Ammonia molecule symmetry operations

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). 

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. 

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... 

The ammonia molecule NH3 belongs, in its ground-state equilibrium geometry, to the C3V point group. Its symmetry operations consist of two C3 rotations, C3, C32... 

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 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... 

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... 

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... 

Now we are in a position to decide whether the integral of a product of functions and operators will vanish. For our examples, we will continue to use orbitals, operators, and coordinates from the ammonia molecule. Some of these quantities, segregated according to symmetry, are given in Table 13-26. 

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

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... 

In the example of H2O above we used the idea of a global axis system, X, Y, Z. This axis system is used to define the positions of the symmetry elements of the molecule and, once set, the global axis system is not moved by any operations that are carried out. This means that the symmetry elements should be considered immovable and symmetry operations only move the atoms in the molecule. This becomes especially important when molecules with more symmetry elements are considered. For example, ammonia (NH3) has a principal axis of order 3 and three vertical mirror planes, as shown in Figure 2.3. 

Symmetry operations for (a) the water molecule of point group C2V. (b) f be ammonia molecule of point group C, and (c) the PtH4 ion of point group D,o -... 

Operation with the symmetry operations for the ammonia molecule of point group Csv... 

Prior to interpreting the character table, it is necessary to explain the terms reducible and irreducible representations. We can illustrate these concepts using the NH3 molecule as an example. Ammonia belongs to the point group C3V and has six elements of symmetry. These are E (identity), two C3 axes (threefold axes of rotation) and three crv planes (vertical planes of symmetry) as shown in Fig. 1-22. If one performs operations corresponding to these symmetry elements on the three equivalent NH bonds, the results can be expressed mathematically by using 3x3 matrices. ... 

An example of the latter can be given by some internal coordinate operator, such as the inversion coordinate operator q for an ammonia-type molecule (cf. Fig. 3). The decomposition of the thermal molecular state into (strictly) symmetry-adapted eigenstates such as and T leads to expectation values of q that are equal to zero ... 

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