# Linear and Asymmetric Rotors

Linear rotors are linear molecules, for example CO_{2}, C_{2}H_{2} (ethyne) and all diatomic molecules. The moment of inertia of a linear molecule about an axis that lies along the molecular axis is necessarily zero, as the atoms all lie on this axis and so are at zero distance from it. The component of angular momentum about this axis must therefore also be zero, by definition. (Recall J = Iω.)

We may now treat this as a special case of a symmetric rotor – the moments of inertia about two axes perpendicular to the molecular axis are both equal, and different from the moment of inertia about the principle axis. However, in this case, the moment of inertia around the principle axis (which is equivalent to the molecular axis) is always zero. This implies that the quantum number K is also always zero. Substituting these results into the expression for the energy term of a symmetric rotor (on the previous page) gives:

Note that this is precisely the same expression as for a spherical rotor.

However, in this case the K term disappears because K is necessarily zero. In the case of a spherical rotor, the K term disappears because the rotational constants A and B are equal.

Asymmetric rotors are molecules for which all three moments of inertia are different. All molecules that do not fall into one of the other categories are classified in this group. Many common molecules, such as water and ethanol, fall into this group. The energy levels of asymmetric rotors are rather too complex to go into here.