= Solution
A useful vibration-rotation expression, with energies measured from the potential minimum, is
$$
\boxed{\frac{E(v,J)}{hc}=\widetilde\nu_e(v+\tfrac12)-\widetilde\nu_ex_e(v+\tfrac12)^2+B_vJ(J+1),\qquad
B_v\simeq B_e-\alpha_e(v+\tfrac12).}
$$
Here $v=0,1,\ldots$ is the vibrational quantum number, $J=0,1,\ldots$ the rotational angular-momentum quantum number, $\widetilde\nu_e$ the harmonic <spectroscopic wavenumber>, $x_e$ a dimensionless anharmonic coefficient, and $B_v$ the vibrational-state-dependent <rotational constant>. $B_e$ is its equilibrium value and $\alpha_e$ describes vibration-rotation coupling. All wavenumber constants use the same inverse-length units; $h$ is the <Planck constant> and $c$ the <speed of light>.
The elementary harmonic <rigid rotor> result follows by setting $x_e=\alpha_e=0$, with $\widetilde\nu_e=(2\pi c)^{-1}\sqrt{k/\mu}$ and $B_e=h/(8\pi^2cI)$, where $I=\mu r_e^2$. The molecule's electronic energy would add a common constant.
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