Pentagonal Faces Research Articles

Leapfrog fullerenes and Wiener index

Fullerene graphs are cubic, 3-connected planar graphs with only pentagonal and hexagonal faces. A fullerene is called a leapfrog fullerene, Le(F), if it can be constructed by a leapfrog transformation from other fullerene graph F. Here we determine the relation between the Wiener index of Le(F) and the Wiener index of the original graph F. We obtain lower and upper bounds of the Wiener index of Lei(F) in terms of the Wiener index of the original graph. As a consequence, starting with any fullerene F, and iterating the leapfrog transformation we obtain fullerenes, Lei(F), with Wiener index of order O(n2.64) and Ω(n2.36), where n is the number of vertices of Lei(F). These results disprove Hua et al. (2014) conjecture that the Wiener index of fullerene graphs on n vertices is of order Θ(n3).

Practical synthesis of 9-methylthio-7,8-nido-carborane [9-MeS-7,8-C2B9H11]-. Some evidences of BH···X hydride-halogen bonds in 9- XCH2(Me)S-7,8-C2B9H11 (X = Cl, Br, I)

Convenient method of synthesis of 9-methylthio-7,8-nido-carborane [9-MeS-7,8-C2B9H11]- based on demethylation of the dimethylsulfonium derivative 9-Me2S-7,8-C2B9H11 with sodium amide in toluene was developed. A series of alkylmethylsulfonium derivatives of nido-carborane containing heteroatoms with lone electron pairs at α-position of alkyl were prepared 9-XCH2(Me)S-7,8-C2B9H11 (X = Cl, Br, I, OMe) by alkylation of [9-MeS-7,8-C2B9H11]-. The 11B NMR spectra of these compounds demonstrate splitting signals corresponding to B(5) and B(10) atoms that can be explained by the weak intramolecular B-H···X interactions in the diastereomers containing alkyl groups directed downwards and upwards relatively to the nido-carborane pentagonal face. The existence of such interactions was supported by quantum chemical calculations.

Five-fold local symmetries in metallic liquids and glasses

In this article, we review the experimental, theoretical and simulation studies on five-fold local symmetries in metallic liquids and glasses. In the early study on simple liquid structure, it has been realized that five-fold local symmetry plays a key role in irregular structures, supercooling and crystallization of simple liquids. In particular, icosahedral short-range order, representative of five-fold local symmetry, has attracted much attention. In addition, researches proposed a dense random packing model for simple liquid structure in 1959, and found a wide variety of polyhedra and absolute predominance of pentagonal faces in simple liquids, and also pointed out that pentagonal arrangements can only occur in very complex structures such as some of the alloy structures. Based on the Frank's hypothesis of icosahedral short-range order as blocking unit in a simple liquid, a lot of theoretical and experimental efforts have been made to confirm its existence in simple liquids, metallic liquids and glasses. So far, several theoretical methods have been developed for characterizing local atomic structures in simple liquids, such as bond-orientational order parameter, Honeycutt-Andersen index, and Voronoi tessellation. Although the local atomic symmetries in atomic structures in metallic liquids and glasses can be characterized by these methods and the geometries of the atomic structures in liquids and glasses have received much more attention, an atomic cluster model has been developed for establishing the structure-property relationship in metallic liquid and glass. Due to the diversity of the atomic clusters in both type and population of different metallic liquids and glasses, the atomic cluster model could not present a simple description of structure-property relationship. Based on the fundamental characteristics of metallic liquids and glasses, five-fold local symmetry, the structure-property relationship in metallic liquids and glasses, such as dynamic crossover, glass transition, liquid-liquid phase transition, and deformation can be well described in simple, quantitative and unified ways, and therefore a clear physical picture can be provided. All these studies indicate that five-fold local symmetry as a structural parameter is simple, general and effective.

Symmetry of the Pyritohedron and Lattices

The pyritohedron consisting of twelve identical but non regular pentagonal faces and its dual pseudoicosahedron that possess the pyritohedral (Th) symmetry play an essential role in understanding the crystallographic structures with the pyritohedral symmetry. The pyritohedral symmetry takes a simpler form in terms of quaternionic representation. We discuss the 3D crystals with the pyritohedral symmetry which can be derived from the Coxeter-Dynkin diagram of D3.

Structural analysis of the epitaxial interface Ag/ZnO in hierarchical nanoantennas.

Detectors, photo-emitter, and other high order radiation devices work under the principle of directionality to enhance the power of emission/transmission in a particular direction. In order to understand such directionality, it is important to study their coupling mechanism of their active elements. In this work, we present a crystalline orientation analysis of ZnO nanorods grown epitaxially on the pentagonal faces of silver nanowires. The analysis of the crystalline orientation at the metal-semiconductor interface (ZnO/Ag) is performed with precession electron diffraction under assisted scanning mode. In addition, high resolution X-ray diffraction on a Bragg-Brentano configuration has been used to identify the crystalline phases of the arrangement between ZnO rods and silver nanowires. The work presented herein provides a fundamental knowledge to understand the metal-semiconductor behavior related to the receiving/transmitting mechanisms of ZnO/Ag nanoantennas.

