# 25: Summary of applying group theory to molecular motions

- 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.
- Molecular rotations transform in the same way as the \(R_x\), \(R_y\), \(R_z\) functions listed in the character tables.
- 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: \[\begin{array}{ll} \text{Operation:} & \text{List the symmetry operations in the point group} \\ \Gamma_{\text{Cart}} & \text{List the characters for} \: x + y + z \: \text{(from the character table) for each operation} \\ N_{\text{unshifted}} & \text{List the number of atoms in the molecule that are unshifted by each symmetry operation} \\ \Gamma_{3N} & \text{Take the product of the previous two rows to give the characters for} \: \Gamma_{3N} \end{array}\]
- 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 \(\Gamma_{\text{vib}}\).
- 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.
- 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.

### Contributors

Claire Vallance (University of Oxford)