Topological analysis of some special graphs: III. Regular polyhedra

A regular polyhedron is isomorphic to a cluster on which every face has same number of bonds and every atom has an equal number of coordinating atoms. A general strategy for generating the eigenvectors and the eigenvalues of regular polyhedra is given. Not sign analyses are also performed on the eigenvectors of regular polyhedra. The results provide us a quick way to grasp the topological feature of the electronic structure of clusters having interesting topology.     

Eigenvector and eigenvalues of some special graphs. IV. Multilevel circulants

A multilevel circulant is defined as a graph whose adjacency matrix has a certain block decomposition into circulant matrices. A general algebraic method for finding the eigenvectors and the eigenvalues of multilevel circulants is given. Several classes of graphs, including regular polyhedra, suns, and cylinders can be analyzed using this scheme.     

Nonbonding orbitals in fullerenes: nuts and cores in singular polyhedral graphs

A zero eigenvalue in the spectrum of the adjacency matrix of the graph representing an unsaturated carbon framework indicates the presence of a nonbonding pi orbital (NBO). A graph with at least one zero in the spectrum is singular; nonzero entries in the corresponding zero-eigenvalue eigenvector(s) (kernel eigenvectors) identify the core vertices. A nut graph has a single zero in its adjacency spectrum with a corresponding eigenvector for which all vertices lie in the core. Balanced and uniform trivalent (cubic) nut graphs are defined in terms of (-2, +1, +1) patterns of eigenvector entries around all vertices. In balanced nut graphs all vertices have such a pattern up to a scale factor; uniform nut graphs are balanced with scale factor one for every vertex. Nut graphs are rare among small fullerenes (41 of the 10 190 782 fullerene isomers on up to 120 vertices) but common among the small trivalent polyhedra (62 043 of the 398 383 nonbipartite polyhedra on up to 24 vertices). Two constructions are described, one that is conjectured to yield an infinite series of uniform nut fullerenes, and another that is conjectured to yield an infinite series of cubic polyhedral nut graphs. All hypothetical nut fullerenes found so far have some pentagon adjacencies: it is proved that all uniform nut fullerenes must have such adjacencies and that the NBO is totally symmetric in all balanced nut fullerenes. A single electron placed in the NBO of a uniform nut fullerene gives a spin density distribution with the smallest possible (4:1) ratio between most and least populated sites for an NBO. It is observed that, in all nut-fullerene graphs found so far, occupation of the NBO would require the fullerene to carry at least 3 negative charges, whereas in most carbon cages based on small nut cubic polyhedra, the NBO would be the highest occupied molecular orbital (HOMO) for the uncharged system.     

Chemical applications of topology and group theory. 17. An information theoretical approach to polyhedral symmetry [1]

Information theoretic parameters are described which measure the asymmetry of polyhedra based on partitions of their vertices, faces, and edges into orbits under action of their symmetry point groups. Such asymmetry parameters are all zero only for the five regular polyhedra and are all unity for polyhedra having no symmetry at all, i.e. belonging to the C ₁ symmetry point group. In all other cases such asymmetry parameters have values between zero and unity. Values for such asymmetry parameters are given for all topologically distinct polyhedra having five, six, and seven vertices; all topologically distinct eight-vertex polyhedra having at least six symmetry elements; and selected polyhedra having from nine to twelve vertices. Effects of polyhedral distortions on these asymmetry parameters are examined for the tetrahedron, trigonal bipyramid, square pyramid, and octahedron. Such information theoretic asymmetry parameters can be used to order site partitions which are incomparable by the chirality algebra methods of Ruch and co-workers.     

Codes in platonic, Archimedean, Catalan, and related polyhedra: a model for maximum addition patterns in chemical cages

The notion of d-code is extended to general polyhedra by defining maximum sets of vertices with pairwise separation > or =d. Codes are enumerated and classified by symmetry for all regular and semiregular polyhedra and their duals. Partial results are also given for the series of medials of Archimedean polyhedra. In chemistry, d-codes give a model for maximal addition to or substitution in polyhedral frameworks by bulky groups. Some illustrative applications from the chemistry of fullerenes and boranes are described.     

Ba3Y2B6O15, a novel cubic borate

Single crystals of tribarium diyttrium hexaborate, which crystallized in the cubic system, have been obtained by spontaneous crystallization from a high‐temperature melt using Li₂O–BaO–B₂O₃ as flux. Its structure is composed of isolated [B₂O₅]⁴⁻ groups, irregular BaO₉ polyhedra and regular YO₆ polyhedra which occupy alternate sites running along the [111] direction. Irregular BaO₉ polyhedra and regular YO₆ polyhedra construct a three‐dimensional framework, which is reinforced by [B₂O₅]⁴⁻ groups.     

A concurrent algorithm for parallel calculation of eigenvalues and eigenvectors of real symmetric matrices

Elementary Jacobi Rotations are used as the basic tools to obtain eigenvalues and eigenvectors of arbitrary real symmetric matrices. The proposed algorithm has a complete concurrent structure, that is: every eigenvalue-eigenvector pair can be obtained in any order and in an independent way from the rest. Examples based on diagonally dominant real symmetric matrices are given.     

Symmetry factoring of the characteristic equations of graphs corresponding to polyhedra

A systematic procedure is described which uses two-and three-fold symmetry elements in graphs to reduce their adjacency matrices to lead to corresponding factorings of their characteristic polynomials. A graph splitting algorithm based on this matrix reduction procedure is described. Applications of these methods to the factoring of the characteristic polynomials of 28 polyhedra with nine or less vertices are given. General expressions for the eigenvalues of prisms, pyramids, and bipyramids in terms of the eigenvalues of their basal or equatorial regular polygons are calculated by closely related matrix methods.     

