# Rovibrational coupling

{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} Template:Coupling in molecules Rovibrational coupling is a coupled rotational and vibrational excitation of a molecule. It is different from rovibronic coupling, which involves a change in all of electronic, vibrational, and rotational states simultaneously.

## Rotational-Vibrational Spectroscopy

{{#invoke:see also|seealso}} Generally vibrational transitions occur in conjunction with rotational transitions. Consequently, it is possible to observe both rotational and vibrational transitions in the vibrational spectrum. Although many methods are available for observing vibrational spectra, the two most common methods are infrared spectroscopy and Raman spectroscopy.

The energy of rotational transitions is on the order of ${\displaystyle 10^{-23}}$ J whereas vibrational transitions have energies on the order of ${\displaystyle 10^{-20}}$ J. Therefore, highly-resolved vibrational spectra will contain fine structure corresponding to the rotational transitions that occur at the same time as a vibrational transition. Although molecular vibrations and rotations do have some effect on one other, this interaction is usually small. Consequently, the rotational and vibrational contributions to the energy of the molecule can be considered independently to a first approximation:

where ${\displaystyle n}$ is the vibrational quantum number, ${\displaystyle J}$ is the rotational quantum number, h is Planck's constant, ${\displaystyle \nu _{0}}$ is the frequency of the vibration, ${\displaystyle c}$ is the speed of light, and ${\displaystyle {\bar {B}}}$ is the rotational constant.

### Evaluating Spectra

Part of the rotational-vibrational spectrum of methane (CH4) gas (from FTIR), showing the presence of P-, Q- and R- branches (purple, top) and a simulation in PGOPHER (black, bottom). Frequency is on the x-axis, and transmittance on the y-axis.
An energy level diagram showing some of the transitions involved in the IR vibration-rotation spectrum of a linear molecule: P branch (where ${\displaystyle \Delta J=-1}$), Q branch (not always allowed, ${\displaystyle \Delta J=0}$) and R branch (${\displaystyle \Delta J=1}$)

The strict selection rule for the absorption of dipole radiation (the strongest component of light) is that ${\displaystyle \Delta J=0,\pm 1}$. This is because of the vector addition properties of quantum mechanical angular momenta, and because light particles (photons) have angular momenta of 1. In spectroscopy, the transitions where ${\displaystyle \Delta J=-1}$ are referred to as the P-branch. Transitions with ${\displaystyle \Delta J=0}$ are referred to as Q-branch, and ${\displaystyle \Delta J=1}$ as R-branch. For linear molecules the most commonly observed case is that only transitions with ${\displaystyle \Delta J=\pm 1}$ are observed. This is only possible when the molecule has a "singlet" ground state, that is there are no unpaired electron spins in the molecule. For molecules that do have unpaired electrons, Q branches (see below) are commonly observed.

The gap between the R- and P-branches is known as the Q-branch. A peak would appear here for a vibrational transition in which the rotational energy did not change (${\displaystyle \Delta J=0}$). However, according to the quantum mechanical rigid rotor model upon which rotational spectroscopy is based, there is a spectroscopic selection rule that requires that ${\displaystyle \Delta J=\pm 1}$. This selection rule explains why the P- and R-branches are observed, but not the Q-branch (as well as branches for which ${\displaystyle \Delta J=\pm 2}$, ${\displaystyle \Delta J=\pm 3}$, etc.).

The positions of the peaks in the spectrum can be predicted using the rigid rotor model. One prediction of the rigid rotor model is that the space between each of the peaks should be ${\displaystyle 2{\bar {B}}}$ where ${\displaystyle {\bar {B}}}$ is the rotational constant for a given molecule. Experimentally, it is observed that the spacing between the R-branch peaks decreases as the frequency increases. Similarly, the spacing between the P-branch peaks increases as the frequency decreases. This variation in the spacing results from the bonds between the atoms in a molecule not being entirely rigid.

This variation can be mostly accounted for using a slightly more complex model that takes into account the variation in the rotational constant as the vibrational energy changes. Using this model, the positions of the R-branch peaks are predicted to be at:

where ${\displaystyle {\bar {B}}_{0}}$ is the rotational constant for the ${\displaystyle n=0}$ vibrational level and ${\displaystyle {\bar {B}}_{1}}$ is the rotational constant for the ${\displaystyle n=1}$ vibrational level. Likewise, the P-branch peaks are predicted to be at:

Rotational-vibrational spectra will also show some fine structure due to the presence of different isotopes in the spectrum. In the spectrum shown above, all of the rotational peaks are slightly split into two peaks. One peak corresponds to 35Cl and the other to 37Cl. The ratio of the peak intensities corresponds to the natural abundance of these two isotopes.