C2v character table

A property such as a vibrational wave function of, say, H2O may or may not preserve an element of symmetry. If it preserves the element, carrying out the corresponding symmetry operation, for example (t , has no effect on the wave function, which we write as 

We have seen that any two of the C2, ( Jxz), (r Jyz) elements may be regarded as generating elements. There are four possible combinations of + 1 or — 1 characters with respect to these generating elements, + 1 and + 1, + 1 and -1,-1 and +1,-1 and —1, with respect to C2 and (tJxz). These combinations are entered in columns 3 and 4 of the C2 character table in Table A.l 1 in Appendix A. The character with respect to / must always be + 1 and, just as (r Jyz) is generated from C2 and (tJxz), the character with respect to (r Jyz) is the product of characters with respect to C2 and (tJxz). Each of the four rows of characters is called an irreducible representation of the group and, for convenience, each is represented by a symmetry species Aj, A2, or B2. The A] species is said to be totally symmetric since all the characters are + 1 the other three species are non-totally symmetric. 

The symmetry species labels are conventional A and B indicate symmetry or antisymmetry, respectively, to C2, and the subscripts 1 and 2 indicate symmetry or antisymmetry, respectively, to n (xz). 

The symbol F, in general, stands for representation of... . Flere it is an irreducible representation or symmetry species. In Sections 6.2.3.1 and 7.3.2, where we derive the 

In the final column of the character table are given the assignments to symmetry species and These are the components of the symmetric polarizability tensor 

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Reduction of T using the C2v character table shows that ... 

The classification of the orbitals of the oxygen atom is a matter of looking them up in the C2v character table, a full version of which is included in Appendix 1. The 2s(0) orbital transforms as an a, representation, the 2px(0) orbital transforms as a b, representation, the 2p (0) orbital transforms as a b2 representation and the 2p.(0) orbital transforms as another aj representation. 

The C2v character table shows that none of the nitrogen valence orbitals transforms as a2. However, px transforms as bt and therefore can participate in tt bonding. 

As shown in Table 6.3.1, we can see that the C2v character table is divided into four parts Areas I to IV. We now discuss these four areas one by one. 

Additionally, it is noted that, mathematically, each irreducible representation is a square matrix and the character of the representation is the sum of the diagonal matrix elements. In the simple example of the C2v character table, all the irreducible representations are one-dimensional i.e., the characters are simply the lone elements of the matrices. For one-dimensional representations, the character for operation R, x(R), is either 1 or -1. 

Area III. In this part of the table, there will always be six symbols x, y, z, Rx, Ry,Rz, which may be taken as the three components of the translational vector (x, y, z) and the three rotations around the x, y, z axes (Rx, Ry, Rz). For the C2V character table, z appears in the row of A. This means that the z component of the translational vector has A symmetry. Similarly, the x and y components have B and B2 symmetry, respectively. The symmetries of rotations Rx, Ry, and Rz can be seen from this table accordingly. 

Before we leave the C2V character table, it is noted that the characters of all groups satisfy the orthonormal relation ... 

Using the C2v character table (Table A. 11 in Appendix A) the characters of the vibrations under the various symmetry operations can be classified as follows ... 

Looking at the C2v character table, we can say that v, and v2 belong to the totally symmetric irreducible representation A, and v3 belongs to52. 

The ground state a b has only fully occupied orbitals, so its symmetry is A. The first excited state, a b a2, has one fully occupied orbital, a, so this is not considered. The symmetry of this state will be given by the direct product By A2. Table 6-2 lists the direct products under the C2v character table. The symmetry of the state is B2. The other excited state in our example has the configuration a b b2. The direct product is given in Table 6-2 the state symmetry is A2. 

Table 6-2. C2v Character Table and Some Direct Product Representations...

As environmental symmetry decreases, the orbitals will become split to an increasing extent. In the C2v point group, for example, all atomic orbitals will be split into nondegenerate levels. This is not surprising since the C2v character table contains only one-dimensional irreducible representations. This result shows at once that there are no degenerate energy levels in this point group. This has been stressed in Chapter 4 in the discussion of irreducible representations. 

Because all nine direction vectors are included in this representation, it represents all the motions of the molecule, three translations, three rotations, and (by difference) three vibrations. The characters of the reducible representation F are shown as the last row below the irreducible representations in the C2V character table. 

The H2O molecule has C2v S5anmetry (Figure 3.3) and we now show how to use this information to develop an MO picture of the bonding in H2O. Part of the C2V character table is shown below ... 

This matches the row of characters for symmetry type Bi in the C2V character table, and the 2p orbital therefore possesses bi symmetry. The 2py orbital is left unchanged by the E operator and by reflection through the a. (pz) plane, but rotation about the C2 axis and reflection through the... 

Similarly, by using the B2 representation in the C2V character table, we can write down equation 4.12. Equation 4.13 gives the equation for the normalized wavefunction. 

The symmetries of oxygen s 2s and 2p atomic orbitals can be assigned and confirmed using the C2v character table. The x, y, and z axes and the more complex functions assist in assigning representations to the atomic orbitals. In this case ... 

The SO2 molecule belongs to the 2 point group, and in this section we look again at the three normal modes of vibration of SO2, but this time use the C2v character table to determine ... 

The C2V character table is shown below, along with a diagram that relates the SO2 molecule to its C2 axis and two mirror planes we saw earlier that the z axis coincides with the C2 axis, and the molecule lies in the yz plane. 

This is known as a reducible representation and can be rewritten as the sum of rows of characters from the C2v character table. Inspection of the character table reveals that summing the two rows of characters for the Ai and B2 representations gives us the result we require, i.e. ... 

Now compare this row of characters with the rows in the C2V character table. There is a match with the row for symmetry type Ai, and therefore the symmetric stretch is given the Ai symmetry label. Now consider the asymmetric stretching mode of the SO2 molecule. The vectors (Figure 4.12b) are unchanged by the E and cr (yz) operations, but their directions are altered by rotation about the C2 axis and by reflection through the a (xz) plane. Using the notation that a T means no change , and a -1 means... 

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