Rovibrational Spectra
In this chapter, we will transfer what we have learnt so far to the analysis of rovibrational spectra. Rovibrational transitions are transitions that have a different initial and final vibrational state. At the same time, their rotational state can change. Typically, rovibrational spectra are located in the infrared region.
So far, we have only analyzed pure-rotational spectra. There, we could also see patterns of vibrationally excited state but the initial and final vibrational state were the same. Since rovibrational spectra consist of transitions between two different vibrational states, we will also learn how to set up models with multiple vibrational states. You will see that, with only a few modifications, we can transfer what we have learnt so far to rovibrational spectra.
Theory
According to the Born-Oppenheimer approximation, the complete wavefunction $\psi_\text{total}$ of a molecule can be treated as the product of an electronic, vibrational, rotational, and nuclear spin wavefunction $$ \psi_\text{total} = \psi_\text{elec} \psi_\text{vib} \psi_\text{rot} \psi_\text{ns} $$ Therefore, the rotational and vibrational Hamiltonians can be treated independently $$ \mathcal{H}_\text{rovib} = \mathcal{H}_\text{rot} + \mathcal{H}_\text{vib} $$ The vibrational motion has an effect on the rotational constants since the vibrational motion changes the geometry of the molecule which in turn changes the rotational parameters. These changes are given here exemplarily for the $B$ rotational constant $$ B_v = B_e - \sum_r \alpha_r^B \left(v_r + \frac{1}{2} \right) + \sum_{r \geq s} \gamma_{rs}^B \left(v_r + \frac{1}{2} \right) \left(v_s + \frac{1}{2} \right) + ... $$ The $\alpha$'s and $\gamma$'s are the so-called rotation-vibration interaction constants and $\gamma$ is small compared to $\alpha$. $B_e$ is the rotational constant along the $b$-axis in the equilibrium geometry. $r$ and $s$ are running indices over the vibrational modes of the molecules with the respective vibrational quantum numbers $v_r$ and $v_s$.
Starting with two vibrational states, we can set up a multi-state Hamiltonian as follows $$ \mathcal{H} = \begin{pmatrix} \mathcal{H}_\text{rot}^{v_a} + E_v^{v_a} & 0 \\ 0 & \mathcal{H}_\text{rot}^{v_b} + E_v^{v_b} \end{pmatrix} $$ The diagonal entries of the total Hamiltonian are the rovibrational Hamiltonians for the respective states, consisting of the vibrational energies and the rotational Hamiltonians.
In contrast to the pure rotational spectrum, the intensity of a rovibrational band depends on the change of the dipole moment. The selection rules can be easily deduced from the character table of the molecule by multiplying the irreducible representations (irreps) of the initial and final vibrational state. The product will be another irrep and the linear functions belonging to this irrep indicate the allowed transition types.
Example for C$_\text{2v}$ Point Group
| E | C$_2(z)$ | $\sigma_v(xz)$ | $\sigma_v(yz)$ | linear | |
|---|---|---|---|---|---|
| A$_1$ | 1 | 1 | 1 | 1 | b |
| A$_2$ | 1 | 1 | -1 | -1 | |
| B$_1$ | 1 | -1 | 1 | -1 | c |
| B$_2$ | 1 | -1 | -1 | 1 | a |
For example, the rovibrational band $\nu_{21}$ goes from $v=0$ (A$_1$) to $v_{21}=1$ (B$_2$). Multiplying these irreps results in A$_1 \times$ B$_2 = $ B$_2$. The B$_2$ irrep includes the linear function $a$ and therefore the band is an $a$-type band. Similar to the pure rotational case, this means that the selection rules for $K_a$ and $K_c$ are $$ \begin{align} \Delta K_{a}&= 0 \ (\pm2, \pm4,...) \\ \Delta K_{c}&=\pm1 \ (\pm3, \pm 5,...) \end{align} $$
Deducing the fundamental vibrations of a molecule can be non trivial. Quantum chemical calculations can be a great help in this regard and will often also specify the symmetries/irreps of the fundamental modes.
