Ammonia symmetry elements

It is assumed that the reader has previously learned, in undergraduate inorganie or physieal ehemistry elasses, how symmetry arises in moleeular shapes and struetures and what symmetry elements are (e.g., planes, axes of rotation, eenters of inversion, ete.). For the reader who feels, after reading this appendix, that additional baekground is needed, the texts by Cotton and EWK, as well as most physieal ehemistry texts ean be eonsulted. We review and teaeh here only that material that is of direet applieation to symmetry analysis of moleeular orbitals and vibrations and rotations of moleeules. We use a speeifie example, the ammonia moleeule, to introduee and illustrate the important aspeets of point group symmetry. 

Many transition states of chemical reactions contain symmetry elements not present in the reactants and products. For example, in the umbrella inversion of ammonia, a plane of symmetry exists only in the transition state. 

H2O2 (hydrogen peroxide) chirality, 80 symmetry elements, 82 NF3 (nitrogen trifluoride) dipole moment, 98, 99 NFl3 (ammonia)... 

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... 

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. ... 

The symmetry elements of Fl20 are shown in Fig. 13.4. The molecule belongs to the symmetry group C2v with a twofold axis of rotation C2 and two vertical mirror planes cry and cr. In contrast to the case of ammonia, these two reflections are in different classes, since one lies in the plane of the molecule while the other bisects the plane. 

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 ... 

Moreover, all the symmetry elements intersect at a single point (e.g. the oxygen atom in dimethylether (6-1), the nitrogen atom in ammonia (6-7), the middle of the carbon-carbon bond in ethylene (6-5)). The expression paint-group symmetry is therefore used. For each molecule, there is a corresponding point group that completely characterizes its symmetry properties. 

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 ... 

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... 

To emphasize the difference between elements and operations further, consider the structure of ammonia shown in Figure 1.7. A C3 axis is present the symmetry element is a line running through the nitrogen atom and the centre of the triangle formed by the three hydrogen atoms. 

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. 

Figure 2.3 The symmetry elements for ammonia (NH ) (a) viewed with the principal axis in the plane of the page (b) viewed along the principal axis with the mirror planes labelled following the text.

Mossbauer spectroscopy is a specialist characterization tool in catalysis. Nevertheless, it has yielded essential information on a number of important catalysts, such as the iron catalyst for ammonia and Fischer-Tropsch synthesis, as well as the CoMoS hydrotreating catalyst. Mossbauer spectroscopy provides the oxidation state, the internal magnetic field, and the lattice symmetry of a limited number of elements such as iron, cobalt, tin, iridium, ruthenium, antimony, platinum and gold, and can be applied in situ. 

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