Symmetry elements horizontal

Now examine the symmetry elements for the cubic lattice. It is easy to seethat the number of rotation elements, plus horizontal and vertical symmetry elements is quite high. This is the reason why the Cubic Structure is placed at the top of 2.2.3. E)ven though the lattice points of 2.2.1. are deceptively simple for the cubic structure, the symmetry elements are not... 

Note that each of the C2 axes not only is coincident with a B-F bond but also is the line of intersection of the horizontal plane with one of the vertical planes. It is generally true that the intersection of a vertical plane of symmetry with a horizontal plane generates a C2 axis. The list of symmetry elements that we have found for the BF3 molecule includes one C3 axis, three vertical planes (horizontal plane (ah). A molecule possessing these symmetry elements, such as BF3, S03, C03 , and N03 , is said to have l)ih symmetry. In the cases of H20, C1F3, H2CO, and NH3, the symmetry elements included only a C axis and n vertical planes. These molecules belong to the general symmetry type known as C . Molecules that have a Cn axis and also have n C2 axes perpendicular to the C axis are known as Dn molecules. 

These symmetries are then compared with figure 4c which gives the live possible DF symmetries as well as the symmetry elements present in the specimen. The second case is particularly interesting. It corresponds to a two-fold rotation inside a disk (named 1r) due to a horizontal mirror present in the specimen. 

We have now reached a point of departure in the process of adding further symmetry elements to a C axis. We shall consider (1) the addition of different kinds of symmetry planes to the C axis only, and (2) the addition of symmetry planes to a set of elements consisting of the C axis and the n C2 axes perpendicular to it. In the course of this development it will be useful to have some symbols for several kinds of symmetry planes. In defining such symbols we shall consider the direction of the C axis, which we call the principal axis or reference axis, to be vertical. Hence, a symmetry plane perpendicular to this axis will be called,a horizontal plane and denoted ah. Planes that include the C axis are generally called vertical planes, but there are actually two different types. In some molecules all vertical planes are equivalent and are symbolized av. In others there may be two different sets of vertical planes (as in PtClJ" cf. page 32), in which case those of one set will be called ov and those of the other set crrf, the d standing for dihedral. It will be best to discuss these differences more fully as we meet them. 

There is an S4 axis. There are no additional independent symmetry elements the set of methyl groups destroys all the vertical planes and horizontal C2 axes that exist in C8H8 itself. The group is therefore S4. 

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.

This group has symmetry element E, a rotational axis C , and a horizontal plane <7h perpendicular to C . Note that S also exists as a consequence of the elements already present (C and of,). Also, when n is even, the presence of i is again a necessary consequence. Examples include trans-N2F2 (C 21, Fig. 6.1.6), all-trans-1,5,9-cyclododecatricne (C 3h Fig. 6.1.7), and boric acid B(OH)3 (C h Fig. 13.5.1). 

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]. 

Both diastereomers of [CrCl2(NH3)4]+ are shown below, these are usually differentiated by the stereodescriptors cis and trans. The trans isomer belongs to the symmetry point group D4h. The symmetry elements are the main fourfold axis of symmetry C4, a horizontal plane of symmetry ah (perpendicular to the C4 axis), four C2 axes also perpendicular to the C4 axis and four planes of symmetry av the intersection of which is the main axis of symmetry. The cis isomer belongs to the symmetry point group C2v. The associated symmetry elements are a C2 axis and two vertical planes of symmetry av intersecting at the C2 axis. Verify this using the flow chart in the appendix. 

The crucial question now is whether there exists a set of n C2 axes perpendicular to the Cn axis. If so, we proceed to step 5. If not, the molecule belongs to one of the groups Cn, Cnv, and C If there are no symmetry elements except the Cn axis, the group is Cn. If there are n vertical planes, the group is C. If there is a horizontal plane, the group is Cnh. 

Figure 9-22. The crystal structure of the rock salt after Shubnikov and Koptsik [32], (a) A unit cell (b) Projection of the structure along the edges of the unit cell onto a horizontal plane (c) Projection of some symmetry elements of the Fm3m space group onto the same plane. The vertical screw axes 2 and 42 are marked by their respective symbols. Used with permission.

The directions of transition moments in every chromophore are dictated by molecular symmetry. For the cis tautomers of porphycene ( 2 point group), only three mutually orthogonal transition moment directions are allowed. On the other hand, the trans form is of 2 symmetry and, therefore, any direction in the molecular plane is possible, as well as the direction perpendicular to the plane. The determination of transition moment directions in such low symmetry molecules is not an easy task. However, in the case of narcissistic type of reactions exemplified by trans-trans conversion in 1, one can take advantage of an additional symmetry element introduced by the tautomerization process. Double hydrogen transfer converts the molecule into its image, with the horizontal and vertical mirror symmetry planes perpendicular to the molecular plane (Fig. 8.13). Thus, tautomerization results in the rotation of each in-plane transition moment direction. The angle of rotation is twice the value of the angle formed by a particular transition moment with the horizontal (or vertical) in-plane axis. It can be shown [80] that, for a fast excited state process, which results in equal population of both trans tautomers, the measured fluorescence anisotropy r will be expressed by the formula ... 

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