We choose a basis depending on our particular analysis goal. For vibrational normal modes we would like to separate translation and rotation and then to identify the irreducible representations of the modes. To accomplish this we use the basis of Cartesian coordinates on each atom in the molecule. Once we have determined the normal mode symmetries (irreps) we can use the internal coordinate basis for further symmetry analysis. That basis consists of:
Bond stretches
Angle bends
Torsions
Wags
For analysis of the molecular orbitals in the linear combination of atomic orbitals (LCAO) approach we can use the hydrogen atom basis functions with shapes of s, p, and d functions. We can see how each of these transform under the symmetry operations of the group to determine the reducible representation.
