ASAP

The Automated Spectral Assignment Procedure (ASAP; see the original paper) is an analysis method for rovibrational spectra if the rotational manifold of one of the two vibrational states is known to high accuracy, for example from a pure rotational analysis.

The knowledge of the rotational levels of one of the two states greatly simplifies and speeds up the analysis as all transitions into/from this level can be cross-correlated.

Theory

ASAP only requires three prerequisites which are frequently met:

The absorbance spectrum has to be background corrected
The rotational levels of either the upper or lower state have to be accurately described
The selection rules for the IR band of interest have to be known

The background correction is a default step for most infrared measurements. The second and third requirements can be satisfied by a rotational analysis of the ground vibrational state and knowledge of the symmetry of the molecule and the band of interest. The rotational analysis of the ground vibrational state is often straightforward as it is typically the most prominent pattern in the rotational spectrum and for many molecules ground vibrational state analyses are available in the literature.

ASAP then simplifies the rovibrational analysis by cross-correlation as is shown exemplarily in the following figure for the $\nu_{21}$ band of cyclopentadiene.

Working principle of ASAP. The energies of the $\nu_0$ levels are known while the energy of the target state $J_{K_a,K_c} = 25_{15, 10}$ of $\nu_{21}$ shall be determined (left side of the plot). The six allowed transitions from $\nu_0$ into $\nu_{21}$ all have the same offset from their predicted position which equals the energy offset between the actual and predicted energy of the target level: $\Delta \nu_i = \Delta \nu =\Delta E/hc$ (in this example $\Delta\nu \approx -3.1\text{ cm}^{-1}$). When plotting the six transitions in Loomis-Wood fashion (with the predicted positions as the reference frequencies), the six experimental lines are aligned vertically at $\Delta \nu$ (see middle and right column). Identifying the correct lines is simplified by cross-correlating (multiplying) the six spectra which yields the cross-correlation peaks in red on the bottom. Only a single strong peak at $\Delta \nu$ remains drastically simplifying the analysis process.

The rotational levels of the ground vibrational state are known from previous rotational analyses and thus also their energies. We focus on one energy level of upper vibrational state, called target level, and all transitions from the lower vibrational state into this target level. In the left column, the target level $J_{K_a, K_c} = 25_{15, 10}$ of $v_{21}=1$ is shown in red and the energy levels of $\nu_0$ are given in black. Because the relative energies of the lower levels are known, the positions of all transitions into the target level are known except for an energy offset $\Delta E = E^{obs} - E^{calc}$ which is the difference between the actual and calculated position of the target level $$ \tilde{\nu}^{\text{obs}}_{i} = \tilde{\nu}^{\text{calc}}_{i} + \Delta E/hc $$ Here $\tilde{\nu}$ denotes the wavenumber and $i$ the running index over the different transitions from the ground vibrational state into the target level.

Depiction of the correlation of multiple transitions into the same target level. The three transition frequencies (represented by the lengths of the three red arrows) have the same relative offset to each other, no matter the energy of the target level.

Extending ASAP to whole bands

It is important to understand, that ASAP works only with a single target level at a time, because the uncertainties of the rovibrational levels of the upper state are not correlated:

We cannot cross-correlate the transitions of multiple states of the unknown state (here the upper state) as their uncertainties are not correlated. Only transitions going into the same target state are correlated with each other.

Their relative offsets are not fixed as the energy differences between their upper levels are not known.

However, if both the lower and upper vibrational state are known from pure rotation (indicated by black and blue arrows, respectively), all transitions of the band can be cross-correlated with each other:

If both the lower and upper vibrational states are known from pure rotation (indicated by black and blue arrows, respectively), only the vibrational energy difference between the states remains to be determined and all transitions of the band can be cross-correlated with each other.

Here, all transitions of the band can be cross-correlated with each other to obtain a single value for the vibrational energy separation. This procedure is called ASAP$^2$ and can be seen in more detail in the literature. ASAP$^2$ can analyze bands within seconds but requires rotational models for both, the upper and lower, vibrational states.

The spectrum is plotted around the predicted positions of these transitions and the excerpts are aligned vertically (see middle and right column of the first figure). This representation ensures that the six transitions of interest have the same offset from the center. The offset can be found easily by cross-correlating the sub-plots which means multiplying their intensities at each respective offset $$ I_\text{cc}(\tilde{\nu}) = \prod_i I_i(\tilde{\nu}_{i,\text{pred}} + \tilde{\nu}) $$ Here, the cross-correlated intensity $I_\text{cc}(\tilde{\nu})$ (at a certain wavenumber offset $\tilde{\nu}$) is the product of the individual spectra intensities $I_i$ at the same offset $\tilde{\nu}$ from their predicted positions $\tilde{\nu}_{i,\text{pred}}$. As the experimental spectrum typically consists of intensities at discrete (and equidistant) wavenumbers, the intensities between these points are interpolated. In ideal cases, only a single peak appears in the cross-correlation spectrum, which is the offset of the target level's actual energy from its predicted energy (see the red plots on the bottom of the middle and right column). The center frequency of the cross-correlation peak can be fitted and used to create assignments for the corresponding transitions and the energy level itself.