Resonance graphs of fullerenes

A fullerene G is a 3 -regular plane graph consisting only of pentagonal and hexagonal faces. The resonance graph R ( G ) of G reflects the structure of its perfect matchings. The Zhang-Zhang polynomial of a fullerene is a counting polynomial of resonant structures called Clar covers. The cube polynomial is a counting polynomial of induced hypercubes in a graph. In the present paper we show that the resonance graph of every fullerene is bipartite and each connected component has girth 4 or is a path. Also, the equivalence of the Zhang-Zhang polynomial of a fullerene and the cube polynomial of its resonance graph is established. Furthermore, it is shown that every subgraph of the resonance graph isomorphic to a hypercube is an induced subgraph in the resonance graph. For benzenoid systems and tubulenes each connected component of the resonance graph is the covering graph of a distributive lattice; for fullerenes this is not true, as we show with an example.

The Clar and Fries structures of a fullerene I

Given the Kekulé structure of a fullerene that gives its Clar structure, the Kekulé edges that do not lie on any benzene ring of the Clar structure lie on open chains that pair up the pentagonal faces and perhaps some closed chains. In this paper, we introduce Fries chains and show that given the Kekulé structure of a fullerene that gives its Fries structure, the Kekulé edges that do not lie on two benzene rings lie on the union of a set of these Fries chains. The edges that lie on exactly one benzene ring belong to exactly one of these chains while edges that lie on no benzene ring belong to exactly two of these chains. We will see that the Fries chains will include the Clar chains in some cases and will be quite distinct from the Clar chains in other cases.Examples of fullerenes with the property that the set of benzene rings that give its Clar number are not a subset of the benzene rings that give its Fries number have been known for a few years. However, in all of these known examples, almost all of the benzene rings of its Clar structure are among benzene rings of its Fries structure. In this paper, we describe a class of fullerenes with the property that the set of benzene rings of its Clar structure and the set of benzene rings of its Fries structure practically disjoint.

Investigation of the possibility of functionalization of C20 fullerene by benzene via Diels–Alder reaction

C20 fullerene, this novel species with all its pentagonal faces has displayed some unique operations in making fast pericyclic reactions. As an example, the high dienophile character of the C20 fullerene and the ability of this species in making an ultra-fast Diels–Alder reaction with 1,3-butadiene, has been recently reported. Moreover, new experimental reports claim that the C60 fullerene, one of the fullerene family, could make a Diels–Alder reaction with the central ring of anthracene and make the ring non-aromatic. These reports may encourage researchers to do more studies on the properties of this small carbon cage.To address this question, the present research has discussed all the reaction channels of the Diels–Alder cycloaddition of benzene molecule as a 1,3-diene with the C20 fullerene in order to answer this question: “Is C20fullerene able to make a Diels–Alder reaction with this molecule?”.

Mathematical aspects of fullerenes

Fullerene graphs are cubic, 3-connected, planar graphs with exactly 12 pentagonal faces, while all other faces are hexagons. Fullerene graphs are mathematical models of fullerene molecules, i.e., molecules comprised only by carbon atoms different than graphites and diamonds. We give a survey on fullerene graphs from our perspective, which could be also considered as an introduction to this topic. Different types of fullerene graphs are considered, their symmetries, and construction methods. We give an overview of some graph invariants that can possibly correlate with the fullerene molecule stability, such as: the bipartite edge frustration, the independence number, the saturation number, the number of perfect matchings, etc.

Classification of hydrogen bond flips in small water polyhedra applied to concerted proton tunneling.

Recently a new mechanism of proton tunneling in a prism-like water hexamer was revealed [Richardson et al., Science, 2016, 351, 1310]. The tunneling motion involves the concerted breaking of two hydrogen bonds and rotations of two nearest water molecules. Eventually, this structural transformation means flipping one of the hydrogen bonds without the creation of defects in the hydrogen bond network. On the surface of polyhedral water clusters, there are five essentially different types of hydrogen bonds, and only two of them can be changed in this manner. In this article, the topological classification of such transformations for five small water polyhedra: triangular, pentagonal, and hexagonal prisms as well as cube and polyhedron 4454, consisting of four square and four pentagonal faces, is presented. Our classification includes the enumeration of all possible one-bond-flips with consideration of the types of hydrogen bonds on the polyhedral surface. Attention is paid to the most stable proton configurations which can be studied in experiments. It was established that a number of one-bond-flip transitions between the low energy configurations are possible in clusters in the shape of triangular and pentagonal prisms.

