# Molecular Symmetry

## Description

The usual rotation group, SO(3), is conveniently used to describe the symmetry of the rotational wave functions of molecules. Internal rotation, on the other hand, is described by SO(2) symmetry since the rotation axis is fixed with respect to the rigid framework. Combining these two motions leads to the formulation in a five-dimensional rotational symmetry group SO(5). Applied to the example of CH

_{5}^{+}, this group is capable of describing the permutation symmetry elements of the molecular symmetry group in terms of five-dimensional rotations. We showed that an "equivalent rotation" treatment is actually impossible in three dimensions implying a conventional description of the molecular wave functions in terms of rotational wave functions to fail.

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The SO(5) symmetry leads to the formulation of a new kinetic energy, which, as a first approximation, depends only on a single parameter. This energy depends on the quantum numbers

* n*_{1},n_{2}≤n_{1}=0,1,... of the new theory. They define a so-called super-angular momentum. Lie-algebra theory explains the need of

*two* such numbers to describe properly the states of SO(5) symmetry.

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