The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

Applications of finite reflection groups in Fourier analysis and symmetry breaking of polytopes

Cette thèse présente une étude des applications des groupes de réflexion finis aux problems liés aux réseaux bidimensionnels et aux polytopes tridimensionnels. Plusieurs familles de fonctions orbitales, appelées fonctions orbitales de Weyl, sont associées aux groupes de réflexion cristallographique. Les propriétés exceptionnelles de ces fonctions, telles que l’orthogonalité continue et discrète, permettent une analyse de type Fourier sur le domaine fondamental d’un groupe de Weyl affine correspondant. Dans cette considération, les fonctions d’orbite de Weyl constituent des outils efficaces pour les transformées discrètes de type Fourier correspondantes connues sous le nom de transformées de Fourier–Weyl. Cette recherche limite notre attention aux fonctions d’orbite de Weyl symétriques et antisymétriques à deux variables du groupe de réflexion cristallographique A2. L’objectif principal est de décomposer deux types de transformations de Fourier–Weyl du réseau de poids correspondant en transformées plus petites en utilisant la technique de division centrale. Pour les cas non cristallographiques, nous définissons les indices de degré pair et impair pour les orbites des groupes de réflexion non cristallographique avec une symétrie quintuple en utilisant un remplacement de représentation-orbite. De plus, nous formulons l’algorithme qui permet de déterminer les structures de polytopes imbriquées. Par ailleurs, compte tenu de la pertinence de la symétrie icosaédrique pour la description de diverses molécules sphériques et virus, nous étudions la brisure de symétrie des polytopes doubles de type non cristallographique et des structures tubulaires associées. De plus, nous appliquons une procédure de stellation à la famille des polytopes considérés. Puisque cette recherche se concentre en partie sur les fullerènes icosaédriques, nous présentons la construction des nanotubes de carbone correspondants. De plus, l’approche considérée pour les cas non cristallographiques est appliquée aux structures cristallographiques. Nous considérons un mécanisme de brisure de symétrie appliqué aux polytopes obtenus en utilisant les groupes Weyl tridimensionnels pour déterminer leurs extensions structurelles possibles en nanotubes.This thesis presents a study of applications of finite reflection groups to the problems related to two-dimensional lattices and three-dimensional polytopes. Several families of orbit functions, known as Weyl orbit functions, are associated with the crystallographic reflection groups. The exceptional properties of these functions, such as continuous and discrete orthogonality, permit Fourier-like analysis on the fundamental domain of a corresponding affine Weyl group. In this consideration, Weyl orbit functions constitute efficient tools for corresponding Fourier-like discrete transforms known as Fourier–Weyl transforms. This research restricts our attention to the two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2. The main goal is to decompose two types of the corresponding weight lattice Fourier–Weyl transforms into smaller transforms using the central splitting technique. For the non-crystallographic cases, we define the even- and odd-degree indices for orbits of the non-crystallographic reflection groups with 5-fold symmetry by using a representation-orbit replacement. Besides, we formulate the algorithm that allows determining the structures of nested polytopes. Moreover, in light of the relevance of the icosahedral symmetry to the description of various spherical molecules and viruses, we study symmetry breaking of the dual polytopes of non-crystallographic type and related tube-like structures. As well, we apply a stellation procedure to the family of considered polytopes. Since this research partly focuses on the icosahedral fullerenes, we present the construction of the corresponding carbon nanotubes. Furthermore, the approach considered for the non-crystallographic cases is applied to crystallographic structures. We consider a symmetry-breaking mechanism applied to the polytopes obtained using the three-dimensional Weyl groups to determine their possible structural extensions into nanotubes

Brisure de la symétrie icosaédrique du C60 vers des fullerènes plus grands et les nanotubes apparentés

La symétrie icosaédrique exacte du fullerène C60 est vue comme une orbite du groupe de Coxeter H3. Cette orbite est décomposable en orbites des sous-groupes symétriques de rangs inférieurs. Les orbites forment un empilement de couches parallèles centrées sur un axe traversant le C60 de part en part. En insérant au milieu de l'empilement un certain nombre d'orbites du sous-groupe étudié, on peut retrouver la structure d'un fullerène plus grand. En répétant l'insertion, on peut obtenir des fullerènes et nanotubes de toute longueur souhaitable. Ce mémoire présente les cas où les sous-groupes utilisés sont notés A2 et A1×A1, car ils sont respectivement isomorphes aux groupes de Weyl des algèbres de Lie simples A2 et A1×A1. Ces deux cas permettent d'obtenir à terme des nanotubes de type zigzag et chiral. Des avenues de généralisation de la méthode sont discutées dans la dernière partie de ce mémoire.Exact icosahedral symmetry of C60 fullerene is viewed as an orbit of H3 Coxeter group. This orbit is decomposable into orbits of symmetrical subgroups of lower ranks. The orbits form a stack of parallel layers centered on an axis traversing the C60 from side to side. By inserting in the middle of the stack a certain number of orbits of the studied subgroup, one can find the structure of a larger fullerene. By repeating the insertion, fullerenes and nanotubes of any desirable length can be obtained. This thesis presents the cases where the subgroups used are denoted A2 and A1×A1, since they are respectively isomorphic to the Weyl groups of simple Lie algebras A2 and A1×A1. These two cases make it possible eventually to obtain nanotubes of the zig-zag and chiral type. Avenues of generalization of the method are discussed in the last part of this thesis

