Ammonia reflection operation

Every symmetry operation in the group has an inverse operation that is also a member of the group. In this context, the word inverse should not be confused with inversion. The mathematical inverse of an operation is its reciprocal, such that A A = A A = , where the symbol A represents the inverse of operation A. The identity element will always be its own inverse. Likewise, the inverse of any reflection operation will always be the original reflection. The inversion operation (/) is also its own inverse. The inverse of a C proper rotation (counterclockwise) will always be the symmetry operation that is equivalent to a C rotation in the opposite direction (clockwise). No two operations in the group can have the same inverse. The list of inverses for the symmetry operations in the ammonia symmetry group are as follows ... 

EXAMPLE 13-2 BH3, like ammonia, has three planes of reflection symmetry that contain the three-fold rotational axis. However, BH3, being planar, also has a reflection operation through the molecular plane. Is this reflection operation in the... 

There are five kinds of symmetry operations that one can utilize to move an object through a maximum number of indistinguishable configurations. One is the trivial identity operation E. Each of the other kinds of symmetry operation has an associated symmetry element in the object. For example, our ammonia model has three reflection operations, each of which has an associated reflection plane as its symmetry element. It also has two rotation operations and these are associated with a common rotation axis as symmetry element. The axis is said to be three-fold in this case because the associated rotations are each one-third of a complete cycle. In general, rotation by iTt/n radians is said to occur about an -fold axis. Another kind of operation—one we have encountered before is inversion, and it has a point of inversion as its symmetry element. Finally, there is an operation known as improper rotation. In this operation, we first rotate the object by some fraction of a cycle about an axis, and then reflect it through a plane perpendicular to the rotation axis. The axis is the symmetry element and is called an improper axis. 

For example, the ammonia molecule belongs to the point group Csv and so, in addition to the principal axis, there are three vertical mirror planes. The result of a reflection operation is shown in Figure 4.7 to see the how the transformation affects the basis set, one of the three degenerate mirror planes has been chosen as an example. The mirror contains the 3 vector, and so the superscript C has been added to the mirror plane symbol following the convention employed in earlier chapters. 

The group developed above to describe the symmetry of the ammonia molecule consisted only of the permutation operations. However, if the triangular pyramid corresponding to this structure is flattened, it becomes planer in me limit. The RF3 molecule shown in Fig. lb is an example of this symmetry. In this case it becomes possible to invert the coordinate perpendicular to the plane of the molecule, the z axis. Obviously, the operation of reflection in the (horizontal) plane of the molecule, <7h> is identical. It is easy, then, to identify the irreducible representations A and A" as symmetric or antisymmetric, respectively, under the coordinate inversion. The group composed of the identity and the inversion of the z axis is then <5 = s> whose character table is of the form of Table 7. 

When there is a rotation operation C and a reflection in a plane containing the axis, there must be altogether n such reflections in a set of n planes separated by angles of 2ir/2n, intersecting along the C axis (recall a(1) x C3 = ct(2) for the ammonia molecule). 

In many cases, the symmetry of a molecule provides a great deal of information about its quantum states and allowed transitions, even without explicit solution of the Schrodinger equation. A geometrical transformation which turns a molecule into an indistinguishable copy of itself is called a symmetry operation. A symmetry operation can consist of a rotation about an axis, a reflection in a plane, an inversion through a point, or some combination of these. In this chapter, we will consider in detail the symmetry groups of ammonia and water, Csv and C2v, respectively. 

Table C.l. Symmetry operations of the ammonia molecule (the reflections pertain to the mirror planes perpendicular to the triangle, Fig. C.2, and go through the centre of the triangle)...

EXAMPLE 13-1 BH3, like ammonia, has a three-fold symmetry axis. However, BH3 is planar. As a result, there is an extra symmetry operation—reflection through the molecular plane—that does not move any nuclei. Thus, both the identity operation and this reflection leave all nuclei unmoved. Is this an example of redundant operations ... 

In discussing the concept of class, it is unnecessary to postulate parallel universes, and the reader should not be disturbed by this pedagogical device. The people in the other universes are merely working with ammonia models that have been reflected or rotated with respect to the model orientation that we chose in Fig. 13-3. Operations in the same class are simply operations that become interchanged if our coordinate system is subjected to one of the symmetry operations of the group. 

Another example of degeneracy is found in the Csv example of the N—H bonds of ammonia that we used in Sections 4.5. 8. From Figure 4.7, the vertical reflection plane swaps the N—H bond basis vectors bi and 62 but leaves unchanged. Similar diagrams can be drawn for reflections through the equivalent planes and [Pg.98]

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