Here the coordinates are again relative to the centre of mass. By choosing a suitable coordinate transformation, this matrix may be diagonalized (Section 16.2), with the eigenvalues being the moments of inertia and the eigenvectors called principal axes of inertia.
For a general polyatomic molecule, the rotational energy levels cannot be written in a simple form. A good approximation, however, can be obtained from classical mechanics, resulting in the following partition function.
Here Ii are the three moments of inertia. The symmetry index s is the order of the rotational subgroup in the molecular point group (i.e. the number of proper symmetry operations); for H2O it is 2, for NH3 it is 3, for benzene it is 12, etc. The rotational partition function requires only information about the atomic masses and positions (eq. (13.27)), i.e. the molecular geometry.
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