Structure prediction of nano materials

This thesis is predominantly on the exploration of configurational space of different materials of different dimensions using ab initio techniques. We focus on understanding the predicted geometrical configurations and their electronic properties. For this purpose, we employed MHM to explore the PES as well as identify putative low lying energy structures

On Mathematical Properties of Buckminsterfullerenef

Some of the mathematical properties of buckminsterfullerene are considered, that is, geometrical, topological, group-theoretical and graph-theoretical properties. These mathematical properties are used to predict several structural and chemical properties of buckminsterfullerene

From mathematical models to quantum chemistry in cluster science : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Chemistry at Massey University, Albany, New Zealand

Listed in 2019 Dean's List of Exceptional ThesesThe structures and stabilities of hollow gold clusters are investigated by means of density functional theory (DFT) as topological duals of carbon fullerenes. Fullerenes can be constructed by taking a graphene sheet and wrapping it around a sphere, which requires the introduction of exactly 12 pentagons. In the dual case, a (111) face-centred cubic (fcc) gold sheet can be deformed in the same way, introducing 12 vertices of degree five, to create hollow gold nano-cages. This one-to-one relationship follows trivially from Euler’s polyhedral formula and there are as many golden dual fullerene isomers as there are carbon fullerenes. Photoelectron spectra of the clusters are simulated and compared to experimental results to investigate the possibility of detecting other dual fullerene isomers. The stability of the hollow gold cages is compared to compact structures and a clear energy convergence towards the (111) fcc sheet of gold is observed. The relationship between the Lennard-Jones (LJ) and sticky-hard-sphere (SHS) potential is investigated by means of geometry optimisations starting from the SHS clusters. It is shown that the number of non-isomorphic structures resulting from this procedure depends strongly on the exponents of the LJ potential. Not all LJ minima, that have been discovered in previous work, can be retrieved this way and the mapping from the SHS to the LJ structures is therefore non-injective and non-surjective. The number of missing structures is small and they correspond to energetically unfavourable minima on the energy landscape. The optimisations are also carried out for an extended Lennard-Jones potential derived from coupled-cluster calculations for the xenon dimer, and, although the shape of the potential is not too different from a regular (6,12)-LJ potential, the number of minima increases substantially. Gregory-Newton clusters, which are clusters where 12 spheres surround and touch a central sphere, are obtained from the complete set of SHS clusters. All 737 structures result in an icosahedron, when optimised with a (6,12)-LJ potential. Furthermore, the contact graphs, consisting only of atoms from the outer shell of the clusters, are all edge-induced sub-graphs of the icosahedral graph. For higher LJ exponents the symmetry of the potential energy surface breaks away from the icosahedral motif towards the SHS landscape, which does not support a perfect icosahedron for energetic reasons. This symmetry breaking is mainly governed by the shape of the potential in the repulsive region, with the long-range attractive region having little influence

