Toroidal forms of graphitic carbon
Physical Review B, 1993
A series of the toroidal cage forms of graphitic carbon is proposed. They have local topological structures of positive Gaussian curvature (for instance C«) and of negative Gaussian curvature. The toroidal forms consist of fivefold, sixfold, and sevenfold carbon rings. The cohesive energies of the tori (C", where n ranges from 120 to 1920), which are calculated by molecular dynamics, indicate that these structures are energetically more stable than C«. These structures are also found to be thermodynamically stable by means of a finite-temperature simulation.
Fullerenes as Tilings of Surfaces
Journal of Chemical Information and Modeling, 2000
If a fullerene is defined as a finite trivalent graph made up solely of pentagons and hexagons, embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective (elliptic) plane. The usual spherical fullerenes have 12 pentagons; elliptic fullerenes, 6; and toroidal and Klein-bottle fullerenes, none. Klein-bottle and elliptic fullerenes are the antipodal quotients of centrosymmetric toroidal and spherical fullerenes, respectively. Extensions to infinite systems (plane fullerenes, cylindrical fullerenes, and space fullerenes) are indicated. Eigenvalue spectra of all four classes of finite fullerenes, are reviewed. Leapfrog fullerenes have equal numbers of positive and negative eigenvalues, with 0, 0, 2, or 4 eigenvalues zero for spherical, elliptic, Klein-bottle, and toroidal cases, respectively.
Journal of Chemical Information and Modeling, 2014
The structure and properties of the three 10 smallest nonface-spiral (NS) fullerenes NS-T-C 380 , NS-D 3 -11 C 384 , NS-D 3 -C 440 , and the first isolated pentagon NS-fullerene, 12 NS-D 3 -C 672 , are investigated in detail. They are constructed by 13 either a generalized face-spiral algorithm or by vertex 14 insertions followed by a force-field optimization using the 15 recently introduced program Fullerene. The obtained structures 16 were then further optimized at the density functional level of 17 theory and their stability analyzed with reference to I h -C 60 . The 18 large number of hexagons results in a higher stability of the 19