Vibrational States
In the previous chapter, we have seen that isotopologues possess fingerprints that are very similar to the main isotopologue but weaker in intensity. Their intensities depend on the isotopologue ratios in the sample which often can be estimated from the natural abundances of the isotopes. However, there are also ways to create isotopically enriched samples, which can help tremendously with the analysis of otherwise very weak isotopologues.
Vibrationally excited states are another possibility for similar but weaker fingerprints. In pure-rotational spectra, these are transitions where the rotational quantum numbers do change but the vibrational quantum numbers do not change (e.g. $J = 1 \leftarrow 0$ but $v_1=1 \leftarrow 1$). Due to the vibration, the (time averaged) mean geometry of the molecule changes which results in slightly changed rotational constants. For the rotational constant $B$, this change is expressed via the rotation-vibration interaction constants $\alpha$ and $\gamma$ $$ B_v = B_e - \alpha_e \left(v + \frac{1}{2} \right) + \gamma_e \left( v + \frac{1}{2} \right)^2 + ... $$ where $v$ is the quantum number of the respective vibrational mode. Often, the last term can be neglected and $\alpha$ suffices for initial predictions of the rotational spectrum. Another important point is that even for $v=0$ the rotational constant $B_{v=0}$ is not equal to the equilibrium rotational constant $B_e$. When determining structures, it is therefore important to correct the experimental values for their zero-point vibrations.
The rotational spectrum of a vibrationally excited state is weaker than the spectrum of the ground vibrational due to its lower population. This can be caculated via the vibrational Boltzmann factor $$ e^{-E_\text{vib}/kT} $$ where $E_\text{vib}$ is the energy of the vibrationally excited state, $k$ the Boltzmann constant, and $T$ the temperature. To build some intuition, a few Boltzmann factors are given in the following table for room temperature.
Boltzmann Factors
| Rel. Int. | $E$ [cm$^{-1}$] |
|---|---|
| 90% | 22 |
| 80% | 47 |
| 70% | 74 |
| 60% | 107 |
| 50% | 145 |
| 40% | 191 |
| 30% | 251 |
| 20% | 336 |
| 10% | 480 |
| 5% | 625 |
| 2% | 816 |
| 1% | 960 |
Creating initial Models for OCS
The first exercise is to create initial models for the different vibrational states of OCS. The results of quantum chemical calculations by Sven Thorwirth (CCSD(T)/cc-pV(Q+d)Z niveau) are summarized in the following table. Since OCS is a linear molecule consisting of three atoms, there are $3N-5$ vibrational modes (with the number of atoms $N=3$). Two of the bending vibrations are degenerate, a topic we will address later. The remaining three fundamental modes are specified here in the format $(v_1, v_2, v_3)$.
| State | E [cm$^{-1}$] | $\alpha$ [MHz] |
|---|---|---|
| (0,0,0) | 0 | 0 |
| (1,0,0) | 2064 | 36.96310 |
| (0,1,0) | 522 | -10.64110 |
| (0,0,1) | 857 | 19.82430 |
With this information, we can calculate the relative intensities to be $5\cdot 10^{-3}\ \%$, $8.2\ \%$, and $1.6\ \%$ for $v_1=1$, $v_2=1$, and $v_3=1$, respectively. A synthetic spectrum consisting of OCS in the ground vibrational state, and its three singly vibrationally excited states is available here.
Next, we will create predictions for all four states. For the ground vibrational state, we can use the model we obtained in the previous sections. The *.var file is shown in the following:
OCS Mon May 12 17:45:47 2025
3 77 10 0 0.0000E+000 1.0000E+037 -1.0000E+000 1.0000000000
l -1 1 0 0 0 1 1 1 0 -1
100 6.081492117938989E+003 4.96677822E-006 /B
200 -1.301428323813037E-003 3.14237140E-009 /-D
300 -8.943126294981807E-011 3.29687176E-013 /H
0.5149825 0.5144969-0.4570838-0.6856282 0.8894236-1.0000000
For the vibrationally excited states, we will extend the model by including the vibrational energy and the rotation-vibration interaction constants. This is shown exemplarily for $v_3=1$:
OCS v3=1 Mon May 12 17:45:47 2025
5 77 10 0 0.0000E+000 1.0000E+037 -1.0000E+000 1.0000000000
l -1 1 0 0 0 1 1 1 0 -1
00 2.569222e+7 1e-37 /E
100 6.081492117938989E+003 1e-37 /B
200 -1.301428323813037E-003 1e-37 /-D
300 -8.943126294981807E-011 1e-37 /H
100 -19.82430 1e-37 /-alpha
The differences between the two *.var files are:
- The title has been changed to reflect the vibrational state
- The number of parameters has been increased in line 2
- The energy of the vibrational state has been added (in MHz)
- The rotation-vibration interaction constant for $v_3=1$ has been added
- The correlation matrix was deleted (last line in the ground vibrational state file)
These changes reflect our idea of the pure-rotational spectrum of $v_3 = 1$. We expect the spectrum to look similar to the ground vibrational state fingerprint, but with a slight difference in $B$, as specified by the rotation-vibration interaction constant $\alpha$, and with a relative intensity given by the vibrational energy.
Copy or create a respective *.int file and create initial predictions by running SPCAT. Then, load the spectrum and the predictions into LLWP and search for the fingerprint of the vibrationally excited state. You should see a Loomis-Wood comparable to the following figure.
Compare the relative intensities of the two series you see in the resulting Loomis-Wood plot. Also load either the assignments or the predictions for the ground vibrational state to confirm which series is which. If you load the predictions (meaning you have loaded two sets of predictions with the same quantum numbers), choose the correct file in the Reference Series window under File. From here on on, the same steps as for the isotopologues apply. Assign the correct peaks in the spectrum, improve the model, and verify the final model (RMS, parameter uncertainties, ...). Then repeat the same procedure for $v_1 = 1$ before you go to $v_2=1$. Typically, we would start with the most prominent fingerprint, meaning the energetically lowest vibrational state. However, we have saved $v_2=1$ for last. Create some predictions for $v_2=1$ and find out why!
The Degenerate $v_2=1$ State
When you are looking for $v_2=1$ in the Loomis-Wood plot, you should see the following:
You'll see two series which are symmetrically split around the predictions. In addition, the $J=1\leftarrow 0$ transition is missing. This results from the degeneracy of the two bending modes of OCS. There are two perpendicular bending motions. A linear combination of these two motions results in a vibrational angular momentum $l$ (imagine a bending swirl around the axis). This angular momentum $l$ interacts with the molecules overall rotation - particularly for rotations around axes perpendicular to the bond (which is always the case for linear molecules). The coupling between the two motions lifts the degeneracy between the $+l$ and $-l$ components. Therefore, this splitting is called $l$-type doubling.
The rotational Hamiltonian is expanded by a term for the $l$-type doubling $$ \mathbf{H}_\text{rot} = \frac{\hat{J}^2}{2I_b} + \frac{\hat{J}_z^2}{2} \left( \frac{1}{I_a} - \frac{1}{I_b} \right) $$ where $\hat{J}_z$ has eigenvalues $\hbar^2 l^2$ and $I_a$ is the effective moment of inertia perpendicular to the molecule axis. Because $J$ is the total angular momentum and $|l| = 1$, there is no $J=0$ level. The splitting between the two $l$-components is expressed by $$ \Delta E_{|l|=1} = h \frac{q_i}{2} (v_i + 1) J(J+1) $$ where $i$ denotes the doubly degenerate mode. Additionally, the rotational constant $B$, and the centrifugal distortion parameters $D$, $H$, ... are slightly altered ($J(J+1)$ is replaced by $J(J+1) - l^2$).
There is an easy way to implement this in the *.var file. We implement a symmetric top molecule and limit its values of $K$ to $\pm 1$.
OCS, v2 = 1 Fri May 9 09:42:04 2025
7 67 3 0 0.0000E+00 1.0000E+03 -1.0000E+00 1.0000000000
l -1 1 1 1 0 1 0 1 0 -1
0 1.560795250000000E+07 1.00000000E-37 /E2
100 6.092080204713206E+03 3.78512499E-05 /B
200 -1.322560382007065E-03 2.99040695E-08 /-D
300 -7.459290961846832E-11 4.97384073E-12 /H
40000 3.180706217543557E+00 4.02419841E-07 /q/2
$K_\text{min}$ and $K_\text{max}$ (the fourth and fifth parameters in row 3) are set to 1. $q_i/2$ is added as the parameter with the parameter code 40000. This means it is multiplied with $J_+^2 + J_-^2$ (where $J_+$ and $J_-$ are the raising and lowering operators, respectively). Therefore, it couples states with $\Delta K = \pm 2$ which in our case is $\Delta l = \pm 2$. Copy the *.var file and try to fit the $v_2=1$ doublets with it. This is the final exercise for this chapter.
Summary
In this chapter, we have learnt to identify vibrationally excited states in the rotational spectrum. The vibrational excitation causes slight deviations in the rotational parameters and the lower population of these states reduces their intensity. The procedure to identify them is very similar to the procedure for isotopologues and often the combination of ground vibrational state parameters and calculated rotation-vibration interaction parameters yields good initial models. Degenerate modes can lead to a splitting of the respective transitions which can be treated in SPFIT like a symmetric top with $K = \pm 1$.