Ammonia symmetry

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 ... 

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)... 

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. 

It will be shown that, upon interaction with water or ammonia, the T -like symmetry of the Ti(IV) centers in TS-1 is strongly distorted, as testified by UV-Vis, XANES, resonant Raman spectroscopies [45,48,52,58,64,83,84], and by ab initio calculations [52,64,74-76,88]. As in Sect. 3 for the dehydrated catalyst, the discussion follows the different techniques used to investigate the interaction. 

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... 

In principle, geometry optimization carried out in the absence of symmetry, i.e., in Ci symmetry, must result in a local minimum. On the other hand, imposition of symmetry may result in a geometry which is not a local minimum. For example, optimization of ammonia constrained to a planar trigonal geometry (Dbi, symmetry) will result in a geometry which is an energy maximum in one dimension. Indeed,... 

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