The rotational and vibrational energy levels of diatomic molecules can be approximated as,

\[ \begin{equation} \large E_{\text{vib}}= hc\omega_e(\nu+1/2)-hc\omega_e x_e(\nu+1/2)^2, \end{equation} \] \[ \begin{equation} \large E_{\text{rot}}= hc((B_e - \alpha_e(\nu+1/2))J(J+1)+D_e(J(J+1))^2), \end{equation} \]

where $\omega_e$, $x_e$, $B_e$, $\alpha_e$, and $D_e$ are spectroscopic constants. The quantum numbers $\nu$ and $J$ can take on integer values, $\nu, J=0,1,2,\cdots$. Here $h$ is Planck's constant and $c$ is the speed of light in vacuum. The units of all of the spectroscopic constants are cm^-1 except for $x_e$ which is unitless. The rotational and vibrational energy levels $E_{\nu J}=E_{\text{vib}} + E_{\text{rot}}$ are plotted in the bond potential on the left. An enlargement of the energy level spacing is shown on the right. The rotational levels have a degeneracy of $(2J+1)$.

Vibration-rotation energy levels of H₂

$U(r)$ [eV] $r$ [Å] Bond length: 0.74144 Å. 0.74144 eV.

[eV]

$\omega_e$ =   cm-1   $\omega_e x_e$ =   cm-1  $B_e$ =   cm-1  $\alpha_e$ =   cm-1  $D_e$ =   cm-1  $U_0$ =   eV  $r_e$ =   Å  $\nu_{\text{min}}$ =  $\nu_{\text{max}}$ =  $J_{\text{max}}$ =

---

The spectroscopic constants can be found in:

• Demtröder, Kapitel 9.5 Atome, Moleküle und Festkörper
• CRC Handbook of Chemistry and Physics
• K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979.