Rovibrational Spectra
This chapter discusses how we can transfer what we have learnt so far to the analysis of rovibrational spectra which are typically found in the infrared region. In rovibrational spectra, the transitions go from an initial vibrational state into a different final vibrational state. At the same time, their rotational levels can change. Up to this point we have analyzed pure rotational spectra (transitions with the same initial and final vibrational state) and only used single state fits (a single vibrational state in each *.par file). To analyze rovibrational spectra, we will have to set up multi state fits. However, you will see that with only a few modifications we can transfer all our learnings 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 vibration has a effect on the rotational constants as the vibrational motion changes the geometry of the molecule which in turn changes the rotational parameters, 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 configuration. $r$ 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 are 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 |
We look at the rovibrational band $\nu_{21}$, which goes from $v=0$ (A$_1$) to $v_{21}=1$ (B$_2$). By multiplying their irreps we get A$_1 \times$ B$_2 = $ B$_2$. The B$_2$ irrep includes the linear function $a$ and thus 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.
With this knowledge we can calculate the energy levels and know which transitions between these levels are allowed. This is everything we need to predict or fit the positions of the lines in our rovibrational spectrum.
How to code this in SPFIT/SPCAT
We will stick here with the $\nu_{21}$ band of cyclopentadiene as an example. 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 distinctions to the *.par file for only a single vibrational state.
- The third parameter in the third row, which specifies the number of vibrational states is set to two
- The tenth parameter in the third row is set to -1, indicating that more lines for vibrational states are following (positive value in the fourth row to indicate the last line for setting up vibrational states)
- The fourth row specifies how the $\nu_{21}$ state deviates from the default settings in the third row; 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 due $\nu_{21}$ belonging to the B$_2$ irrep
- The vibrational energy of $v_{21}=1$ is defined in the fifth row
- 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 row 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 cannot be determined)
The *.int file is almost identical to the rotational case with the small difference that the third row 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 (which otherwise could be easily excluded).
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 will create 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. In some instances, it can be benificial to do the rovibrational analysis first and then let it guide the pure rotational analysis.
You can also convert the units directly in LLWP by going to the Files > Edit Files window and clicking the cog icon behind the file. Enter in the x-Transformation field the corresponding transformation, with $x0$ being the original values, and then press update. I often want to convert my *.cat files from MHz to cm$^{-1}$, which requires the following input
x0 / 29979.2458
The same principles apply for extending the Hamiltonian to three or more states and for hot-bands (rovibrational transitions starting from a vibrationally excited state).
The CDMS offers worked examples for analyses including infrared data on its Numerous Examples page.