PHY.K02UF Molecular and Solid State Physics

## Rotational and vibrational energy levels of diatomic molecules

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

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

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: