Zigzags, railroads, and knots in fullerenes

Two connections between fullerene structures and alternating knots are established. Knots may appear in two ways: from zigzags, i.e., circuits (possibly self-intersecting) of edges running alternately left and right at successive vertices, and from railroads, i.e., circuits (possibly self-intersecting) of edge-sharing hexagonal faces, such that the shared edges occur in opposite pairs. A z-knot fullerene has only a single zigzag, doubly covering all edges: in the range investigated (n /= 38, all chiral, belonging to groups C(1), C(2), C(3), D(3), or D(5). An r-knot fullerene has a railroad corresponding to the projection of a nontrivial knot: examples are found for C(52) (trefoil), C(54) (figure-of-eight or Flemish knot), and, with isolated pentagons, at C(96), C(104), C(108), C(112), C(114). Statistics on the occurrence of z-knots and of z-vectors of various kinds, z-uniform, z-transitive, and z-balanced, are presented for trivalent polyhedra, general fullerenes, and isolated-pentagon fullerenes, along with examples with self-intersecting railroads and r-knots. In a subset of z-knot fullerenes, so-called minimal knots, the unique zigzag defines a specific Kekulé structure in which double bonds lie on lines of longitude and single bonds on lines of latitude of the approximate sphere defined by the polyhedron vertices.     

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.     

Hypoelectronic dirhenaboranes having eight to twelve vertices: internal versus surface rhenium-rhenium bonding

Fehlner, Ghosh, and their co-workers have synthesized a series of dirhenaboranes Cp(2)Re(2)B(n-2)H(n-2) (n = 8, 9, 10, 11, 12) exhibiting unprecedented oblate (flattened) deltahedral structures. These structures have degree 6 and/or 7 rhenium vertices at the flattest regions on opposite sides of an axially compressed deltahedron thereby leading to Re═Re distances in the range 2.69 to 2.94 ? suggesting internal formal double bonds. These experimental oblate (flattened) deltahedral structures are shown by density functional theory to be the lowest energy structures for these dirhenaboranes. In some cases the energy differences between such oblate deltahedral structures and the next higher energy structures are quite considerable, that is, up to 25 kcal/mol for the nine-vertex Cp(2)Re(2)B(7)H(7) structures. The higher energy Cp(2)Re(2)B(n-2)H(n-2) structures are of the two types: (1) Most spherical (closo) deltahedra having unusually short 2.28 to 2.39 ? Re-Re edges with unusually high Wiberg bond indices suggesting formal multiple bonds on the deltahedral surface; (2) Deltahedra having one or two degree 3 vertices and 2.6 to 2.9 ? Re-Re edges. The latter deltahedra are derived from smaller deltahedra by capping Re(2)B faces with the degree 3 vertices.     

Face-spiral codes in cubic polyhedral graphs with face sizes no larger than 6

According to the face-spiral conjecture, first made in connection with enumeration of fullerenes, a cubic polyhedron can be reconstructed from a face sequence starting from the first face and adding faces sequentially in spiral fashion. This conjecture is known to be false, both for general cubic polyhedra and within the specific class of fullerenes. Here we report counterexamples to the spiral conjecture within the 19 classes of cubic polyhedra with positive curvature, i.e., with no face size larger than six. The classes are defined by triples {p ₃, p ₄, p ₅} where p ₃, p ₄ and p ₅ are the respective numbers of triangular, tetragonal and pentagonal faces. In this notation, fullerenes are the class {0, 0, 12}. For 11 classes, the reported examples have minimum vertex number, but for the remaining 8 classes the examples are not guaranteed to be minimal. For cubic graphs that also allow faces of size larger than 6, counterexamples are common and occur early; we conjecture that every infinite class of cubic polyhedra described by allowed and forbidden face sizes contains non-spiral elements.     

Van der Waals surface graphs and molecular shape

A van der Waals surface graph is the graph defined on a van der Waals surface by the intersections of the atomic van der Waals spheres. A van der Waals shape graph has a vertex for each atom with a visible face on the van der Waals surface, and edges between vertices representing atoms with adjacent faces on the van der Waals surface. These are discrete invariants of three‐dimensional molecular shape. Some basic properties of van der Waals surface graphs are studied, including their relationship with the Voronoi diagram of the atom centres, and a class of molecular embeddings is identified for which the dual of the van der Waals surface graph coincides with the van der Waals shape graph. This revised version was published online in July 2006 with corrections to the Cover Date.     

Truncating and chamfering diagrams of regular polyhedra

Truncating the vertex and chamfering the edges of a polyhedron is the classic way to obtain Archimedean polyhedra taking a Platonic solid as the starting point. We have considered the set of polyhedra obtained by this method and have found singular diagrams that show a special relation between these Archimedean polyhedra.     

Topological aspects of chemically significant polyhedra

This paper considers both static and dynamic properties of chemically significant polyhedra. Static properties of polyhedra consider relationships between the numbers and degrees/sizes of polyhedral vertices, edges, and faces; polyhedral symmetries; and numbers of topologically distinct polyhedra of various types. Dynamic properties of polyhedra involve studies of polyhedral isomerizations from both macroscopic and microscopic points of view. Macroscopic aspects of polyhedral isomerization can be described by graphs called topological representations in which the vertices correspond to different permutational isomers and the edges to single degenerate polyhedral isomerization steps. Such topological representations are presented for isomerizations of polyhedra having five, six, and eight vertices. Microscopic aspects of polyhedral isomerizations arise from consideration of the details of polyhedral topology, such as the topological aspects of diamond-square-diamond processes. In this connection, Gale diagrams are useful for describing isomerizations of five- and six-vertex polyhedra, including the Berry pseudorotation of a trigonal bipyramid through a square pyramid intermediate and the Bailar or Ray and Dutt twists of an octahedron through a trigonal prism intermediate.     

A “vertex electron pair” scheme (VEPS) for describing the skeletal electron distribution in borane-type clusters

A predominantly localized electron pair scheme is outlined for describing the electron distribution and bonding in closo borane anions B_nH_n²⁻ and related electron deficient deltahedral clusters, in which a skeletal electron pair is assigned to each vertex, one pair being regarded as delocalized just inside the roughly spherical surface on which the skeletal atoms lie. The scheme gives a clearer picture of the electron distribution than is conveyed by resonating 2- and 3-centre bonds in the polyhedron edges and faces, and allows the bond orders of the polyhedron edge links to be calculated readily. The consequence of formal removal of BH²⁺ units from closo species B_nH_n²⁻ to generate nido species B_n−1H_n−1⁴⁻ and arachno species B_n−2H_n−2⁶⁻ is explored, and seen to allow rationalization of two features of such deltahedral-fragment clusters: (i) why a high-connectivity vertex is left vacant and (ii) why the frontier orbitals of such species concentrate electronic charge around their open faces. Moreover, in the case of D_4‘h B₄H₄⁶⁻ (cf. C₄H₄²⁻) and D_5h B₅H₅⁶⁻ (cf. C₅H₅⁻), the approach leads directly to the familiar picture for aromatic ring systems in which the highest filled, doubly degenerate π-bonding molecular orbital concentrates electronic charge in rings above and below the polygon on which the skeletal nuclei lie. It also leads to the expectation that arachno clusters with non-adjacent vacant vertices will be more stable than those with adjacent vacant vertices.     

二十面体壳层结构：纪念C60发现10周年

圆顶建筑、病毒衣壳和Ｃ６０分子笼的生成机制和尺寸差别很大，但都属二十面体壳层，遵循同一几何规律。圆顶建筑和病毒衣壳由近似一等的三角面构成，有１２个五键加３＾５顶，１０（Ｔ－１）个六键连３＾６顶，其中三角面数Ｔ＝ｈ＾２＋ｈｋ＋ｋ＾２是二十面体的一个三角面中的小三角面数，ｈ，ｋ是六角坐标系中３＾５顶的坐标。另一方面，全碳分子笼Ｃｎ（包括其原型Ｃ６０），由１２个五角面与１０（Ｔ－１）个六角面围成，全是三     

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

Possible convex polyhedra for three-dimensional water networks in clathrate and semiclathrate hydrates are discussed in this paper. All such polyhedra have all vertices of order three. Therefore, the number of vertices (v), edges (e), and faces (f) must satisfy the equalities e=3v/2 and f=(4+v)/2. Possible polyhedra of this type with exclusively quadrilateral, pentagonal, and hexagonal faces and with up to 18 faces are examined. Many of these polyhedra are duals of various triangulated coordination polyhedra studied in previous papers of this series. In order to minimize angular strain, polyhedra with the maximum number of pentagonal faces are favored for water networks in clathrate and semiclathrate hydrates subject to the presence of sufficiently large cavities to accommodate the guest molecule.

---

Zusammenfassung In dieser Arbeit werden mögliche konvexe Polyeder für dreidimensionale Wasser-Netzwerke in Klathrat- und Semiklathrathydraten diskutiert. Daher muß die Anzahl der Scheitelpunkte (v), Kanten (e) und Flächen (f) den Gleichungen e=3v – und f=(4+v) – genügen. Es werden mögliche Polyeder dieses Typs mit bis zu 18 Flächen, die ausschließlich quadrilateral, pentagonal und hexagonal sein sollen, untersucht. Viele dieser Polyeder sind Zwillinge von verschiedenen, aus Dreiecken zusammengesetzten Koordinationspolyedern, die in früheren Arbeiten dieser Reihe untersucht wurden. Um die Winkeldeformation auf ein Mindestmaß zurückzuführen, werden im Falle von Wassernetzwerken in Klathrat- und Semiklathrathydraten Polyeder mit der maximalen Anzahl von pentagonalen Flächen bevorzugt, weil so ausreichend große Hohlräume zur Aufnahme des Gastmoleküls entstehen.

     

Endohedral nickel and palladium atoms in metal clusters: analogy to endohedral noble gas atoms in fullerenes in polyhedra with five-fold symmetry

Nickel and palladium atoms with their closed-shell d(10) electronic configurations are encapsulated in the icosahedral clusters [Ni@Ni(10)E(2)(CO)(18)](4-)(E = Sb, Bi, Sb[rightward arrow]Ni(CO)(3), CH(3)Sn and n-C(4)H(9)Sn) and the geometrically related pentagonal antiprismatic cluster Pd@Bi(10)(4+) found in Bi(14)PdBr(16). Such endohedral d(10) atoms in pentagonal antiprismatic clusters are donors of zero skeletal electrons and interact only weakly with the atoms in the surrounding polyhedron so that they may be regarded as analogous to endohedral noble gases in fullerenes such as He@C(60). On the other hand, endohedral nickel and palladium atoms in 10- and 11-vertex flattened deltahedral bare metal clusters of group 13 metals without five-fold symmetry, such as Ni@E(10)(10-) found in Na(10)NiE(10)(E = Ga, In) and Pd@Tl(11)(7-) found in A(8)Tl(11)Pd (A = Cs, Rb, K), interact significantly with the cluster atoms, particularly those at the flattened vertices of the deltahedron. The role of endohedral d(10) atoms Ni and Pd in polyhedra with five-fold symmetry as "pseudo-noble-gases" can be related to their positions at the "composite divide" of the "Metallurgists' Periodic Table" proposed by H. E. N. Stone on the basis of alloy systematics as well as the equivalence of the five d orbitals in polyhedra with five-fold symmetry.     

Distance-restricted matching extendability of fullerene graphs

A fullerene graph is a 3-connected cubic plane graph whose all faces are bounded by 5- or 6-cycles. In this paper, we show that a matching M of a fullerene graph can be extended to a perfect matching if the following hold: (i) three faces around each vertex in \(\{x,y:xy\in M\}\) are bounded by 6-cycles and (ii) the edges in M lie at distance at least 13 pairwise.     

Supraicosahedral polyhedra in carboranes and metallacarboranes: The role of local vertex environments in determining polyhedral topology and the anomaly of 13-vertex closo polyhedra

Metal-free carboranes having 13 vertices are anomalous since their closo polyhedra having the expected 28 skeletal electrons are not the usual deltahedra with exclusively triangular faces but instead polyhedra with one or two trapezoidal faces obtained by removal of one or more edges from the corresponding 13-vertex deltahedron. Removal of such edges converts degree 6 boron vertices in the 13-vertex deltahedron into more favorable degree 5 boron vertices while lowering the degree of nearby carbon vertices. Thus the anomaly of the 13-vertex carborane closo polyhedron can be rationalized by the preference of boron for degree 5 vertices. The 12-vertex tetracarbon carborane (CH₃)₄C₄B₈H₈ with a nido electron count of 28 skeletal electrons but with two quadrilateral faces has a solid state structure derived from a 13-vertex “closo” polyhedron with one quadrilateral face by removal of a degree 4 vertex to give the second quadrilateral face. However, the corresponding tetraethyl derivative (C₂H₅)₄C₄B₈H₈ has a different solid state structure derived from removal of a degree 6 vertex from an unusual 13-vertex deltahedron with three degree 6 vertices to give an open hexagonal face rather than two quadrilateral faces. In contrast to the 13-vertex closo polyhedra, the 14-vertex closo polyhedron is a true deltahedron, namely the D_6d bicapped hexagonal antiprism, which is found in a carborane derivative as well as in several dimetallacarboranes with the metal atoms always at the degree 6 vertices. However, the 15-vertex closo polyhedron, so far found only in the metallaborane 1,2-μ-(CH₂)₃C₂B₁₂H₁₂Ru(η⁶-p-cymene), is a non-deltahedron with one quadrilateral face.     

Spiral codes and Goldberg representations of icosahedral fullerenes and octahedral analogues

An icosahedral fullerene may be considered as a tessellation of the sphere specified by an ordered pair of integers, or as a tightly wound spiral of faces. Explicit analytical relations for interconverting the two representations are given, enabling the canonical spiral code to be constructed for an icosahedral fullerene of any size. Analogous relations hold for the octahedral square + hexagon polyhedra that have been mentioned as possible candidates for boron-nitride "fullerenes".     

Graphs whose vertices are graphs with bounded degree: Distance problems

By an f-graph we mean a graph having no vertex of degree greater than f. Let U(n,f) denote the graph whose vertex set is the set of unlabeled f-graphs of order n and such that the vertex corresponding to the graph G is adjacent to the vertex corresponding to the graph H if and only if H is obtainable from G by either the insertion or the deletion of a single edge. The distance between two graphs G and H of order n is defined as the least number of insertions and deletions of edges in G needed to obtain H. This is also the distance between two vertices in U(n,f). For simplicity, we also refer to the vertices in U(n,f) as the graphs in U(n,f). The graphs in U(n,f) are naturally grouped and ordered in levels by their number of edges. The distance nf/2 from the empty graph to an f-graph having a maximum number of edges is called the height of U(n,f). For f =2 and for f≥(n-1)/2, the diameter of U(n,f) is equal to the height. However, there are values of the parameters where the diameter exceeds the height. We present what is known about the following two problems: (1) What is the diameter of U(n,f) when 3≥f<(n-1)/2? (2) For fixed f, what is the least value of n such that the diameter of U(n,f) exceeds the height of U(n,f)?     

Nesting of fullerenes and Frank-Kasper polyhedra

The duality relationship between fullerenes and Frank-Kasper polyhedra suggests that these two families of polyhedra appear nested in solid state structures. Magic numbers, described by simple mathematical relationships, identify four families of fullerenes with tetrahedral and icosahedral symmetries.     

The Chirality of Icosahedral Fullerenes: a Comparison of the Tripling (leapfrog), Quadrupling (chamfering), and Septupling (capra) Transformations

Fullerene polyhedra of icosahedral symmetry have the midpoints of their 12 pentagonal faces at the vertices of a macroicosahedron and can be characterized by the patterns of their hexagonal faces on the (triangular) macrofaces of this macroicosahedron. The numbers of the vertices in fullerene polyhedra of icosahedral symmetry satisfy the Goldberg equation v=20(h ²+hk+k ²), where h and k are two integers and 0 <h≥ k≥ 0 and define a two-dimensional Goldberg vector G = (h, k). The known tripling (leapfrog), quadrupling (chamfering), and septupling (capra) transformations correspond to the Goldberg vectors (1, 1), (2, 0), and (2, 1), respectively. The tripling and quadrupling transformations applied to the regular dodecahedron generate achiral fullerene polyhedra with the full I _h point group. However, the septupling transformation destroys the reflection operations of the underlying icosahedron to generate chiral fullerene polyhedra having only the I icosahedral rotational point group. Generalization of the quadrupling transformation leads to the fundamental homologous series of achiral fullerene polyhedra having 20 n ² vertices and Goldberg vectors (n, 0). A related homologous series of likewise achiral fullerene polyhedra having 60 n ² vertices and Goldberg vectors (n, n) is obtained by applying the tripling transformation to regular dodecahedral C₂₀ to give truncated icosahedral C₆₀ followed by the generalized operations (as in the case of quadrupling) for obtaining homologous series of fullerenes. Generalization of the septupling (capra) transformation leads to a homologous series of chiral C_20m fullerenes with the I point group and Goldberg vectors G=(h, 1) where m=h ²+h+1.     

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.