Ammonia symmetry operations

Fig. 9.1 The ammonia symmetry operations ab and SaC3. The nitrogen atom lies above the xy plane.

The ammonia moleeule NH3 belongs, in its ground-state equilibrium geometry, to the C3v point group. Its symmetry operations eonsist of two C3 rotations, C3, 3 ... 

To illustrate the application of Eq. (37), consider the ammonia molecule with the system of 12 Cartesian displacement coordinates given by Eq. (19) as the basis. The reducible representation for the identity operation then corresponds to the unit matrix of order 12, whose character is obviously equal to 12. The symmetry operation A = Cj of Eq. (18) is represented by the matrix of Eq. (20) whore character is equal to zero. Hie same result is of course obtained for die operation , as it belongs to the same class. For the class 3av the character is equal to two, as exemplified by the matrices given by Eqs. (21) and (22) for the operations C and Z), respectively. The representation of the operation F is analogous to D (problem 12). 

To provide further illustrations of the use of symmetry elements and operations, the ammonia molecule, NH3, will be considered (Figure 5.6). Figure 5.6 shows that the NH3 molecule has a C3 axis through the nitrogen atom and three mirror planes containing that C3 axis. The identity operation, E, and the C32 operation complete the list of symmetry operations for the NH3 molecule. It should be apparent that... 

Figure 6-21. The C3v symmetry operations applied to the three hydrogen atom l v orbitals of ammonia as basis function.

For another simple, but more general, example of a symmetry group, let us recall our earlier examination of the ammonia molecule. We were able to discover six and only six symmetry operations that could be performed on this molecule. If this is indeed a complete list, they should constitute a group. The easiest way to see if they do is to attempt to write a multiplication table. This will contain 36 products, some of which we already know how to write. Thus we know the result of all multiplications involving E, and we know that... 

The simpliest and most important molecule with a low barrier to inversion is ammonia, NH3. In its ground electronic state, NH3 has a pyramidal equilibrium configuration with the geometrical symmetry described by the point group C3V (Fig. 1). Configuration B which is obtained from A by the symmetry operation E is separated from A by an inversion barrier of about 2000 cm . A large amplitude... 

Vfr- Ammonia is an example of a molecule belonging to this point group and it has six symmetry operations which obey the group table introduced in Chapter 3 (Table 3-4.1). If we set up base vectors... 

FIGURE 4-14 Symmetry Operations for Ammonia. (Top view) NH3 is of point group with the symmetry operations , C3, 3, cr, ... 

Figure 2.1 The actions of symmetry operations of the point group, C3V, in the structure of the ammonia molecule giving rise to the permutation representation based on the matrices in the second column of the figure. The principal rotational axis, C3, is normal to the plane of the paper.

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. 

Before applying these symmetry principles, we will discuss a slightly more general example to illustrate what occurs when one (or more) of the irreducible representations is two-dimensional (or three-dimensional). The simplest symmetry group that has a two-dimensional irreducible representation is C3. This group contains the symmetry elements appropriate to the ammonia molecule (Fig. 23.26). The projection of the atoms on the xy plane is shown in Fig. 23.26(b). The symmetry operations are ... 

Let us make a list of all the symmetry operations allowed for the ammonia molecule. To this end, let us label the vortices of the triangle by a, b, c and locate it in such a way as to coincide the triangle center with the origin of the coordinate system, and the y axis indicated the vortex a. 

The parallelepiped in Figure 2 is the unit cell of the ammonia crystal phase I. Thus, the ammonia crystal can be regarded as the combination of a pattern of four ammonia molecules (16 atoms) in the unit cell with all possible translations in a cubic primitive lattice. Considerations about crystalline symmetry lead to the conclusion that ammonia in phase I crystallizes according to space group P2i3. Letter P in the symbol stands for primitive lattice, and the other symbols denote the main symmetry operations. The last element in the symbol, 3, indicates the presence of a three-fold axis not aligned with the principal rotation axis (if it was, it would follow letter P), which further indicates that the lattice is cubic. A cubic unit cell is completely specified by just one... 

When a complete set of all the possible point symmetry operations for a molecule has been identified, the resulting list will form the basis for a mathematical group. Consider the ammonia molecule, NH3, shown in Figure 8.7 (the labels on the H atoms are imaginary and are merely present for bookkeeping purposes). The symmetry operations present in NH3 are the following C3, C3, (- 3 ), (Ty, (7y2, and (Tyy... 

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 ... 

The set of symmetry operations of ammonia is said to form a group, G. This is a fundamental mathematical structure consisting of a set of elements and a multiplication rule with the following characteristics ... 

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)...

If we pick some object or shape of finite size and construct a group from symmetry operations for that shape, we have a symmetry point group for that object. We now illustrate how this is done for a ball-and-stick model of the ammonia molecule in its equilibrium nuclear configuration. It is not difficult to find operations that do no more than interchange identical hydrogen nuclei. There are a number of possible rotations about the z axis (Fig. 13-3). One could rotate by 120°, 240°, 360°, 480°, etc., either... 

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. 

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. 

Once one has recognized the set of symmetry elements associated with a given object, it is a straightforward matter to hst the symmetry operations associated with the set. Simplest are the operations associated with elements a and i, because each such element gives rise to only one operation. Proper and improper axes are somewhat more complicated. Let us return to our ammonia molecule for illustration of this. There we had a threefold axis C3 and we noted that we could rotate by 27t/3 (C ) to get one configuration, and 4 r/3 [C = (C ) ] to get another. Alternatively we could choose... 

FIGURE13.10 C3 is not its own inverse. The ammonia molecule needs another symmetry operation to rotate to its original position. 

FIGURE 13.11 The sixsymmetry operations of ammonia, NH3. One N-H bond lies in each plane of symmetry. Collectively, these six symmetry operations compose the C y point group. 

As example of work, it will be considered the ammonia molecule. Figure 2.8, where the whole set of symmetry operations contain the objects of so-called group... 

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