Planar representation

Haworth formulas (Section 25 6) Planar representations of furanose and pyranose forms of carbohydrates... 

In the Fischer convention, the ermfigurations of other molecules are described by the descriptors d and L, which are assigned comparison with the reference molecule glyceraldehyde. In ertqrloying the Fischer convention, it is convenient to use projection formulas. These are planar representations defined in such a w as to convey three-dimensional structural information. The molecule is oriented with the major carbon chain aligned vertically in such a marmer that the most oxidized terminal carbon is at the top. The vertical bonds at each carbon are directed back, away fiom the viewer, and the horizontal bonds are directed toward the viewer. The D and L forms of glyceraldehyde are shown below with the equivalent Fischer projection formulas. 

Fig. 8. Planar representation of the (9M,0)-(5n,5n) knees, having a 36° bend angle produced by a heptagon-pentagon pair on the equatorial plane. The arrows show the dotted line of bonds where the knee N or N is connected to the corresponding straight tubules (a) knee N for n= 1 (b) stretched knee N , . for h = 1 and c=38 (c) general knees N and jV f.

Fig. 10. (a) Planar representation of a single rotational bond shift at the (9,0) to (9,0) connection of two (9,0)-(5,5) knees. This leads to a 27r/9 rotation out of the upper knee plane (b) single bond shift at the (5,5) to (5,5) connection of two (9,0)-(5,5) knees. This leads to a InjS rotation, out of the lower knee plane. The arrow indicates the location of the bond shift. 

Fig. 15. Growth of a (5ii,5n) tubule on the catalyst surface, illustrated by that of the (5,5) tubule. The central grey circle represents the catalyst particle with 10 coordination sites, and the small grey circles represent the other 10 catalyst coordination sites. The normal and bold lines represent single and double bonds, respectively, while coordinative bonds are represented by dotted lines [(a), (b) and (c)] (a ), (b ) and (c ) are the corresponding planar representations.

Growth mechanism of a (9n,0) tubule, over 24n coordination sites of the catalyst. The growth of a general (9 ,0) tubule on the catalyst surface is illustrated by that of the (9,0) tubule in Fig. 16 which shows the unsaturated end of a (9,0) tubule in a planar representation. At that end, the carbons bearing a vacant bond are coordinatively bonded to the catalyst (grey circles) or to a growing cis-polyacetylene chain (oblique bold lines in Fig. 16). Tlie vacant bonds of the six c/s-polyacetylene chains involved are taken to be coordinatively bonded to the catalyst [Fig. 16(b)]. These polyacetylene chains are continuously extruded from the catalyst particle where they are formed by polymerization of C2 units assisted by the catalyst coordination sites. Note that in order to reduce the number of representations of important steps, Fig. 16(b) includes nine new Cj units with respect to Fig. 16(a). 

Fig. 17. Growth mechanism of a (9/i,0)-(5n,5n) knee involving from 24 to 20 coordination sites of the catalyst, (a)-(g) Planar representation of the successive tubule growing steps (g ) Schlegel diagram representation of the whole knee with the Ci numbering corresponding to that of the individual steps (a)-(g).

In judging hindrance, it is useful to view the molecule in its three-dimensional, folded configuration. For instance, 17 can be reduced without undue difficulty, whereas 18 requires extreme conditions (Raney Ni, 2(WC, 200 atm) (7), a difference not expected from planar representations of the molecule. Saturation of A -octalin (17) may largely go through a prior isomerization to A -octa in, despite an unfavorable equilibrium 121). 

Figure 7-1. Planar representation of the "three-point attachment" of a substrate to the active site of an enzyme. Although atoms 1 and 4 are identical, once atoms 2 and 3 are bound to their complementary sites on the enzyme, only atom 1 can bind. Once bound to an enzyme, apparently identical atoms thus may be distinguishable, permitting a stereospecific chemical change.

Chemically, aldrin is l,2,3,4,10,10-hexachloro-l,4,4a,5,8,8a-hexahydro-l,4,5,8-di-methanonaphthalene and possesses, in planar representation, the structure ... 

The planar representation shown serves, in part, to obscure somewhat the complexity of the structural problem involved. Actually, four stereoisomers are represented by the simple planar representation shown it is not yet known with certainty which is the one corresponding to aldrin. Physically pure aldrin is a white crystalline solid with the properties set forth in Table I. 

Fig. 2.1-3. Hypothetical parent n/do-B Hn4 structures [n = 4-12 ni(do)A to n/-12] in perspective and planarized representation bold lines mark the apertures (Roman numbers). The perspective representations are fragments of the corresponding closo-...

Fig. 3.4-5. Perspective and planarized representation of the ni-7 structure of hypothetical B7H74 (derived from cl-8 minus B6, aperture in bold lines).

Fig. 3.4-7. Perspective and planarized representations of the ni-9 (derived from cl-10 minus BIO) structures of hypothetical BgHg4 (apertures in bold lines).

When the significant eigenvectors are more than 2 or 3, the information cannot be easily visualized by few eigenvector plots. In these cases the use of nonlinear mapping (NLM) can give a planar representation of the objects with greater fidelity to the structure of the information in the hyperspace of the variables... 

In the most general case, stresses on any of the six control-volume faces can potentially contribute to a force in any direction. In a cartesian coordinate system, only stresses in a certain direction can contribute to a force in that direction. In cylindrical coordinates and other noncartesian systems, the situation is more complex. As an example of this point, consider Fig. 2.15, which is a planar representation of the z face of the cylindrical differential element. Notice two important points that are revealed in this figure. One is that the the area of the 0 face varies from rdO on one side to (r + dr)dO on the other. Therefore, in computing net forces, the area s dependence on the r coordinate must be included. Specifically,... 

Fig. 2.10 Planar representation of a silicon crystal doped with P5+ giving rise to a Ps defect.

Figure 1.17. The four classical stereoisomers of a CTV-Cgo tris-adduct conjugate with an e.e.e addition pattern, and their planar representations. Each structure corresponds to a unique topological stereoisomer.

Figure 1.18. The four classical stereoisomers of a CTV-Cgo tris-adduct conjugate with a trans-3, trans-3,trans-3 addition pattern, and their planar representations. Topologically, (/, ft)-44 is equivalent (can be deformed without breaking any bond) to (M, k)-45, and (M, fC)-44 is equivalent to (/ , fC)-45. It appears that the four classical stereoisomers can be reduced to only two different topological stereoisomers.

Common planar representations of chemical structures account at best qualitatively for the basic properties of such encoded compounds. For example, one can expect a sour taste for a compound formula, which contains a carboxylic group. However, to deduce which one of two given compounds, possessing the same carboxylic group, will be more acidic and how many times more, is a matter of a more or less scientific guess. Three-dimensional representations readily provided by modern molecular modelling software are not those helpful in this respect. 

There have been numerous other earlier attempts to extract more detailed representations of the pair distribution function from computer simulations. These include calculations of radial functions along vectors (directions) away from the molecule [15], the accumulation of two-dimensional slices of the local density around a molecule [16], and the projection of the full three-dimensional structure onto a two-dimensional (planar) representation [17,18]. These approaches have had some success in providing more detailed structural information and often appeared to represent necessary compromises required by limiting (at that time) computational resources. 

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