Direct transition mechanism for molecular diffusion in gas hydrates

In this work, we use dual cage explicit atomic systems to demonstrate theoretically that direct transitions are feasible through hexagonal and pentagonal faces in type I hydrate without compromising the overall structure integrity.

Tetracarbaboranes: nido structures without bridging hydrogens.

The structures and energetics of the tetracarbaboranes C4Bn-4Hn (n = 6 to 13) have been investigated by density functional theory and coupled cluster calculations. In general, the lowest energy structures of the tetracarbaboranes C4Bn-4Hn minimize the number of C-C polyhedral edges as well as the degrees of the carbon vertices. For the C4B2H6 and C4B3H7 systems the lowest energy structures are pyramidal structures having all four carbon atoms located on the base of the pyramid. The lowest energy structure for the 9-vertex C4B5H9 system is a capped square antiprism. The frameworks of the lowest energy C4B4H8 and C4B6H10 structures resemble those of the isoelectronic experimentally known B8H12 and B10H14 structures. However, an experimentally known S4 adamantane-like 10-vertex structure found in Me4C4B6Et6 based on a tetracapped octahedron lies only ∼7 kcal mol-1 in energy above the lowest energy structure. The lowest energy structures for the 11- to 13-vertex C4Bn-4Hn (n = 11, 12, 13) systems can be derived from an (n + 1)-vertex closo deltahedron by removing a high-degree vertex. At least three of the four carbon atoms are located on edges of the resulting pentagonal or hexagonal open face in the low-energy structures. However, the structures of the experimentally known R4C4B8H8 (R = Me, Et) obtained from the dimerization of R2C2B4H42- differ from these low-energy structures. The Me4C4B8H8 polyhedron has a C4 chain and two tetragonal faces whereas the Et4C4B8H8 polyhedron has a hexagonal face with two C2 units. These structures lie within 2 kcal mol-1 of each other thereby accounting for the fluxional properties of these systems observed by NMR spectroscopy.

Tetracarbalane structures: nido polyhedra and non-spherical deltahedra

Uhl and coworkers have synthesized and characterized structurally the 11-vertex tetracarbalanes (AlMe)7(CEt)4(μ-H)2 and (AlEt)7(CCH2Ph)4(C[triple bond, length as m-dash]CPh)(μ-H). In order to understand the nature of the unusual C4Al7 polyhedra in such systems, the complete series of permethylated tetracarbalanes C4Aln-4Men (n = 6 to 14) as model compounds have been investigated using density functional theory. An overriding factor in determining the polyhedra for the lowest energy structures of a wide range of tetracarbalanes C4Aln-4Men (n = 6 to 14) is the availability of vertices of degrees 3 and 4 for all four carbon atoms. For the 11-vertex C4Al7Me11 system the experimentally observed C4Al7 deltahedron with carbon atoms at the four degree 4 vertices is found in the lowest energy structure but severely distorted from ideal C2h to Cs symmetry. This distortion is removed by adding two bridging hydrogens to the C4Al7Me11 structure in the same locations as in the experimental structures. The lowest energy structures of the tetracarbalanes C4Aln-4Men (n = 12, 13, 14) with more than 11 vertices are deltahedra having the carbon atoms at degree 4 vertices similar to the lowest energy C4Al7Me11 structures. The lowest energy structures for the smaller tetracarbalanes C4Aln-4Men (n = 6, 7, 8) are nido structures with pentagonal or hexagonal open faces related to nido borane structures. The intermediate 9- and 10-vertex tetracarbalanes C4Aln-4Men (n = 9, 10) have relatively complicated energy surfaces.

Molecular dynamics simulations of the formation of ethane clathrate hydrates

This study details molecular dynamics simulations for the formation of ethane clathrate hydrates. Ethane hydrate nucleation shares with methane hydrate nucleation the formation of coordinated guest clusters in water, resulting in structures such as guest-saturated pentagonal and hexagonal faces. Ethane hydrate also forms structure I, which consists of two 512 and six 51262 cages in its unit cell. Observations from the simulations reveal differences between methane and ethane in the nucleation process, including the types and occupancies of cages. While the formation of occupied 512 cages dominates methane hydrate nucleation, nucleation of ethane hydrates predominately exhibits the 4151062 cage as both the first and most common early cage type. These 4151062 cages do not occur in the crystal structure, and due to their oblong shape, restrict the orientation of the coordinated guest ethane molecule. The initial kinetically favored 4151062 cages are observed to undergo transformation into different cage types, such as the thermodynamically favored 51262 cage.

