Polygonal Complexes and Graphs for Crystallographic Groups

Pellicer, Daniel; Schulte, Egon

doi:10.1007/978-1-4939-0781-6_16

“…Here we generalise Ringel's approach in two directions. If we do not insist that each symbol appears exactly twice, we may use such schemes to describe the combinatorial structure of more general polygonal complexes in the sense of Schulte et al [53,54,55,72]. On the other hand, if we allow symbols with a single appearance, we may describe chemical structures, such as benzenoids as graphs embedded in a surface with a boundary.…”

Section: Methodsmentioning

confidence: 99%

Catacondensed Chemical Hexagonal Complexes: A Natural Generalisation of Benzenoids

Anstöter¹,

Bašić²,

Fowler³

et al. 2021

Preprint

0

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Catacondensed benzenoids (those benzenoids having no carbon atom belonging to three hexagonal rings) form the simplest class of polycyclic aromatic hydrocarbons (PAH). They have a long history of study and are of wide chemical importance. In this paper, mathematical possibilities for natural extension of the notion of a catacondensed benzenoid are discussed, leading under plausible chemically and physically motivated restrictions to the notion of a catacondensed chemical hexagonal complex (CCHC). A general polygonal complex is a topological structure composed of polygons that are glued together along certain edges. A polygonal complex is flat if none of its edges belong to more than two polygons. A connected flat polygonal complex determines an orientable or nonorientable surface, possibly with boundary. A CCHC is then a connected flat polygonal complex all of whose polygons are hexagons and each of whose vertices belongs to at most two hexagonal faces. We prove that all CCHC are Kekulean and give formulas for counting the perfect matchings in a series of examples based on expansion of cubic graphs in which the edges are replaced by linear polyacenes of equal length. As a preliminary assessment of the likely stability of molecules with CCHC structure, all-electron quantum chemical calculations are applied to molecular structures based on several CCHC, using either linear or kinked unbranched catafused polyacenes as the expansion motif. The systems examined all have closed shells according to Hückel theory and all correspond to minima on the potential surface, thus passing the most basic test for plausibility as a chemical species. Preliminary indications are that relative energies of isomers are affected by the choice of the catafusene motif, with a preference shown for kinked over linear polyacenes, and for attachment by angular connection at the branching hexagons derived from the vertices of the underlying cubic structure. Avoidance of steric crowding of H atoms appears to be a significant factor in these preferences.

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“…Here we generalise Ringel's approach in two directions. If we do not insist that each symbol appears exactly twice, we may use such schemes to describe the combinatorial structure of more general polygonal complexes in the sense of Schulte et al [56][57][58]75] On the other hand, if we allow symbols with a single appearance, we may describe chemical structures, such as benzenoids as graphs embedded in a surface with a boundary.…”

Section: Methodsmentioning

confidence: 99%

“…[28] Example 1. A typical example of a polygonal complex in the sense of Schulte et al [56][57][58]75] is a 2-dimensional skeleton of the tesseract (the 4-dimensional cube, see Figure 1). This skeleton is composed of 16 vertices, 32 edges and 24 quadrilateral faces.…”

Section: Methodsmentioning

confidence: 99%

Catacondensed Chemical Hexagonal Complexes: A Natural Generalisation of Benzenoids

Anstöter

¹

,

Bašić

²

,

Fowler

³

et al. 2020

View full text Add to dashboard Cite

Catacondensed benzenoids (those benzenoids having no carbon atom belonging to three hexagonal rings) form the simplest class of polycyclic aromatic hydrocarbons (PAH). They have a long history of study and are of wide chemical importance. In this paper, mathematical possibilities for natural extension of the notion of a catacondensed benzenoid are discussed, leading under plausible chemically and physically motivated restrictions to the notion of a catacondensed chemical hexagonal complex (CCHC). A general polygonal complex is a topological structure composed of polygons that are glued together along certain edges. A polygonal complex is flat if none of its edges belong to more than two polygons. A connected flat polygonal complex determines an orientable or nonorientable surface, possibly with boundary. A CCHC is then a connected flat polygonal complex all of whose polygons are hexagons and each of whose vertices belongs to at most two hexagonal faces. We prove that all CCHC are Kekulean and give formulas for counting the perfect matchings in a series of examples based on expansion of cubic graphs in which the edges are replaced by linear polyacenes of equal length. As a preliminary assessment of the likely stability of molecules with CCHC structure, all-electron quantum chemical calculations are applied to molecular structures based on several CCHC, using either linear or kinked unbranched catafused polyacenes as the expansion motif. The systems examined all have closed shells according to Hückel theory and all correspond to minima on the potential surface, thus passing the most basic test for plausibility as a chemical species. Preliminary indications are that relative energies of isomers are affected by the choice of the catafusene motif, with a preference shown for kinked over linear polyacenes, and for attachment by angular connection at the branching hexagons derived from the vertices of the underlying cubic structure. Avoidance of steric crowding of H atoms appears to be a significant factor in these preferences.

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“…In [8] 14 semiregular periodic nets are identified which also have the (sphere packing) property that there are no intervertex distances less than the common edge length (in a maximum-symmetry embedding). See also the detailed discussion in Pellicer and Schulte [19] and Schulte [25].…”

Section: Regular Discrete Netsmentioning

confidence: 99%

String-Node Nets and Meshes

Power

¹

,

Schulze

²

2017

Discrete Comput Geom

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New classes of infinite bond-node structures are introduced, namely stringnode nets and meshes, a mesh being a string-node net for which the nodes are dense in the strings. Various construction schemes are given including the minimal extension of a (countable) line segment net by a countable scaling group. A linear mesh has strings that are straight lines and nodes given by the intersection points of these lines. Classes of meshes, such as the regular meshes in R 2 and R 3 , are defined and classified. Stringlength preserving motions are also determined for a number of fundamental examples and contrasting flexing and rigidity properties are obtained with respect to noncrossing motions in the space of smooth meshes.

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Polygonal Complexes and Graphs for Crystallographic Groups

Cited by 4 publications

References 31 publications

Catacondensed Chemical Hexagonal Complexes: A Natural Generalisation of Benzenoids

Catacondensed Chemical Hexagonal Complexes: A Natural Generalisation of Benzenoids

Catacondensed Chemical Hexagonal Complexes: A Natural Generalisation of Benzenoids

String-Node Nets and Meshes

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