The total angular momentum can change by $\Delta J = -1, 0, +1$ (due to the angular momentum carried by a single photon). Transitions with $\Delta J = -1$ belong to the so-called P-branch, $\Delta J = 0$ transitions to the Q-branch, and $\Delta J = +1$ transitions to the R-branch.
Based on this knowledge we can calculate the energy levels and know which transitions between these levels are allowed. This is all we need to predict or fit the positions of the lines in our rovibrational spectrum.
How to code this in SPFIT/SPCAT
We will use the $\nu_{21}$ band of cyclopentadiene as an example. You can download the corresponding infrared spectrum in units of wavenumbers or MHz. The *.par file looks as follows (again the sextic and higher-order parameters have been omitted)
c-C5H6
25 6794 20 0 0.0000E+00 5.0000E+00 -1.0000E+00 1.0000000000
a 1 -2 0 80 0 2 7 9 -1 -1 0
1 1 0 80 0 2 9 7 1
11 2.882386606316676E+07 1.00000000E+37 /E_21
10000 8.426108836654932E+03 1.00000000E+37 /A_0
10011 8.415726897530922E+03 1.00000000E+37 /A_21
20000 8.225640359463139E+03 1.00000000E+37 /B_0
20011 8.244088848620053E+03 1.00000000E+37 /B_21
30000 4.271437282769861E+03 1.00000000E+37 /C_0
30011 4.278340137646470E+03 1.00000000E+37 /C_21
200 -2.692734182009970E-03 1.00000000E+37 /-DJ_0
211 -2.824310576562861E-03 1.00000000E+37 /-DJ_21
1100 4.059625037116573E-03 1.00000000E+37 /-DJK_0
1111 4.212855250360758E-03 1.00000000E+37 /-DJK_21
2000 -1.682736841929652E-03 1.00000000E+37 /-DK_0
2011 -1.850652885295154E-03 1.00000000E+37 /-DK_21
40100 -4.221850551738846E-05 1.00000000E+37 /d1_0
40111 -1.329121728420262E-04 1.00000000E+37 /d1_0
50000 -6.013191777326903E-07 1.00000000E+37 /d2_0
-50011 -6.013191777326903E-07 1.00000000E-37 /d2_21
There are some important differences compared to the *.par file for a single vibrational state.
- The third parameter in the third line, which specifies the number of vibrational states, is set to two
- The tenth parameter in the third line is set to -1, indicating that more lines for vibrational states are following (positive value in the fourth line to indicate the last line for setting up vibrational states)
- The fourth line specifies how the $\nu_{21}$ state deviates from the default settings in the third line; it is given the identifier 1 (meaning the ground vibrational state is identified by a 0 and $\nu_{21}$ by an 1) and the weights for even and odd states are swapped since $\nu_{21}$ belongs to the B$_2$ irrep
- The vibrational energy of $v_{21}=1$ is defined in the fifth line
- Parameter IDs ending with 11 belong to $v_{21}=1$ whereas parameters ending with 00 belong to $v=0$
- The last parameter ID has a negative sign, this indicates that the parameter is kept fixed to the parameter in the previous line at their current ratio (this is used here to fix the $d_2$ constant of $v_{21}=1$ to that of the ground vibrational state as it could not be determined)
The *.int file is almost identical to the rotational case with the small difference that the third line specifies the transition strength of the $\nu_{21}$ band
c-C5H6
1 66520 50808.4397 0 80 -7.0 -200 10e+7
011 0.062908075199908
The code 011 specifies (from left to right) that the initial state is $v=0$, the final state is $v_{21}=1$, and transitions are of $a$-type. In addition, I have set the second cutoff value to -200 and the frequency cutoff to 10e+7 GHz to include all rovibrational lines.
If you also want to calculate the pure rotational transitions at the same time you can add the following lines for $v=0$ and $v_{21}=1$, respectively
002 0.419
112 0.419
Running spcat on these files creates a *.cat file with entries that look like this
28836388.4034 0.1233 -3.1885 3 0.0000 21 665201404 1 0 1 1 0 0 0 0
The last eight numbers are the quantum numbers $J$, $K_a$, $K_c$, and the identifier for the vibrational states. Analogously, entries in the *.lin file should have the following format
2 0 2 1 1 0 1 0 962.1701938350 -2.0e-04 1.0000
if you want to specify the center position in cm$^{-1}$ as indicated by the negative uncertainty (-2.0e-4) or
2 0 2 1 1 0 1 0 2.884514e+7 6 1.0000
if you want to specify the center position in MHz. Similar to the *.int file, where you can add rows for both the pure rotational spectrum and the rovibrational spectrum, you can also add pure rotational lines to the *.lin file. This is a common practice (if pure rotational data is available) to improve the accuracy of the rotational parameters.
