Normal Vibrations by Symmetry

The 3N- 6(5) normal vibrations of an TV-atom molecule can be classified into symmetry species of a point group according to their symmetry properties. As an example, consider the displacements of individual atoms of the H2O molecule (C2V) using the Cartesian coordinates shown below  

the symmetry species of the three displacements of the oxygen atom are readily determined as shown below  

Here the + sign denotes the out-of-plane displacement in the +x direction. 

Since the two hydrogen atoms are equivalent, we consider symmetry species of six linear combinations of their displacements  

Since these nine displacements include three translational (7) and three rotational (R) motions of the whole molecule, we must substract them from our calculations. It is readily seen that Tx, Ty and Tz belong to the B, B2 and A, whereas Rx, R and Rz belong to the B2, B and A2 species, respectively. Table 1 -8 summarizes these results. Thus, we find that two vibrations belong to the A and one vibration belongs to the B2 species. The approximate vibrational modes of the two A type vibrations can be derived by combining z, (z2 + z3) and (y2 - y3) as follows  

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