Normalized irreducible character

The normalized irreducible characters Xy express the way in which these specific features propagate when N and S change. Therefore they axe called propagation coefficients [65]. 

We then discover an extremely important fact each normal coordinate belongs to one of the irreducible representations of the point group of the molecule concerned and is a part of a basis which can be used to produce that representation. Because of their relationship with the normal coordinates, the vibrational wavefunctions associated with the fundamental vibrational energy levels also behave in the same way. We are therefore able to classify both the normal coordinates and fundamental vibrational wavefunctions according to their symmetry species and to predict from the character tables the degeneracies and symmetry types which can, in principle, exist. 

For certain point groups, we have one-dimensional (irreducible) representations with complex characters. Suppose that the normal coordinate Qx transforms according to the one-dimensional (irreducible) representation T some of whose characters are complex numbers. We then have for any symmetry operation R... 

Consider next the water molecule. As we have seen, it has a dipole moment, so we expect at least one IR-active mode. We have also seen that it has CIt, symmetry, and we may use this fact to help sort out the vibrational modes. Each normal mode of iibratbn wiff form a basis for an irreducible representation of the point group of the molecule.13 A vibration will be infrared active if its normal mode belongs to one of the irreducible representation corresponding to the x, y and z vectors. The C2 character table lists four irreducible representations A, Ait Bx, and B2. If we examine the three normal vibrational modes for HzO, we see that both the symmetrical stretch and the bending mode are symmetrical not only with respect to tbe C2 axis, but also with respect to the mirror planes (Fig. 3.21). They therefore have A, symmetry and since z transforms as A, they are fR active. The third mode is not symmetrical with respect to the C2 axis, nor is it symmetrical with respect to the ojxz) plane, so it has B2 symmetry. Because y transforms as Bt, this mode is also (R active. The three vibrations absorb at 3652 cm-1, 1545 cm-1, and 3756 cm-, respectively. 

Here x(C4), for instance, means the character for the covering operation consisting of rotation about one of the fourfold or principal cubic axes (normals to cube faces) by 2a-/4. Any rotation about such an axis leaves two atoms invariant, and hence x(Cs) = x(C4) = 2- On the other hand, x((Y)=x(Q)—0 since no atoms are left invariant under rotations about the twofold or secondary cubic axes (surface diagonals) or about the threefold axes (body diagonals). Inversion in the center of symmetry is denoted by I. By using tables of characters for the group Oh, one finds that the irreducible representations contained in the character scheme (2) are, in MuIIiken s notation,4... 

For a doubly degenerate normal mode, both components must be used together as the basis of a two-dimensional irreducible representation. For example, the operations C2 and ctv on the two normal vibrations that constitute the i>6 mode lead to the character (sum of the diagonal elements of the corresponding 2x2 matrix) of -2 and 0, respectively, as illustrated below. Working through the remaining symmetry operations, the symmetry species of can be identified as Eu. 

These 12 irreducible representations account for the 12 degrees of motional freedom of HNNH. Subtracting the irreducible representations corresponding to the translation and rotation of the molecule (see C2h character table, Table 5-2) leaves us the symmetry species of the normal modes of vibration ... 

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

The deficiency is rectified in Figure 3.24 by taking suitable sums and differences of the motions of Figure 3.23. For molecular motions these transformations correspond to the orthog-onalization of the like symmetry linear combinations, which occur as repetitions. Note that this particular example is an especially favourable one, as it is a single-orbit problem and is also one where the vibrational character contains not more than one copy of any irreducible symmetry, so the forms of all the vibrational modes are entirely determined by symmetry. In a more general case, symmetry considerations provide a basis for the normal modes rather than the modes themselves. 

The normal modes of ammonia, a C v molecule, are shown in Fig. 14.4. All modes for NH3 are fR active. For molecules of any symmetry, the IR active modes of vibration can be determined from the character table of its symmetry group. Modes which are infrared active belong to the same irreducible representation as one of the cartesian coordinates x, y or z, shown in one of the right-hand columns of the character table. Check this for the C2v, C v and T>ooh molecules we have considered. 

The characters of the irreducible representation satisfy the normalization condition... 

Let a representation be written with the 3N rectangular coordinates of an A -atom molecule as its basis. If it is decomposed into its irreducible components, the basis for these irreducible representations must be the normal coordinates, and the number of appearances of the same irreducible representation must be equal to the number of normal vibrations belonging to the species represented by this irreducible representation. As stated previously, however, the 3A rectangular coordinates involve six (or five) coordinates, which correspond to the translational and rotational motions of the molecule as a whole. Therefore, the representations that have such coordinates as their basis must be subtracted from the result obtained above. Use of the character of the representation, rather than the representation itself, yields the same result. 

To determine the number of normal vibrations belonging to each species, the x( ) thus obtained must be resolved into the X ( ) of Ih irreducible representations of each species in Table 1.6. First, however, the characters corresponding to the translational and rotational motions of the molecule must be subtracted from the result shown in Eq. 1.86. 

From the Czv character table, the three translations span Ai, Bi, and Bj, and the three rotations span Ai, Bi, and Bi. If these are taken away from eqn [75], this leaves 2Ai and Bi- The three normal vibrations of the bent triatomic molecule, therefore, span these irreducible representations. (See Figure 9 the antisymmetric stretch is Bi.)... 

From the representation of the normal modes of a symmetric linear molecule shown in Figure 14.30, draw the changes in the vectors upon operation of each symmetry element and assign irreducible representation labels to the normal modes of CO2. You will have to use the character table in Appendix 3. 

Check the character table in Appendix 3. In the rightmost column, the T2 irreducible representation has the x, y, and z labels. Therefore, only the T2-labeled vibrations will be IR-active, and the conclusion is that CCI4 will have only two IR-active vibrational modes. These modes, being triply degenerate, represent six of the nine normal vibrations of CCI4. 

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