Hexagonal system classes

The exclusion of nonplanar helicenic and hollow coronoid species from the class of benzenoids was maybe not fully justified from a chemist s point of view, but there were good and convincing mathematical reasons for this anyway we use the term benzenoid in the same sense as in the book [3]. On the other hand, we find that it serves no purpose to strictly distinguish between benzenoid hydrocarbons (chemical objects) and benzenoid systems (mathematical objects), since this distinction is always obvious from the context. We note in passing that what we call benzenoid system is the same as hexagonal system or hexagonal animal in the mathematical literature. 

Non-zao and independoit tensor elemmts as in the classes of the hexagonal system. 

The hexagonal system has a sixfold axis of symmetry along [0001] and two sets of three twofold rotational axes, one set along the (2110) family (only one shown in Figure 4.10f) and the other along the (1100) family (only one shown). There are two sets of three mirror planes, 2110 (only one shown) and [1100] (only one shown), as well as the (0002) mirror plane. The Schoenflies symbol for this class is implying the horizontal mirror plane in addition to the mirror symmetry implied in Dg. The full international symbol is 6/m2/m2/m abbreviated to 6/mmm, indicating a mirror plane normal to the sixfold axis and mirror planes parallel to the sixfold axis. 

In this paper, we review progress in the experimental detection and theoretical modeling of the normal modes of vibration of carbon nanotubes. Insofar as the theoretical calculations are concerned, a carbon nanotube is assumed to be an infinitely long cylinder with a mono-layer of hexagonally ordered carbon atoms in the tube wall. A carbon nanotube is, therefore, a one-dimensional system in which the cyclic boundary condition around the tube wall, as well as the periodic structure along the tube axis, determine the degeneracies and symmetry classes of the one-dimensional vibrational branches [1-3] and the electronic energy bands[4-12]. 

The term crystal structure in essence covers all of the descriptive information, such as the crystal system, the space lattice, the symmetry class, the space group and the lattice parameters pertaining to the crystal under reference. Most metals are found to have relatively simple crystal structures body centered cubic (bcc), face centered cubic (fee) and hexagonal close packed (eph) structures. The majority of the metals exhibit one of these three crystal structures at room temperature. However, some metals do exhibit more complex crystal structures. 

Fig. 35. Hexagonal and trigonal systems. (See also Figs. 24, 27 and 29.) a. Hexagonal- type unit cell. 6. Apatite, 3Ca3(P04)I.CaFB. Class 6/rm c. riydrocinchonine sulphate hydrate, (C19H940N9)9.H2S04. llHaO. Class 6m. d. Rhombohedral-type uriit cell, e. A habit of calcite, CaC()s. Class 3m. /. KBr03. Class 3m.

The information obtainable from the Laue symmetry is meagre it consists simply in the distinction, between crystal classes, and then only in the more symmetrical systems—cubic, tetragonal, hexagonal, and trigonal (see Table VI). But it is useful in cases in which morphological features do not give clear evidence on this point. 

It is noteworthy that the adducts of M with CA and TCA constitute members of a rare class of organic solids containing channels.12 Trimesic acid is a good example of a system forming hexagonal channels.13 The other example of such a solid is the adduct of TCA with 4,4 -bipyridyl.14 Experiments to explore whether the channels in the CA-M adduct can be filled by cations and other species are in progress. 

The opening lines of the chapter on enumeration from the monograph of Gutman and Cyvin [22] read By enumeration of benzenoid systems, the counting of all possible non-isomorphic members within a class of benzenoids is understood. Usually, but not always, the number of hexagons (h) is the leading parameter. Thus the enumeration for h — 1,2, 3, 4, etc. is to be executed. ... 

The distinction between catacondensed and pericondensed systems is applicable to all the classes of polyhexes treated above (Fig. 2). A catacondensed polyhex is defined by the absence of internal vertices. An internal vertex is a vertex shared by three hexagons. A pericondensed polyhex possesses at least one internal vertex. In terms of dualists a pericondensed polyhex reveals itself by the presence of at least one three-membered cycle (triangle). A dualist of a circulene has a cycle larger than a triangle. 

As far back as 1968, Balaban and Harary [13] were aware of the unique position of zethrene, which was placed in a class of its own under the pericondensed benzenoids with h = 6. It is an essentially disconnected benzenoid. However, these authors did not sort out the other two essentially disconnected benzenoids (annelated perylenes) with the same numbers of hexagons. Neither did they sort out perylene itself, which is the unique essentially disconnected benzenoid with h = 5. In the table we are referring to [13], the entry for the classified pericondensed system with h = 3 is misplaced. 

The title class refers to benzenoids which have the maximum A value, A = Amas, for a given number of hexagons (h). For A > 0 they are obvious non-Kekuleans. These systems are treated in detail by Brunvoll et al. [101]. 

In most of the enumerations of benzenoid systems the number of hexagons, h, has been used as a leading parameter. This is to say that the numbers of non-isomorphic benzenoids with a given h have been determined, and this set has occasionally been subdivided into different classes see e.g. a consolidated report [30] with supplements [31]. Also when special classes of benzenoids have been generated specifically the numbers of benzenoids were produced as a function of h. 

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