Ammonia molecule symmetry

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). 

As a second example of molecular symmetry, consider the ammonia molecule. It has three symmetrically equivalent hydrogen atoms, but it is not... 

Fig, 3 Cartesian displacement coordinates for the ammonia molecule, The Z(CO a is perpendicular to the plane of the paper (which is not a plane of symmetry)... 

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... 

It is assumed that the reader has previously learned, in undergraduate inorganic or physical chemistry classes, how symmetry arises in molecular shapes and structures and what symmetry elements are (e.g., planes, axes of rotation, centers of inversion, etc.). For the reader who feels, after reading this appendix, that additional background is needed, the texts by Cotton and EWK, as well as most physical chemistry texts can be consulted. We review and teach here only that material that is of direct application to symmetry analysis of molecular orbitals and vibrations and rotations of molecules. We use a specific example, the ammonia molecule, to introduce and illustrate the important aspects of point group symmetry. 

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... 

The pentaamminechlorocoball(lll) canon has idealized C4 symmetry, thal is, the random orientation of the hydrogen atoms resulting from free rotation of the ammonia molecules is often ignored for simplicity. 

FIG. A5-3. Hie ammonia molecule, showing its threefold symmetry axis Cj, and one of its three planes of symmetry passes through H, and N and bisects the H2—H line. 

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... 

Before we apply the formalism developed in Section 3 to the vibration—inversion-rotation spectra of ammonia, we shall discuss in this section certain group theoretical problems concerning the classification of the states of ammonia, the construction of the symmetry coordinates, the symmetry properties of the molecular parameters, and the GF matrix problem for the ammonia molecule. 

A simple example provides useful illustration of these observations. Consider the Csv point symmetry of the ammonia molecule. The group character table is... 

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.

There is partial localization of the valence density in methane. The condensation into four partially localized pairs of electrons arranged along four tetrahedral axes is a result of the combined effects of the ligand field and the Pauli exclusion principle described above. Most important is that this partial localization of the pair density is reflected in the properties of the VSCC of the carbon atom which undergoes a corresponding condensation into four local concentrations of electronic charge. These properties of the pair density are not just the result of the tetrahedral symmetry of the ligand field in methane because, as we will now see, the Fermi hole exhibits the same behaviour in the ammonia molecule. 

FIGURE 21.4 An ammonia molecule has a 3-fold axis of rotation and three mirror planes of symmetry. Each of the N —H bonds lies in a mirror plane. 

If the ammonia molecule is drawn as a pyramid with the nitrogen atom at the top (Fig. 21.4), then the only axis of rotational symmetry is a 3-fold axis passing downward through the N atom. Three mirror planes intersect at this 3-fold axis. 

The pyramidal ammonia molecule of C3v symmetry has an experimental interbond angle of 26 107° (Herzberg, 1956). Its geometry is depicted in Figure 2.22. It is convenient to work in terms of the angle y that each NH bond makes with the z symmetry axis, whose value is related to half the valence angle 6 by the relation ... 

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