# 3.9: Rotation of Linear Molecules

The rotational motion of a linear polyatomic molecule can be treated as an extension of the diatomic molecule case. One obtains the $$Y_{J,M} (\theta,\phi)$$ as rotational wavefunctions and, within the approximation in which the centrifugal potential is approximated at the equilibrium geometry of the molecule $$(R_e)$$, the energy levels are:

$E^0_J = \hbar^2 \dfrac{J(J+1)}{2I}.$

Here the total moment of inertia I of the molecule takes the place of $$\mu R_e^2$$ in the diatomic molecule case

$I = \sum\limits_a m_a (R_a - R_{CofM})^2;$

$$m_a$$ is the mass of atom a whose distance from the center of mass of the molecule is $$(R_a - R_{CofM}).$$ The rotational level with quantum number J is (2J+1)-fold degenerate again because there are (2J+1) M- values.