Dihedral plane

The rotation axis of highest order is called the principal axis of rotation it is usually placed in the vertical direction and designated the z-axis of the molecule. Planes of reflection which are perpendicular to the principal axis are called horizontal planes (h). Planes of reflection which contain the principal axis are called vertical planes (v), or dihedral planes (d) if they bisect 2 twofold axes. 

The differences between horizontal, vertical and dihedral planes of symmetry... 

Dihedral plane ca Reflect the molecule through a plane which bisects two C2 axes... 

J 90° and 180° both produce equivalent configurations of the molecule. A rotation through 270° also produces an equivalent configuration, but that is equivalent to a rotation through 90° in the opposite direction and so does not indicate an extra type of axis of symmetry. The molecular plane contains two C2 axes (the superscript prime is to distinguish them from the C2 axis which is coincident with the C4 axis), both of which contain the iodine atom and two diametrically opposed chlorine atoms. The two C2 axes are contained respectively by the two vertical planes which also contain the major axis. There are, in addition, two C2 axes (coincident with the x and y axes) which are contained by the horizontal plane, are perpendicular to the major axis, and also bisect the respective Cl-I-Cl angles. The double-prime superscript serves to distinguish these axes from the C2 and C2 axes. The C2 axes are contained by two extra planes of symmetry (they are the xz and yz planes), which are termed dihedral planes since they bisect the two C2 axes. The dihedral planes contain the major axis, C4. 

Figure 2.12 (Left) The allene molecule which has a C2 axis (major) coincident with the CCC axis with two dihedral planes which contain that axis and both contain and bisect respectively the two CH2 groups...

Let us look next at the consequences of adding a vertical plane to the C axis. First we recall (Section 3.5) that the operations generated by C when n is odd will require that an entire set of n such vertical planes exist. All of these planes are properly called vertical planes and symbolized ov. When n is even, however, we have seen (Section 3.5) that only nil planes of the same type will exist as a direct consequence of the C axis. However, we have also shown (Section 3.7) that another set of nil vertical planes must exist as the various products C gv. These vertical planes in this second set are usually called dihedral planes, since they bisect the dihedral angles between members of the set of a s, and they are denoted ad. Obviously, it is completely arbitrary which set is considered vertical and which dihedral. In either case, n even or n odd, the set of operations generated by the C and by all of the cr s constitutes a complete set, and such a group is called C v. 

Our next and final task is to consider the consequences of adding to C and the n CVs a set of dihedral planes, vertical planes that bisect the angles between adjacent pairs of C2 axes. The groups generated by this combination of symmetry elements are denoted Dttd, The products of a Gd with the various C operations are all other operations. However, among... 

The subscript d denotes the presence of dihedral planes which bisect the angles between C 2 axes that are normal to the principal axis. If n is even,... 

Exercise 2.3-1 The projection diagram is given in Figure 2.17. The dihedral planes are rrx, crb, and rrc, where ax bisects the angle between b and c, ab bisects the angle between x and c, and rrc bisects the angle between x and b. 

Figure 2.17. Projection diagram for the point group D3d = D3(S)C, (see eq. (2.3.9)). For example, JC2b =

Determine correlation relations between the IRs of (a) Td and C3v, and (b) Oh and D3d. [Hints. Use character tables from Appendix A3. For (a), choose the C3 axis along [111] and select the three dihedral planes in Td that are vertical planes in C3v. For (b), choose one of the C3 axes (for example, that along [11 1]) and identify the three C2 axes normal to the C3 axis.]... 

Example 12.6-2 The classes of Td are E 4C3 3C2 6S4 6binary rotations are BB rotations. The six dihedral planes occur in three pairs of perpendicular improper BB rotations so both 3C2 and 6rrd are irregular classes. There are therefore Nv 5 vector representations and Ns = 3 spinor representations. 

Exercise 17.4-5 AtX, P(k) is C2v the character table for which is shown in Table 17.7. The basis functions shown are those for the IRs of C2v when the principal axis is along a. Table 17.8 contains the character table for C3V with basis functions for a choice of principal axis along [1 1 1], The easiest way to transform functions is to perform the substitutions shown by the Jones symbols in these two tables. The states X2 and S4 are antisymmetric with respect to Jones symbol for ah in Table 17.7). Note that vertical planes at A and, as Table 17.8 shows, A2 is antisymmetric with respect to from Table 17.2 we see that the bases for T j and T2 are antisymmetric with respect to fold axis at T is parallel to kz, and carrying out the permutation y —> z, z —> x, x y on the bases for A[ and A2 gives xy(x2 y2) and x2 y2 for the bases of Tj and T2, which are antisymmetric with respect to ab.)... 

Tm then follows from the character table for Oh and eq. (4.4.20). (The easiest way of determining (r, ) is to look at the figure representing the set of points to be permuted (in this case, a cube) and determine the number of points that are unshifted under one operation from each class. This number is the character for that class in the permutation representation Tm. For example, each of the dihedral planes contains four points which are invariant under reflections in that plane.)... 

On the basis of the idealised structure depicted, copper(I) benzoate belongs to the symmetry point group D2a. The symmetry elements present are three C2 axes, an S4 axis and two dihedral planes of symmetry o (these are referred to as dihedral because they are vertical planes of symmetry which... 

FIG. A5-6. The tetrahedron, showing some of its essential symmetry elements. All St and C3 axes are shown, but only one of the six dihedral planes [Pg.1322]

A plane which includes the main axis is a vertical plane cr, . A vertical plane which bisects two Cj axes which are oriented perpendicularly to the main axis is defined as a dihedral plane aj. 

As noted in the introduction, these techniques provide us with a powerful tool for directly evaluating the transferability of force constants and the corresponding similarity of intramolecular forces in different molecular environments. To demonstrate this, we present the results of a torsional analysis in propane" (Figure 14). In this case there are two types of 1-4 interactions H" H and H -C. The first question to be asked with respect to the 1-4 H—H interaction is, as stated above, whether it is the same as in ethane, i.e., transferable even within the homologous hydrocarbon family. Along with the similarity between the two molecules there is also the difference that in propane, unlike ethane, there is no plane of symmetry that coincides with the dihedral plane at Thh = 0- This point deserves some special attention. 

D d nCj 1 n dihedral planes parallel with A" (C i) and which bisect the angles between those n axes LA" (C )... 

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