The kinetic energy of the internal movement can be separated into two energies, it can be divided into rotational kinetic energy and vibrational kinetic energy. Initially we will make the approximation that both movements are independent, since the rotational movement depends on the angles $\theta$ and $\varphi$ while the vibrational movement only depends on the distance between nuclei $R$.
\begin{equation} \frac{-\hbar}{2\mu}\nabla^2\psi_{rot}=E_{rot}\psi_{rot} \end{equation}
The rigid rotor model gives us the rotational energy of a diatomic molecule:
\begin{equation}\label{4} E_{rot}=\frac{\hbar^2 J(J+1)}{2I_e} \end{equation}
Where $I_e=\mu R_{e}^{2}$ is the equilibrium moment of inertia.
We can also write the rotational energy in terms of the rotational constant $B_e$, where $B_e=\frac{h}{8\pi^2I_e}$
\begin{equation}\label{5} E_{rot}=B_e hJ(J+1) \end{equation}
The rotation quantum number $J$ takes values: 0,1,2,3..... and the rotational levels are $(2J+1)$ times degenerate.