Ammonia point group

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. 

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 ... 

The ammonia molecule is a trigonal pyramid, belonging to the C3v point group. The 2s and 2p orbitals of the nitrogen atom and the Is orbital group combinations of the three hydrogen atoms transform, with respect to the C3v point group, as indicated in Table 6.1. 

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. ... 

Consider now the construction of the A i symmetry group orbital of the hydrogen s atomic orbitals in ammonia as an example of the application of the projection operator. (The various kinds of orbitals will be discussed in detail in Chapter 6.) The projection operator for the A irreducible representation in the C3v point group is... 

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... 

Since we know all the irreducible representations of a point group, we can tell a lot about the possible solutions to eqn (8-2.16), without actually solving it. For example, for ammonia which belongs to the point group, we know from Table 7-7.1 that its electronic and nuclear wavefunctions must fall into the following three categories ... 

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.

Identify the symmetry point groups of formaldehyde [C2v, ammonia [C3J, phenol [CJ, gyloxal C2h and allene [D2hl-... 

An important point put forward by the study of ammonia is the choice of a well-balanced basis set. For the calculation of effects such as barriers it is important to use a basis as saturated" as possible — or, if this cannot be achieved, at least equally well adapted to the different conformations to be considered. This requirement may be a cause of trouble, especially if the different geometrical configurations belong to different symmetry point groups. By symmetry reasons, the mixing of some atomic orbitals in a given molecular orbital may be forbidden for certain conformations and not for others. 

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