# Symmetric Rotors

In a symmetric rotor, two of the moments of inertia are equal, but different from the third. Molecules belonging to this category include ammonia (NH_{3}), benzene (C_{6}H_{6}), and chloromethane(CH_{3}Cl). eg

The unique axis of the molecule (the one about which the moment of inertia is different from the other two) is called its principle axis. In the molecule above, it would be parallel to the C-Cl bond. We shall designate the moment of inertia about the principle axis as I_{p}, and that about the other two axes as I_{o}.

If I_{p} > I_{o}, then the rotor is classed as oblate, and typically has a shape like a frisbee. If the situation is reversed, and I_{o} > I_{p}, then the rotor is called prolate, and has a shape like a telescope. The above example falls into the prolate group; though its shape is not obviously that of a telescope, the greater mass of Cl compared to H ensures that I_{o} > I_{p}.

The classical expression for the rotational energy of the body now becomes:

where *J*_{a} refers to the angular momentum around axis a, etc. The expression can be rewritten using the fact that the total angular momentum of the body, * J* , is given by

**J**^{2}=

*J*

_{a}

^{2}+

*J*

_{b}

^{2}+

*J*

_{c}

^{2}, which gives:

We can write the total angular momentum, * J*, in terms of the quantum number J, as we established for the spherical rotor. However, in the discussion of the quantum mechanics of rotation, we also established that it was possible to specify the component of the angular momentum about one axis in terms of a quantum number that we shall call K.

This quantum number is used to specify the component of the angular momentum about the principle axis of the molecule.

The components of angular momentum about this axis are restricted to values of K, where K = 0, ±1, ±2….±J. If we make the substitution J_{a}^{2} =K^{2}^{2} and replace **J**^{2} with J(J + 1)^{2}, we obtain an expression for the energy of a rotational state in terms of J and K. Division of this by hc gives us the rotational term (the energy level expressed as a wavenumber), which is:

with the rotational constants A and B given by:

Note that K appears as its square in the energy expression. This indicates that the sign of K (which corresponds to the direction of rotation) does not affect the value of the energy, as we would expect.