A Complete Set of Vibrational Modes for

In Chapter 1 it was noted that the number of vibrational modes of a molecule can be calculated by counting the degrees of freedom of the atoms (three per atom for X,Y and Z movement) and subtracting the degrees of freedom for motion of the molecule as a whole, three for its translation and (for nonlinear molecules) three for rotation. This was used in Section 5.2 to arrive at a reducible representation for the basis of nine atomic degrees of freedom for H2O, the classic C2V molecule. The characters for this representation were given in Table 5.1. We can now apply the reduction formula to identify the irreducible representations for the three vibrations of HjO. 

As expected, the nine basis vectors have produced nine irreducible representations, but only 9 — 6 = 3 of these can correspond to molecular vibrations. The others are motions of the molecule as a whole. 

There are three translations and three rotations of the molecule as if it were a rigid body. For any molecule in the point group, the rigid body motions will have the same irreducible representations. In the standard character tables of Appendix 12 the symbols x, y, z andi , Ry, are written in the rightmost columns and can be used to identify the representations for rigid-body movement and rotation respectively. So, most of the time, it is just a matter of referring to the character table to find the irreducible representations that should be removed and so isolate the vibrational mode symbols. 

However, to demonstrate how the rigid-body motion conforms to the irreducible representations, in this example we will go over the effect of symmetry operations on the translational and rotational motion of H2O. 

Problem 5.6 Confirm that movement of the H2O molecule as a whole in the X and Z directions follows the Bi and Aj representations respectively. 

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