Rovibrational Transitions

Ro-Vibrational Transitions

1 Order of Magnitude Energies

The Born-Oppenheimer approximation allows us to treat the electrons in a molecule as a cloud– they are much less massive and therefore have much higher velocities than the nuclei. If $L$ is the molecular size, typical electrons have momentum $\hbar /a$ and the electronic energy spacings can be expressed as:

$E_{elec}\sim {\frac {\hbar ^{2}}{mL^{2}}}\,\!$ The nuclei feel an equivalent potential that only depends on the internuclear distance and the electronic state. The internuclear potential has a minimum, and vibrations about the minimum can be roughly modeled as a harmonic oscillator. This potential is about ${\frac {1}{2}}M\omega ^{2}\xi ^{2}$ , where $\omega$ is the vibration frequency and $\xi$ is the displacement from equilibrium. If the displacement is the same order as the size of the molecule, the electronic energies should be about $\hbar ^{2}/2mL^{2}$ :

${\frac {1}{2}}M\omega ^{2}L^{2}\sim {\frac {\hbar ^{2}}{2mL^{2}}}\,\!$ $E_{vib}\sim \hbar \omega \sim \left({\frac {m}{M}}\right)^{1/2}{\frac {\hbar ^{2}}{mL^{2}}}\sim \left({\frac {m}{M}}\right)^{1/2}E_{elec}\,\!$ The nuclei can also rotate, and have rotational energies that depend on their angular momentum. If $J$ is the quantum angular momentum number,

$E_{rot}\sim {\frac {\hbar ^{2}J(J+1)}{2I}}\,\!$ $I\sim ML^{2}\,\!$ For small $J$ ,

$E_{rot}\sim {\frac {m}{M}}E_{elec}\,\!$ So the electronic, vibrational, and rotational energy states have contributions that scale with the electron-to-nucleus mass ration:

$E_{elec}:E_{vib}:E_{rot}\sim 1:\left({\frac {m}{M}}\right)^{1/2}:\left({\frac {m}{M}}\right)\,\!$ 2 Rotation-Vibration Spectra

While it is possible to have a pure rotational spectrum, a pure vibrational spectrum is very unlikely: energies required to excite vibrations are much larger than those required to excite rotation. However, a combination of rotation and vibrational modes can be excited.

Rotational energies can be described using the angular momentum number $J$ :

$E_{rot}={\frac {\hbar ^{2}}{2I}}J(J+1)\,\!$ where $J=0,\ 1,\ 2,\ ...$ , $I$ is the moment of inertia, and $\mu$ is the reduced mass:

$I=\mu L^{2}\,\!$ $\mu ={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}\,\!$ The vibrations can again be modeled as a harmonic oscillator:

$E_{vib}=\hbar \omega (n+1/2)\,\!$ where $n=0,\ 1,\ 2,\ ...$ and $\omega ={\sqrt {k/m}}$ where $k$ is the effective spring constant.

The total energy of rovibrational transitions, then, is:

$E_{rovibe}=\hbar \omega (n+1/2)+{\frac {\hbar ^{2}}{2I}}J(J+1)\,\!$ The selection rules for rovibrational transitions tell us that $\Delta n=+1$ and $\Delta J=\pm 1$ . $\Delta J=+1$ is called the R branch, and $\Delta J=-1$ is called the P branch. This notation matches the videos above, but is opposite the notation in Rybicki & Lightman. We can use our rovibe energy expression to find the frequency of emitted photons for the R branch:

$\Delta J=+1\,\!$ $E_{2}-E_{1}=\hbar \omega (n+1+1/2)-\hbar \omega (n+1/2)+{\frac {\hbar ^{2}}{2I}}(J+1)(J+2)-{\frac {\hbar ^{2}}{2I}}J(J+1)\,\!$ $E_{2}-E_{1}=\hbar \omega +{\frac {\hbar ^{2}}{I}}(J+1)=\hbar \omega _{obs,R}\,\!$ $\omega _{obs,R}=\omega +{\frac {\hbar }{I}}(J+1)\,\!$ and the P branch:

$\Delta J=-1\,\!$ $E_{2}-E_{1}=\hbar \omega (n+1+1/2)-\hbar \omega (n+1/2)+{\frac {\hbar ^{2}}{2I}}(J-1)(J)-{\frac {\hbar ^{2}}{2I}}J(J+1)\,\!$ $E_{2}-E_{1}=\hbar \omega -{\frac {\hbar ^{2}}{I}}J=\hbar \omega _{obs,P}\,\!$ $\omega _{obs,P}=\omega -{\frac {\hbar }{I}}J\,\!$ However, we should note that the potential is not perfectly harmonic; the slight asymmetries we can see in the potential cause the bond length to increase as $n$ increases. So if $n$ increases, $L$ increases, causing $I$ to increase. Our rovibrational energy expression depends on $I^{-1}$ , so the energy decreases as $n$ increases. This effectively pulls all of our spectral lines to the left, which decreases the separation between lines on the R branch and increases the separation between lines on the P branch. Similar effects can also be found when comparing the expressions for the observed frequencies. Putting everything together, we can plot our rovibrational spectrum.