Molecular graphs planarization

Figure 6.19 Molecular graph (schematically) calculated for the planar biphenyl molecule at the B3LYP/6-31G level.

Figure 14. Top row Molecular structures of (a) twistane, (b) tritwistane, (c) [6]chochin. Bottom row planar images of the corresponding molecular graphs.71 Reprinted with permission from K. Mislow, Croat. Chem. Acta 1996, 69, 485. Copyright 1996, Croatian Chemical Society.

In short, the absence of rigidly achiral presentations, planar or otherwise, is a necessary condition for the topological chirality of a molecular graph. Yet, as we shall later see, it is still not a sufficient one. 

Topological stereoisomers have identical bond connectivity, but they cannot be interconverted one into another by continuous deformation [51]. They are homeomorphic, not isotopic. Their molecular graphs are non-planar, either intrinsically or extrinsically. 

Another topologically chiral K5 molecule is the chiral ferrocenophane ketone derivative 51 (Fig. 8), which is an intermediate in the synthesis of tetrabridged ferrocenophanes [79]. It is important to notice that in order to obtain a non-planar molecular graph, at least four of the ten Fe —C bonds must be taken into account. The K5 subgraph (52) also contains an oriented edge ( —CH2CH2CH2CO-fragment) and a coloured vertex (Fe atom). 

Fio. 3.7. Planar projections of molecular graphs of hydrocarbon molecules generated from theoretical charge distributions. Bond critical points are denoted by black dots. Structures 1 to 4 are normal hydrocarbons from methane to butane, 5 is isobutane, 6 is pentane, 7 is neopentane, and 8 is hexane. The remaining structures are identified in Table 3.2. The structures depicted in these diagrams are determined entirely by information contained in the electronic charge density. 

A natural extension of the work described in Sect. 2.4 was to prepare a Cu(l)-complexed [2]catenane with macrocycles incorporating the Zn and Au porphyrins. [2]catenanes are topologically non-trivial molecules (non-planar molecular graph) in which two rings are interlocked but not linked [23,24]. Therefore, differentiating the rings with Zn- and Au-porphyrins enables us to study photoinduced electron transfer in mechanical bond systems and also to have information on the conformation of the system. 

In Figure 85 we show molecular graphs of a selection of anti-aromatic structures which also have odd-member rings. All the structures are assumed to be planar, and all the CC bonds are assumed to be of similar length. The RE and the degree of antiaromaticity, A, are listed in Table 35. For the last four structures shown in Figure 85, it is difficult to... 

The frequency of path-distance pairs for a single molecule can be viewed by means of three-dimensional plots in which the frequency distribution (z axis) is reported versus distances [x axis) and the path-distance (y axis), see Fig. 5.2. Molecular shape peculiarities are condensed in the frequency distribution graph in which low path-distance values (path = distance), represented by points close to the distance axis of the plot in Fig. 5.2, characterize more planar surfaces, whereas high path-distance values (path > distance) characterize more wrinkled surfaces. 

An important contribution to the enumeration of spanning trees of molecular planar graphs was the theorem by Gutman et al 314... 

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