Vertical mirror plane

The individual ions of [P(CeH5)4]-[Th(N03)5(C3H9P0)2] showing the atomic numbering (viewed along c with the mirror plane vertical) [18]. 

It is the mentioned symmetry properties additional to the discrete translational symmetry that lead to a classification of the various possible point lattices by five Bravais lattices. Like the translations, these symmetry operations transform the lattice into itself They are rigid transforms, that is, the spacings between lattice points and the angles between lattice vectors are preserved. On the one hand, there are rotational axes normal to the lattice plane, whereby a twofold rotational axis is equivalent to inversion symmetry with respect to the lattice point through which the axis runs. On the other hand, there are the mirror lines (or reflection lines), which he within the lattice plane (for the three-dimensionaUy extended surface these hnes define mirror planes vertical to the surface). Both the rotational and mirror symmetry elements are point symmetry elements, as by their operation at... 

The conformational orientation adopted by the allenylidene group =C=C= CR R also merits comment. Thus, in half-sandwich complexes, mainly derived from [M(77 -CxHy)L2] (M = Fe, Ru, Os) metal fragments, a marked preference of the allenylidene group to adopt a vertical orientation in which the ipso carbon atoms of the R /R substituents are contained in the molecular plane (pseudo mirror plane bisecting the half-sandwich metal fragment) is observed. Preference for this conformation arises from the dominant metal ( xy-Cp back donation of the metal-HOMO into the allenylidene-LUMO n orbital (see Fig. 13). In contrast to this general trend, an unusual horizontal orientation of the allenylidene group was... 

Fig. 28. Numbering for peaks of G1 (cf. Fig. 7 b) and isomer distribution in peaks 1-5. Dashed horizontal lines symbolize mirror planes (enantiomers) vertical triple lines indicate equivalent identities...

Again, applying the principle of a summarized chirality as governing the sequence of elution of the 5 peaks of G1 leads to the distribution of components shown in Fig. 28 extending the information from Fig. 27 by addition of mirror planes (dashed lines), indicating equivalent identities (vertical triple lines) and by enclosing meso forms and racemates in boxes to symbolize the relative molar ratios of the various forms. The compounds within one box relate to those in the other boxes as diastereoisomers. 

In the trigonal point group 3, the axis of quantization is chosen along the three-fold axis, while the x and y axes may be selected anywhere in the plane perpendicular to the z axis. In the point group 3m, which occurs for many distorted octahedral complexes, there are also vertical mirror planes. The relation to the cubic axes is described by the transformation... 

In the symmetry-adapted formulation, the 43- term no longer occurs because the d-orbital density contains a vertical mirror plane even if such a plane is absent in the point group. This is illustrated as follows. Point groups without vertical mirror planes differ from those with vertical mirror planes by the occurrence of both dlm+ and d(m functions, with m being restricted to n, the order of the rotation axis. But the coordinate system can be rotated around the main symmetry axis such that P4 becomes zero. As proof, we write the (p dependence as... 

In the new coordinate system, the x axis coincides with one of the vertical mirror planes of the density, reducing the size of M 1 from 5 x 5 to 4 x 4. 

There are two types of Cs-symmetric metallocenes, XXX and XXXI (Table 8-5). Both types contain a mirror plane of symmetry—a horizontal plane in XXX, a vertical plane in XXXI. Both are achiral molecules, but they differ very significantly in stereoselectivity. XXX produces atactic polymer, while XXXI usually forms syndiotactic polymer. 

Mirror planes are denoted by the Greek letter a. Mirror planes in a molecule that has an axis of rotation are further classified according to the relative orientation of the axis and the mirror plane. Because it is usual to draw molecules with the principal rotation axis vertical, a mirror plane containing the principal rotation axis is called a vertical mirror and denoted. A mirror plane perpendicular to the rotation axis is called a horizontal mirror and denoted Oh-... 