Stereochemistry of vanadium in oxygen-containing compounds

Using Voronoi-Dirichlet (VD) polyhedra and the method of overlapping spheres, we analyze the coordination of 714 crystallographically different vanadium atoms in the structure of compounds containing VO _n polyhedra. Vanadium atoms are found to be bonded to 4, 5, 6, or 7 oxygen atoms. The effect of the valence state and coordination number of vanadium atoms on the main parameters of their VD polyhedra is considered. A unified linear dependence between the solid angles of VD polyhedron faces corresponding to V-O bonds and the respective interatomic distances varying in a range of 1.55 Å to 2.79 Å is stated to exist. It is shown that the parameters of VD polyhedra can be used to determine the valence state of vanadium atoms.     

From extended solids to nano-scale actinide clusters

Selected highlights of more than a decade of research efforts concerning the structural chemistry of actinyl materials at the University of Notre Dame is reviewed, with an emphasis on complex topological arrangements of polyhedra to form extended structures and frameworks. Earlier work focused on structures of uranyl minerals and synthetic compounds, with increasing emphasis on neptunyl materials and the importance of cation–cation interactions in their structural details and properties. Much of the research over the past 5 years has examined a growing family of nano-scale clusters of uranyl peroxide polyhedra containing from 16 to 60 polyhedra. These clusters contain topological squares, pentagons and hexagons, and six have adopted fullerene topologies with 12 pentagons and an even number of hexagons.     

Regular Polyhedra Helices

The concept of a regular polyhedra helix is defined and the full set of such moieties are analysed. It is found that apart from trivial cases, eight distinct RPH exist.     

Structural changes in water clusters during methane adsorption

The structure of water clusters that have adsorbed from one to six methane molecules is studied by molecular dynamics simulation. Characteristic structural units of Voronoi and hybrid polyhedra are employed to reveal the structural changes resulting from the attachment of CH₄ molecules to the clusters. The most significant changes in topological properties are associated with variations in the number of faces of simplified polyhedra. A change is unambiguously detected in the small-angle peak intensity in the angular distribution of nearest geometrical neighbors determined with the use of the Voronoi polyhedra. The results of two different calculations of the number of hydrogen bonds in the clusters are compared, and the “nonsphericity” coefficients are calculated for the polyhedra.     

Topology of polyhedral calthrate frameworks. 1. Relationships between basic topological characteristics of regular frameworks

The relations between the numbers of nodes, bonds, cycles, and polyhedral voids in regular tetrahedral frameworks formed of space-filling polyhedra are analyzed. Institute of Inorganic Chemistry, Siberian Branch, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 5, pp. 884–890, September–October, 1995. Translated by L. Smolina     

正二十面体分子对称性轨道

本文应用多面体分子轨道理论中的群重叠法,讨论正多面体分子对称性轨道的一般造法和变换系数S的计算方法。以正二十面体分子为例,构造了三角组分系的对称性轨道,并把S—系数计算结果列于附表。     

More Icosahedral Fulleroids

The discovery of the famous fullerene has raised an interest in the study of other candidates for a modeling of carbon molecules. Motivated by a P. Fowler's question Delgado Friedrichs and Deza defined I(a,b)-fulleroids as cubic convex polyhedra having only a-gonal and b-gonal faces and the symmetry groups isomorphic with the rotation group of the regular icosahedron. In this note we prove that for every n8 there exist infinitely many I(5,n)-fulleroids. This answers positively questions posed recently by Delgado Friedrichs and Deza.     

Partition of <Emphasis Type="Italic">π</Emphasis>-electrons between faces of polyhedral carbon aggregates

We report for the trivalent regular and semiregular polyhedra (three Platonic and seven Archimedean carbon polyhedra) the π-electron partition between rings of various sizes based on considering all their resonance structures. It was found that small odd-membered (3-and 5-membered) faces are assigned a lower share of π-electrons than that corresponding to equipartition (i.e., 1/3 of an electron for carbon atoms shared between three rings). In contrast, 4-membered rings obtain a larger share of π-electrons than that corresponding to equipartition.     

Coordination Polyhedra MC<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> (M Is <Emphasis Type="Italic">s</Emphasis>-Metal Atom) in Crystal Structures

The crystallochemistry of compounds containing s-metal atoms in the carbon environment has been analyzed using the method of Voronoi–Dirichlet intersecting sectors and polyhedra. The chemical nature of the s metal has been found to affect the characteristics of the Voronoi–Dirichlet polyhedra. It is shown that the volume of Voronoi–Dirichlet polyhedron is practically independent of the coordination number of the s-metal atom.     

Structural paradigms in macropolyhedral boranes

Macropolyhedral borane clusters are concave polyhedra constituting fused convex simple polyhedra. They are formally obtained by condensation of simple polyhedral boranes under elimination of between one and four BH(3) or isoelectronic units. The number of eliminated vertexes from simple polyhedra equals the number of shared vertexes in macropolyhedral boranes. For each of the eight classes with general formulae ranging from B(n)H(n-4) to B(n)H(n+10), more than one structure type is possible, differing in the number of shared vertexes and in the types of the two combined cluster fragments. However, only one type of "potential structures" is represented by experimentally known examples and is found to be favored by theoretical calculations. A sophisticated system exists among the favored macropolyhedral borane structures. For each class of macropolyhedral boranes, the number of skeletal electron pairs is directly related to the general formula, the number of shared vertexes and the type of fused cluster fragments. In order to predict the distribution of vertexes among the fused fragments, we propose the concept of preferred fragments. Preferred fragments are those usually present in the thermodynamically most stable structure of a given class of macropolyhedral boranes and are also frequently observed in the experimentally known structures. This allows us to completely predict the cluster framework of the thermodynamically most stable macropolyhedral borane isomers.