First-Principles Study of Lithiation of Type I Ba-Doped Silicon Clathrates

Silicon clathrate materials, previously known for their superconducting and thermoelectric characteristics, have also recently been investigated for their electrochemical properties as anodes for lithium-ion batteries due to their unique cage structure and ability to incorporate extrinsic guest atoms. To better understand the preferred structures for small degrees of lithiation, first-principles density functional theory (DFT) was used to investigate the type I clathrate compounds Si46, LixBaySi46 (0 ≤ x ≤ 8; y = 6, 8), and LixBayAl6Si40 (0 ≤ x ≤ 8; y = 6). The formation energies, electronic band structures, and density of states (DOS) were calculated. Lithium occupation in framework vacancies, empty and Ba-occupied cage cavities, and near the pentagonal and hexagonal faces of the clathrate polyhedra was considered. The data showed that Li insertion into framework or Ba vacancies could stabilize the clathrate structure. Silicon substitution by Al lowered the formation energies of the lithiated compounds a...

Similarity and a Duality for Fullerenes

Fullerenes are molecules of carbon that are modeled by trivalent plane graphs with only pentagonal and hexagonal faces. Scaling up a fullerene gives a notion of similarity, and fullerenes are partitioned into similarity classes. In this expository article, we illustrate how the values of two important fullerene parameters can be deduced for all fullerenes in a similarity class by computing the values of these parameters for just the three smallest representatives of that class. In addition, it turns out that there is a natural duality theory for similarity classes of fullerenes based on one of the most important fullerene construction techniques: leapfrog construction. The literature on fullerenes is very extensive, and since this is a general interest journal, we will summarize and illustrate the fundamental results that we will need to develop similarity and this duality.

Hyperbolic analogues of fullerenes with face-types [formula omitted] and [formula omitted

Mathematical models of fullerenes are cubic polyhedral and spherical maps of face-type (5,6), that is, with pentagonal and hexagonal faces only. Any such map necessarily contains exactly 12 pentagons, and it is known that for any integer α≥0 except α=1 there exists a fullerene map with precisely α hexagons.In this paper we consider hyperbolic analogues of fullerenes, modelled by cubic polyhedral maps of face-type (6,k), where k∈{9,10}, on orientable surface of genus at least two. The number of k-gons in this case depends on the genus but the number of hexagons is again independent of the surface. For every triple k∈{9,10}, g≥2 and α≥0, we determine if there exists a cubic polyhedral map of face-type (6,k) with exactly α hexagons on an orientable surface of genus g. The only unsolved cases are k=10, g=5 and α≤3 when we are not able to say if a hyperbolic fullerene with these parameters exists.

2-resonant fullerenes

A fullerene graph F is a planar cubic graph with exactly 12 pentagonal faces and other hexagonal faces. A set H of disjoint hexagons of F is called a resonant pattern (or sextet pattern) if F has a perfect matching M such that every hexagon in H is M-alternating. F is said to be k-resonant if any i (0≤i≤k) disjoint hexagons of F form a resonant pattern. It was known that each fullerene graph is 1-resonant and there are only nine fullerene graphs that are 3-resonant. In this paper, we show that the fullerene graphs which do not contain the subgraph L or R as illustrated in Fig. 1 are 2-resonant except for the specific eleven graphs. This result implies that each IPR fullerene is 2-resonant.

Electric radiation mapping of silver/zinc oxide nanoantennas by using electron holography.

In this work, we report the fabrication of self-assembled zinc oxide nanorods grown on pentagonal faces of silver nanowires by using microwaves irradiation. The nanostructures resemble a hierarchal nanoantenna and were used to study the far and near field electrical metal-semiconductor behavior from the electrical radiation pattern resulting from the phase map reconstruction obtained using off-axis electron holography. As a comparison, we use electric numerical approximations methods for a finite number of ZnO nanorods on the Ag nanowires and show that the electric radiation intensities maps match closely the experimental results obtained with electron holography. The time evolution of the radiation pattern as generated from the nanostructure was recorded under in-situ radio frequency signal stimulation, in which the generated electrical source amplitude and frequency were varied from 0 to 5 V and from 1 to 10 MHz, respectively. The phase maps obtained from electron holography show the change in the distribution of the electric radiation pattern for individual nanoantennas. The mapping of this electrical behavior is of the utmost importance to gain a complete understanding for the metal-semiconductor (Ag/ZnO) heterojunction that will help to show the mechanism through which these receiving/transmitting structures behave at nanoscale level.

A mathematical programming model for computing the Fries number of a fullerene

A fullerene graph is a cubic 3-connected plane graph with pentagonal and hexagonal faces. The Fries number of a fullerene is the maximum number of benzene-like faces over all possible perfect matchings. The Fries number and its associated Kekulé structure of a fullerene play a key role in molecular energy and stability. In this paper we propose a binary integer linear programming and a quadratic programming model for determining the Fries number of a fullerene. Moreover, interior point approach, as one of the most robust optimization techniques, is implemented to find the optimal solution of the proposed quadratic programming problem in moderate computing time.