A special type of vertical plane of symmetry occurs if there are two or more twofold axes perpendicular to the principal axis, and a mirror plane bisects the angle between two of these twofold axes, as in benzene. Such a plane is called... 

For example, the symmetry group of benzene contains six dyads perpendicular to the principal Cq axis, and six vertical mirror planes containing the principal axis. These mirror planes can be divided into two sets of three those passing through atoms and those passing between atoms. The product of Ce with any of the mirror reflections is another reflection in the same set. It is conventional to distinguish these two sets by calling members of one of them and members of the other cr but the choice of which are the (t and which are the Od is arbitrary. 

Problem 4-2. Classify all the mirror planes of cyclopropane, benzene, and the staggered, eclipsed, and gauche conformations of ethane as vertical, horizontal or diagonal mirrors. 

Fig. 10.5. A CO3 group lying on a mirror plane perpendicular to the page (shown by the vertical line). The two atoms 01 are related by the mirror plane. C and 02 lie on the mirror plane and have site symmetry m (Cs).

Figure 14.4. a) Orbital correlation diagram for n2s + 0J2S cheletropic addition of SO2 to an olefin. The symmetry element preserved is a vertical mirror plane, (b) Orbital correlation diagram for elimination of CO from a norbornadienone. Two vertical planes of symmetry are preserved. 

Figure 14.9. (a) Orbital correlation diagram for the direct insertion of carbene into an olefin to form cyclopropane. Symmetry classification is with respect to the vertical bisecting mirror plane. (b) State correlation diagram showing the intended correlations and the avoided crossing of states So and S2. 

It in addition to the C axis, there is a horizontal plane perpendicular to that axis) the molecule is said to have symmetry. An example of this relatively unimportant group is t/vms-dichloroethene (Fig. 3.13a). If there are r mirror planes containing the rotation axis, Cn, the planes are designated vertical planes, and the molecule has Cm symmetry. Many simple inorganic... 

FIGURE 1.16 Isomers of cyclopropane-1,2-dicarboxylic acid, (a) is-l,2-Dicarboxycyclopropane (trans isomers) (b) Z-l,2-Dicarboxycyclopropane isomer, (meso isomer) (c) Molecular models of E- (trans-) 1,2-dimethylcyclo-propane shown in the tube representation. Rotation of the right hand structure about a vertical axis through the center of the cyclopropane will superimpose the two methyl groups. The methylene of the rotated structure will be in the back, rather than the front, and not superimposed, (d) A mirror plane through the methylene and the back carbon-carbon bond is a plane of symmetry. The two carboxyl groups appear not to reflect each other in the model shown but they can rotate freely and will reflect each other on an instantaneous basis. 

Any set with the four properties (a)-(d) forms a group therefore the set G is a group for which the group elements are point symmetry operators. This point group is called C3v or 3m, because the pyramid has these symmetry elements a three-fold principal axis and a vertical mirror plane. (If there is one vertical plane then there must be three, because of the three-fold symmetry axis.)... 

Figure 2.8. Stereograms for the six D , T, and O point groups from combinations of proper rotations. Those for the three Cni, point groups from combinations of proper rotations and a horizontal mirror plane [in addition to 2 (m or C, ) and 6 (C3/,) shown in Figure 2.7], Those for the Cm, point groups from combinations of proper rotations and a vertical mirror plane.

Figure 3. Top Enantiomorphs (m = projection of mirror plane) of a chiral construction consisting of the product of a polar vector (oriented line segment in a vertical position) combined with an axial vector (oriented circle in a plane perpendicular to the polar vector). Bottom Left-handed (left) and right-handed (right) cylindrical helices derived from the correponding constructions above. The screw axes coincide with the polar vectors.

Figure 29. Enantiomorphous conformations of a [2]catenane that is a topological rubber glove.113 Unlabeled vertices represent carbon atoms, and hydrogen atoms are suppressed for clarity. The enantiomorphs are related by the mirror plane (m) and are interconverted by rotation of the 1,5-dioxynaphthalene moiety about the C-0 bonds (arrows).

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