The same principles apply to extending the Hamiltonian to three or more states and for hot-bands (rovibrational transitions starting from a vibrationally excited state). You can find worked examples for analyses including infrared data in the CDMS on its Numerous Examples page.
Tipps for the Analysis
It can often be benificial to begin the analysis with the pure-rotational analysis of the ground vibrational state. The knowledge of the rotational levels in the ground vibrational state can then be exploited with the Automated Spectral Assignment Procedure (ASAP) to significantly simplify the analysis of the rovibrational spectrum.
If your predictions and your experimental data are in different units, e.g. MHz and cm$^{-1}$, you can convert the units directly in LLWP. Go to the Files > Edit Files window and click the cog icon behind the file. In the x-Transformation field, enter the corresponding transformation, with x0 being the original values, and then press update. For example, if you often want to convert the units in your *.cat file from MHz to cm$^{-1}$, enter the following input
x0 / 29979.2458
Understanding the Spectrum
After we have talked about the theory and how to implement rovibrational spectra in SPFIT/SPCAT, we will examine the predictions to get a more intuitive understanding of the spectrum.
- Create the *.var and *.int files according to above
- Create the *.cat file by running SPCAT on these files
- Open the *.cat file in LLWP and navigate to the band (center of 2.88e7 MHz, width of 3e6 MHz)
- Open the Files Window via Files > Edit Files, navigate to the *.cat tab and click the cog icon behind the *.cat file
- Set the filter to qnu1 == qnl1 and press Update to only see the Q-branch
- Repeat the same for qnu1 == qnl1 + 1 and qnu1 + 1 == qnl1 to see the P- and R-branches respectively
#ff0000; qnu1 == qnl1
#00ff00; qnu1 == qnl1 + 1
#0000ff; qnu1 + 1 == qnl1
Do not forget to reset the filter to see all three branches again:
As you can see, the spectrum can be roughly divided into the P-branch to the left, the intense Q-branch in the center, and the R-branch on the right. You can repeat the same for different values of $K_a$ by using filters like qnu2 == 0 (for the final state $K_a$) or qnl2 == 0 (for the initial state $K_a$). What happens for different values of $K_a$?
Keep in mind, that the contoure of a band heavily depends on the rotational parameters and the transition type. Feel free to change the *.int file to predict $b$- and $c$-type bands and use what you have just learnt to compare them with each other. However, you will also have to change the symmetry of either the upper or lower state to predict $b$-type bands. You can do so by inverting the weights for even and odd states of either the upper or lower state. Otherwise you will see the following warning in the *.out file
WARNING: dipole 2 12 has no matrix elements, (4) transition between states of different weight
This line tells you that you are requesting a transition between levels with different weights, which are in different sub-matrices and therefore cannot be computed. Typically this warning results from incorrect symmetries of your vibrational states, the wrong dipole moment, or the wrong axis for statistical weight.
Assigning the Spectrum
If one of the two vibrational states is already known from pure rotation, you can greatly simplify your analysis by using the Automated Spectral Assignment Procedure (ASAP). Otherwise, the analysis is very similar to the pure rotational analysis of an asymmetric top.
- Find the strongest series in your band and display them in a Loomis-Wood plot
- Assign the series by fitting the appropriate lineshape function to it (maybe also use Fit all to speed up this process
- Save these assignments to your *.lin file
- Improve your model in the *.par file by running SPFIT
- Check if any parameters should be added to or omitted from your model with pyckett_add and pyckett_omit
You will repeat these steps till the desired quantum number range is covered or no more unassigned lines are visible in the spectrum. Double check that there are no outliers or trends in your residuals, that all parameters are physically meaningful and determined, and that the WRMS error of your analysis aligns with your expectations.