To increase the fault-tolerance and speed of the assignment process, multiple cross-correlation spectra for a series of target levels can be plotted in Loomis-Wood fashion. Then, adjacent plots act as reference for each other, resulting in a reliable and efficient assignment process.

Loomis-Wood plot of cross-correlation plots for the $J_{15, J-15}$ series of energy levels. The row on the bottom is equivalent to the cross-correlation spectrum shown in the first figure. The other rows are obtained equivalently by applying the same procedure to the respective target levels. Accidental cross-correlation signals (as for $J=31$) are easily identified as the true cross-correlation signals form an easy to follow trend.

ASAP in LLWP

You can start the ASAP implementation of LLWP by running ASAP in the terminal. Then load

Your experimental spectrum
Your predicted transitions in a *.cat file and the corresponding predicted energy levels in a *.egy file (see the previous chapter)
In the ASAP Settings window, set the reference series to match your molecule
Set the interpolation resolution in the Settings tab to your experimental resolution

The energies in your *.egy file will be in cm$^{-1}$ and the units of your *.cat file and your spectrum can vary. In my workcases, the spectra are also given in cm$^{-1}$ and I only have to convert my *.cat file from MHz to cm$^{-1}$. This can be done by via the Units Cat File input in the Egy File tab. To convert from MHz to cm$^{-1}$, enter the value 3.3356409519815204e-05.

You should also make sure, that only transitions that are strong enough to appear in the spectrum are considered. Either apply an appropriate intensity cutoff in your *.int file or use the Filter tab. There, you can either enter an expression to filter the transitions (e.g., y > 1e-4) or set a relative intensity threshold. The latter is helpful, if you do not have a good feeling about a cutoff intensity. Setting it to 0.1 will consider for each cross-correlation plot only transitions with an intensity of at least 10% relative to the strongest transition contributing to that cross-correlation plot

For an asymmetric top, the reference series will look similar to the following:

Typical reference series of energy levels for an asymmetric top.

The first target level is specified to be the $J_{K_a, K_c} = 10_{10, 0}$ level of the vibrational state with the identifier 1. This should correspond to the unknown vibrational state. Here, this is the upper state as indicated by the checked Upper State Level checkbox. The Inc checkboxes specify how the quantum numbers should change from row to row, here $J$ and $K_c$ should increase by 1. You can also change the quantum numbers by values different to 1 by clicking the ⇆ arrow which will exchange the checkboxes by free inputs.

Even though the most obvious level to start with would be the $J_{K_a, K_c} = 0_{0, 0}$ level, this level often does not result in the strongest cross-correlation signal (no R-branch lines, no lines with $\Delta K_a \gt 0$, ...). Therefore, it is often advantageous to start with somewhat higher $K_a$ values.

After this initial setup, press Calculate Cross-Correlation and the Loomis-Wood plot of cross-correlation spectra should appear. Choose the correct lineshape to fit the cross-correlation peaks (Fit > Choose Fit Function ), check that the default uncertainty (in the New Assignments window) is set to a sensible value (negative values indicate to SPFIT that the values are in cm$^{-1}$) and start assigning the first energy levels. After you have assigned a few initial levels, you can use the Fit all functionality by right clicking on the plot and selecting Fit all.

ASAP implementation in LLWP in action. The central Loomis-Wood plot shows the cross-correlation plot. After assigning an energy level, the corresponding individual lines are shown in the ASAP Detail Viewer similar to the figure for the working principle.

Except for some of the initial set up steps, this is very similar to assigning rotational (or rovibrational) lines in Loomis-Wood plots.

The lines contributing to a cross-correlation peak are shown in the ASAP Detail Viewer window after fitting a peak (or by right-clicking on the row and selecting Show in Detail Viewer). The individual lines can be inspected to judge the quality of the fit/cross-correlation. If you want to exclude a line from the cross-correlation procedure, you can right click it in the ASAP Detail Viewer window and remove it from the cross-correlation. This is reset after recalculating all cross-correlation plots via the Calculate Cross-Correlation button.

ASAP$^2$ in LLWP

ASAP$^2$ analyses can be performed interactively in LLWP. The set up is similar to the ASAP case. Then, open the View > ASAP$^2$ window and set the filter expressions for the energy levels and the transitions. Typically, you want to exclude energy levels that are not covered by the rotational analyses and transitions that are too weak to show up in the experimental spectrum.

Set the width of the cross-correlation plot, and its resolution with the inputs on top. The threshold value sets all data points below this threshold to exactly zero (which can help to exclude accidental cross-correlation signals due to multiplying noise).

After pressing Calc ASAP$^2$, the cross-correlation plot is calculated and displayed. Select an area with the mouse to fit the center position and then press Assign to save the corresponding assignments. Clicking the Export button will save the cross-correlation plot to a *.csv file.

Rovibrational Spectra