Degenerate vibrations potential energy

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

Fig. 1. The variation of the potential energy of an adatom moving parallel to the surface. Em is the barrier height measured from the zero point energy of the two degenerate modes of vibration parallel to the surface, i.e. Em = E — hvp, where is the frequency of vibration of a localised adatom parallel to the surface.

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