# 1.25: Summary of applying group theory to molecular motions

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1. Atomic or molecular translations transform in the same way as the $$x$$, $$y$$, $$z$$ (or $$T_x$$, $$T_y$$, $$T_z$$) functions listed in the character tables.
2. Molecular rotations transform in the same way as the $$R_x$$, $$R_y$$, $$R_z$$ functions listed in the character tables.
3. The irreducible representations spanned by the motions of a polyatomic molecule may be determined using the $$3N$$ Cartesian basis, made up of $$x$$, $$y$$, $$z$$ axes on each atom. The characters of the matrix representatives are best determined using a table as follows: $\begin{array}{ll} \text{Operation:} & \text{List the symmetry operations in the point group} \\ \Gamma_{\text{Cart}} & \text{List the characters for} \: x + y + z \: \text{(from the character table) for each operation} \\ N_{\text{unshifted}} & \text{List the number of atoms in the molecule that are unshifted by each symmetry operation} \\ \Gamma_{3N} & \text{Take the product of the previous two rows to give the characters for} \: \Gamma_{3N} \end{array}$
4. The irreducible representations spanned by the molecular vibrations are determined by first subtracting the characters for rotations and translations from the characters for $$\Gamma_{3N}$$ to give the characters for $$\Gamma_{\text{vib}}$$ and then using the reduction formula or inspection of the character table to identify the irreducible representations contributing to $$\Gamma_{\text{vib}}$$.
5. The molecular displacements for the vibrations of each symmetry may be determined by using projection operators on the $$3N$$ Cartesian basis vectors to generate SALCs.
6. Alternatively, a basis of internal coordinates (bond lengths and angles) may be used to investigate stretching and bending vibrations. Determine the characters, identify the irreducible representations, and construct SALCs.