Horizontal mirror plane

C2h A molecule with a C2 axis and a mirror plane horizontal to this axis is C2h (a C2h object will also perforce have an inversion center). Example ( )-l,2-difluor-oethene. Similarly B(OH)3 is C3, . 

D2h A molecule with a C2 axis and two perpendicular C2 axes (as for D2 above), plus a mirror plane is D2h. Examples ethene, cyclobutadiene. Similarly, a C3 axis (the principal axis), three perpendicular C2 axes and a mirror plane horizontal to the principal axis confer D3h symmetry, as in the cyclopropenyl cation. Similarly, benzene is D6h, and F2 is D. ... 

Figure 1.8. From left to right horizontal two-fold rotation axis (top) and its alternative symbol (bottom), diagonal three-fold inversion axis inclined to the plane of the projection, horizontal four-fold rotation axis, horizontal, and diagonal mirror planes. Horizontal or vertical lines are commonly used to indicate axes located in the plane of the projection, and diagonal lines are used to indicate axes, which form an angle other than the right or zero angles with the plane of the projection.

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...

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-... 

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. 

A stereogram is to be interpreted as a schematic diagram of an object viewed down the principal axis. Horizontal mirror planes and equatorial rotation axes lie in the plane of the diagram. Points in space are designated by x if they lie above the plane of the diagram and by o if they lie below it. 

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... 

Linear molecules with a center of symmetry, such as BeF, and all of the X, molecules (Fig. 3.I4e-g), possess a horizontal mirror plane and an infinite number of C2 axes perpendicular to the principal axis and thus have Oxh symmetry. 

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 7. A78 isomers. The unfolded surface lattice nets at the left are drawn with boundaries along the vectors between nearest neighbour V5s which are marked by the black circular sectors, whereas the boundaries for the nets at the right are along the edges of deltahedral facets. The projected views of the fullerene polyhedra and deltahedra duals in the centre column are all oriented with a corresponding two-fold axis horizontal. For the four mirror-symmetric isomers, there is one mirror plane in the plane of projection and an orthogonal horizontal one. Marking the symmetry elements for each isomer on the deltahedral surface lattice net defines the asymmetric unit.

C2h c2 axis (perpendicular to the c3h C3 axis and horizontal molecular plane) and a horizontal mirror plane... 

The number of classes, equal to 5, is derived considering all the possible conditions for chirotopicity of the catalytic sites corresponding to L. If they are not chirotopic, i.e. if they are achirotopic (e.g. they are bisected by a horizontal mirror plane), there are two possibilities only the two sites are equal (class I catalysts) or different from each other (class II catalysts). If, on the contrary, they are chirotopic, three possibilities exist the two catalytic sites are homotopic (equal) - related by a twofold symmetry axis (class III catalysts), enan-tiotopic - related by a vertical mirror plane (class IV catalysts) or diastereotopic (different from each other) - no symmetry element is present (class V catalysts). As a consequence, only five classes of metallocene catalysts may exist if interconversion among stereoisomers is not taken into account [122]. 

One w-Fold Princinal Rotation Axis C ( Rotation bv 360In Degrees ). Plus Horizontal Mirror Plane... 

Let us take now a more complicated basis, and consider all the nuclear coordinates of HNNH shown in Figure 4-8a. These are the so-called Cartesian displacement vectors and will be discussed in Chapter 5 on molecular vibrations. Let us find the matrix representation of the crh operation (see Figure 4-8b). The horizontal mirror plane leaves all x and y coordinates unchanged while all z coordinates will go into their negative selves. In matrix notation this is expressed in the following way ... 

For alternant conjugated systems (those having no odd-membered rings), the Id values are reflected across a mirror plane placed horizontally, halfway between ip2 and 3, and also across a mirror plane placed vertically, halfway between C-2 and C-3. It is only necessary therefore to calculate four of the 16 numbers in Fig. 1.31, and deduce the rest from the symmetry. 

Figure 4. Various representations of spin-coupled orbital <)>, for benzene. Left contours in the horizontal plane 1 bohr above the molecular plane. Centre contours in a vertical mirror plane. Right a representative isosurface (3-D contour).

E = identity operation, C = n-fold proper rotation axis, S = n-fold improper rotation axis, <7h = horizontal mirror plane, <7v = vertical minor plane, <7d = dihedral minor plane, i = inversion center. 

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