C2 axes, perpendicular

A point group contains a C axis, an S2 axis, n C2 axes perpendicular to C and at equal angles to each other and n planes bisecting the angles between the C2 axes. For n odd the point group contains a centre of inversion i. It also contains other elements which may be generated from these. 

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. 

Figure 2.13 The staggered form of the ethane molecule, looking down the C3 major axis, and showing the three C2 axes perpendicular to it...

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. 

In the first case, the equivalence of the two halves means that any line which is a perpendicular bisector of the molecular axis is a C2 symmetry axis there is an infinite number of such C2 axes. The equivalence of the two halves of the molecule also means that there is a plane of symmetry perpendicular to the molecular axis. Since there is an infinity of rotations about a unique, vertical axis, C, there is also an infinity of C2 axes perpendicular to C , and there is a horizontal plane of symmetry. The group is, very reasonably, designated D h. 

As noted, there is a C2 axis lying) along the C=C=C axis. There is no higher-order proper axis. There are two more C2 axes perpendicular to this one, as shown in Figure 3.11. Thus, tlhe group must be a D type, and we proceed to step 5. 

There is a C5 axis as seen in Figure 3.16. There are five C2 axes perpendicular to the C5 axis. 

There is a C5 axis, as in Parts A and B. There still remain five C2 axes perpendicular to C5. 

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. 

D2 A C2 axis perpendicular to the ring plane, and two C2 axes perpendicular to that axis... 

D2 A molecule with a C2 axis and two more C2 axes, perpendicular to that axis, has D2 symmetry. Example the tetrahydroxycyclobutadiene shown. Similarly, a molecule with a C3 axis (the principal axis) and three other perpendicular C2 axes is D3. 

D2d A molecule is D2d if it has a C2 axis and two perpendicular C2 axes (as for D2 above), plus two dihedral mirror planes these are mirror planes that bisect two C2 axes (in general, that bisect the C2 axes perpendicular to the principal axis). Example allene (propadiene). Staggered ethane is D3d (it has D3 symmetry elements plus three dihedral mirror planes. Dnd symmetry can be hard to spot. 

This group has symmetry element E, a principal Cn axis, n secondary C2 axes perpendicular to Cn, and a ah also perpendicular to C . The necessary consequences of such combination of elements are a S axis coincident with the Cn axis and a set of n ctv s containing the C2 axes. Also, when n is even, symmetry center i is necessarily present. The BrF molecule has point group symmetry D4h, as shown in Fig. 6.1.8. Examples of other molecules belonging to point groups >2h, D3h, Z>5h and D6h are given in Fig. 6.2.6. 

Next, consider the planar AB3 molecule (BF3) shown in Fig. 1-23. This molecule has no special symmetry. It has a C3 axis of rotation without a collinear S6 axis. It has three C2 axes perpendicular to the C3 axis, and therefore falls into the D classification. It has a Ch plane of symmetry perpendicular to the C3 axis and three [Pg.40]

The next example is the hexagonal planar molecule of type A6 or A6B6 (benzene) shown in Fig. 1-24. The molecule is not of a special symmetry. It has a center of symmetry and a C6 axis of symmetry. No S2 axis exists. Since six C2 axes perpendicular to the C6 axis are found, this molecule also falls into the D classification. Since it has a horizontal plane of symmetry perpendicular to the C6 axis, the molecule belongs to the D6h point group. 

One n-Fold Principal Rotation Axis C , n C2 Axes Perpendicular to Cn, n Dihedral Mirror Planes G4... 

For BF3, there is a C3 axis as shown in the structure and the molecule lies in a horizontal plane, designated as <7/,. There are also three vertical planes. Each B-F bond also defines a C2 axis where the 07, intersects each av. Therefore, the trigonal planar BF3 molecule has one C3 axis, one ah, three av, and three C2 axes. A molecule that has a C axis and n C2 axes perpendicular to it resulting from the horizontal plane intersecting the three vertical planes belongs to the D h point group. Therefore, D3h is the appropriate point group for BF3. Note that a structure such as... 

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. 

Finally, if no C2 axes perpendicular to C are present, then the lowest symmetry will be C , when a perpendicular symmetry plane is present, it will be Cnh, and when there are n coinciding symmetry planes, the point group will be Cm. 

Hie Dn Groups. When a vertical C axis is accompanied by a set of n C2 axes perpendicular to it, the group is Z> . Molecules of Dn symmetry are, in general, rare, but there is one very important type, namely, the trischelates (A5-XIX) of D3 symmetry. 

---