What is Normal mode?

Example 5.15 (Water)

$\ce{H2O}$ in $C_{2v}$, molecule in the $xz$ plane. Atoms left in place: 3, 1, 3, 1; $\chi_{xyz}$: $3, -1, 1, 1$; so $\chi_{3N} = 9, -1, 3, 1$. The reduction gives $3A_1 + A_2 + 3B_1 + 2B_2$. Translations $A_1 + B_1 + B_2$ and rotations $A_2 + B_1 + B_2$ removed: $\Gamma_{\mathrm{vib}} = 2A_1 + B_1$. The two $a_1$ modes are the symmetric stretch ($3657\,\mathrm{cm}^{-1}$) and the bend ($1595\,\mathrm{cm}^{-1}$); the $b_1$ mode is the antisymmetric stretch ($3756\,\mathrm{cm}^{-1}$).

Example 5.16 (Four more molecules)

The same procedure (computed and checked in the figure data of this chapter) gives:

For a linear molecule the infinite group is replaced by its subgroup $D_{2h}$, in which the degenerate $\Pi_u$ bend splits into $B_{2u} + B_{3u}$.