E it If IB TBl

from which the quantum mechanical expression may be derived h2 h2 i l l

Similarly for the oblate case where Ia = Ib = Ib, Ic = Ic, the energy levels are found to be h2 h2 l l

Hence the rotational energy levels of symmetric rotors are considerably more complex than those of linear rotors and spherical tops since the energies depend on both J and K.

(iv) Asymmetric rotors. Although the total angular momentum for molecules of this type is well defined, there is no principal axis along which the component La may be defined, which greatly complicates the calculation of energy levels, as is discussed further by Hanel et al. (2003). Water is a typical example of an asymmetric rotor, and rotational energy levels are found to be so complex that they have an apparently random distribution of energy levels.

At higher moments of inertia, the bonds between the atoms of all molecular types become stretched due, essentially, to centrifugal forces. This stretching increases the moment of inertia and thus lowers the rotational energy levels. Including centrifugal effects, the rotational energy levels of, for example, a simple linear rotor are modified as

where D is a small constant. The rotational energy levels of other molecular types are similarly adjusted.

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