Vibrational-Rotational energy level of a diatomic molecule

Vibrational-Rotational energy level of a diatomic molecule is given by,

$$\ E_v = ( v + \frac{1}{2} ) \hbar \omega_0 + \frac {\hbar ^2}{2I} J( J+1 )$$

Where, \(v=0,1,2,3….\) is the vibrational quantum number.

\(\omega_0 = \sqrt \frac{k}{\mu}\) , \(\mu\) is the reduced mass of the diatomic molecule and k is the force constant.

I is the Moment of inertia of diatomoc molecule and J is the Rotational quantum number.