Electron, Photon, and Positron Scattering Dynamics of Complex Molecular Targets

Electron scattering cross sections have been computed for pyridine and pyrimidine using the static-exchange approximation with model potential to account for dynamic electron correlation. To obtain well-converged orbitals, we have expanded all partial waves to a maximum angular momentum of l = 60 for both targets. We have obtained total cross sections for electron scattering energies to 20 eV. Both targets display similar features, namely a dipole-induced increase in the integrated cross section at scattering energies below 5 eV, and peaks corresponding to resonances in b1, a2, and b1 symmetries. These resonances were investigated through a Siegert eigenstate analysis and Breit-Wigner fit of the SECP eigenphase sums. They were also compared to the virtual orbitals obtained from a minimum basis set Hartree-Fock calculation on both targets. We consider electron scattering resonances from cis-diamminedichloroplatinum, [Pt(NH3)2Cl2], the ligand molecular species Cl2 (1Sigma+g ), and the isolated transition metal center Pt in a nondegenerate atomic state (1S) at the SECP level of theory. As a rigorous comparison to the single-state, single-configuration SECP level results of these smaller, yet electron dense targets, we have also considered scattering from ground state Cl2 and Pt in the 1S and 3D states in the multichannel configuration-interaction (MCCI) approximation originally developed for photoionization for scattering up to 10 eV. Photoionization cross sections and angular distributions in the recoil frame (RFPAD) and molecular frame (MFPAD) have been computed for inner-shell C 1s and Cl 2p ionization from the chloroalkanes chloromethane and chloroethane, with ionization leading to a variety of ionic fragment states. We have also computed valence level ionization from the nitro molecule nitromethane CH3NO2 leading to the dissociation of the CN bond. All of these calculations were performed in the frozen-core Hartree-Fock approximation. Even at this level of theory, we obtain computed results that compare well to the photoelectronphotoion coincidence measurements. The fullerene C20 is the smallest fullerene predicted to exist, with most relevant structural calculations suggesting the reduction of the icosahedral symmetry into one in which the target species possesses at maximum only a dihedral axis. We have computed positron scattering cross sections for the molecule in two low-symmetry structural isomers Ci and C2, within the HF approximation. Density functional expressions were used to incorporate important positron-electron interactions within the calculation. We have found similar cross sections and resonance features for both isomers, including a positron scattering resonance whose density is found within the framework of the fullerene cluster

Carbon in the Galaxy: Studies from Earth and Space

Presented here is the text of the invited papers presented during a meeting entitled, Carbon in the Galaxy: Studies from Earth and Space, that was held at NASA Ames Research Center on November 5 and 6, 1987. For completeness, abstracts from all of the poster papers and the text of a paper summarizing what was learned during the course of the meeting are also included. The underlying premise for the meeting was that there is much to be gained by bringing together scientists from very different disciplines, all of whom study carbon in different ways for different reasons. The interchanges took place during the meeting and the contents of the enclosed papers validate that premise

Reactividad química de nanoformas de carbono : reacciones de arilación y cicloadición 1,3-dipolar

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Químicas, Departamento de Química Orgánica I, leída el 08-05-2015La evolución de la sociedad ha estado fuertemente influenciada por el desarrollo de nuevos materiales y aplicaciones, haciendo de la Ciencia de Materiales un importante campo de investigación, donde interactúan tanto técnicas como conocimientos de diferentes aéreas científicas. En este contexto, los últimos descubrimientos en nanoformas de carbono CNF han de jugar un papel fundamental en el presente y futuro, tanto de la ciencia como de la sociedad en general. Si ya el fullereno C y los fullerenos endoédricos sorprendieron a la comunidad científica por su geometría, tamaño nanométrico y propiedades electrónicas, las últimas nanoformas de carbono descubiertas, nanotubos de carbono CNT y grafeno, han causado mayor expectación. CNT y grafeno exhiben sorprendentes propiedades electrónicas, eléctricas, mecánicas o térmicas, mejorando a algunos de los mejores materiales conocidos hasta la fecha como Cu, Ag o Si, o incluso que el kevlar o el acero. En esta tesis se describe el estudio de la reactividad química, estructura y propiedades de estas diferentes nanoestructuras de carbono nanotubos de carbono de pared simple SWNT , de pared múltiple MWNT , grafeno y fullerenos endoédricos. El trabajo se engloba en dos capítulos bien diferenciados que presentan como nexo común la utilización de reacciones de cicloadición 1,3 dipolar. EN el Capítulo 1 se describe la síntesis de nanoconjugados que contienen como nanoestructuras de carbono SWNT, MWNT y grafeno, combinados con p exTTF, mediante reacciones de arilación de tipo Tour seguidas por reacciones de cicloadición 1,3 dipolar catalizadas por cobre, también conocidas como click chemistry. Dichos nanoconjugados han sido caracterizadas mediante diferentes técnicas en ciencia de los materiales XPS, Raman, TGA, UV vis NIR, FTIR . La comunicación electrónica entre exTTF y SWNT se corroboró mediante técnicas electroquímicas y fotofísicas. Del mismo modo, la geometría cóncava del exTTF se utilizó para evaluar su comportamiento en el reconocimiento de fullereno C tanto en estado libre como ancladas sobre las diferentes CNF, demostrándose la formación del complejo receptor fullereno en ambos casos. Se realizaron valoraciones mediante UV vis NIR para evaluar la extensión de la interacción como valoraciones por RMN para desvelar los puntos de reconocimiento de los receptores con el fullereno, que fueron soportados sobre cálculos teóricos.Depto. de Química OrgánicaFac. de Ciencias QuímicasTRUEunpu