Walk-through of Simulating and Fitting a Simple Spectrum

(This plot is obtained with B' (Excited state B) = 0.502, and B" = 0.5, and zooming in to the central part of the spectrum.)

Line Position Assigning

The next step is produce a list of experimental line positions and the quantum numbers for the transition. When following the steps below, expect to have to make some corrections; the program is designed to make mistakes easy to detect and correct.

1. Right click on one of the peaks in the simulated (lower) spectrum that you think you can assign. A Line List Window will appear:
   
   You will normally want to arrange the windows so that the simulation and the line list do not overlap. Selecting View, Arrange Windows from the main menu may help.
2. All the lines close to the mouse (within a few pixels on the screen) will be shown selected, which will often catch more than one rotational transition. The intensity of the individual transitions (shown under Strength) is normally enough to indicate if you have a sensible selection of lines. The next step will ignore the weaker lines, or you can adjust the selected lines with the mouse. You can delete the unwanted transitions by selecting the required rows with the mouse and pressing the delete button, .
3. Measure the corresponding peak in the experimental spectrum by dragging the across the peak with the right mouse button. (Press and hold with the right mouse button to the left of the peak, drag the mouse to the right of the peak, and release). The peak position will be determined from the spectrum and a marker indicating the determined position will be displayed. Note:
• The exact positions you click and release on in the experimental spectrum is not important, provided you do not include any other peak in the range. For closely spaced peaks you may need to try more than once, possibly expanding the scale.
  
• You can remove the markers from the simulation from the Overlays menu - select "Delete Last Peak" or "Clear All Peaks" as required.
4. Provided Auto is selected in the line list window, the "Position" column in the line window of the selected rows will change to the measured position of the experimental peak and the “Std Dev” column will be set to 1 to indicate a measured, rather than a calculated frequency. (If Auto is off press the Assign button to do this manually.)
5. Repeat steps 1 to 4 to assign a few more peaks.
6. To check that you have the correct assignments, press the "Test" button. The program will read and plot your assignments:
• The observed and calculated values are joined with lines above the experimental spectrum.
• The upper end of the line is the experimental frequency, and the lower end is the calculated frequency.
• When the fit is perfect, these will appear as small vertical tick marks. At this stage the lines will probably be sloping, but it is normally obvious if you have miss-assigned something.
• Positioning the mouse over the top of one of the marks will show the number in the line list of the assignment in the status bar below the simulation.
• Right click on the mark to show the row containing the experimental line position in the line list window.
If you enter the assignments indicated in the picture above, the "Test" button should give a picture like this:

7. It is easy to miss-assign peaks, and wrongly assigned peaks can distort the subsequent fit quite badly An error might give a plot such as:
   
   To correct an assignment, select the corresponding rows(s) with the mouse and either:
   • Delete those rows with the delete button
   • Re-measure the peaks as in step 5 by dragging across the peak with the right mouse button.
   • Manually edit the values in the grid. You can temporarily remove an observation by setting the "Std Dev" to zero.
     

Complete the fit by measuring up all the P and R branch lines, or at least a reasonable selection of them. The file at this stage is available as N2Ostage1.pgo. For further refinements of the fit the residuals window is most useful. With all the P and R branch lines in it should look something like this:

Note the choice of x axis made here (the drop down in the top middle). The default is observation number, but the J has been chosen for the plot above. There is clearly a systematic trend in the residuals - perhaps the centrifugal distortion constant, D, should be floated? Floating both the upper and lower state D does indeed improve the fit, but reveals a couple of lines are fitting particularly poorly:

This is with my measurements - yours may well look different. Any specific measurement can be checked by right clicking on the point in the residuals window and selecting  "Show Transition". For best results the plot should be on quite an expanded scale - if "Show Transition" has just been used the expand plot button () will expand the plot about the transition of interest. With the transition indicated with the arrow above, the plot looks something like this:

It is clear from the top mark that I have been careless in measuring this particular transition. To remeasure, right click on the transition in the residuals window again, but this time select "Edit Point". This selects the transition in the line list window, and it can be measured again by right clicking and dragging on the experimental spectrum. Repeating this for the any other poorly fitting transitions and re-fitting should yield a random set of residuals, which is a sign of a good fit:

Various tools are available to understand the simulated spectrum and the underlying pattern of energy levels. Right clicking on an individual line will give the quantum numbers associated with the line (or the strongest lines close to where you click if there is more than one). The information is expressed in more than one way; on the left hand side are the quantum numbers always used internally, with extra information on the right hand side. For a complete list see the line list window documentation.

The Fortrat Diagram

The overall pattern can also be understood by turning on the Fortrat plot with the Fortrat button(). The resulting plot has the same horizontal axis as the main plot, but J as the vertical axis. This allows individual branches to be picked out. A symbol is plotted for each transition, and the size of the symbol proportional to the intensity:

The Q Branch

To finish the fit, we need to fit the central Q branch also. Zooming in it is clear that, while the basic structure is correct, the constants we have so far do not give the correct spacing:

In fact we need to take account of the fact that the upper state is degenerate, and consider l-doubling (known as Λ-doubling if the degeneracy is electronic).  The upper state is a Π state, and thus doubly degenerate, with two different parities for each J. The two different parities have slightly different rotational constants, so the energy level formula becomes:

E(J) = Origin + BJ(J+1)  ± qJ(J+1)

If you now assign some of the Q branch lines and float q (essentially the difference between the two rotational constants) for the upper state, you should be able to fit the entire spectrum. (Note that q for the lower state has no effect, as the lower state has Σ symmetry and is not degenerate.) The energy level plot should now show the splitting in the upper state clearly. For a perfect fit you may need to consider centrifugal distortion of q and float q_D and possibly the higher terms. Note:

• In the residual window, the Q branch will show as having a different parity.
• The default highest J is 50, so you might need to increase Jmax slightly to simulate all the visible lines. This can be set at the state, manifold, molecule or species level, and defaults to 50 at the species level.
  

The file at this stage is available at N2Ostage3.pgo.

Saving the Line List to a Separate File.

As a final step, save the list of assigned line positions in a separate file; this allows more flexibility in fitting than is available from the line list window, and will be important for later. To do this, in the line list window click on "All" then "More", "Save to File","In Simple Format...". For the purposes of this exercise we will use the file name N2OIRv1.obs. This produces a text file, which is most conveniently edited with the Tabslave Editor, distributed with PGOPHER which has some special features to aid editing. (Any plain text file editor could be used, though not word processors as they can spoil the format.) The first few lines of the file are:

NQN 2
 1 e  2 e  587.09236   1  .00079  : N2O_0.001....dpt - N2Onu2.ovr
 2 e  3 e  586.25496   1  .00155  : N2O_0.001....dpt - N2Onu2.ovr
 3 e  4 e  585.41791   1  .00228  : N2O_0.001....dpt - N2Onu2.ovr

 J'p' J"p" Observed    σ  Comment
 1 e  2 e  587.09236   1  .00079  : N2O_0.001....dpt - N2Onu2.ovr

 P(2) 587.09236 1 .00079  : N2O_0.001....dpt - N2Onu2.ovr
 P(3) 586.25496 1 .00155  : N2O_0.001....dpt - N2Onu2.ovr
 P(4) 585.41791 1 .00228  : N2O_0.001....dpt - N2Onu2.ovr

The underlying energy level diagram can also be seen with "View", "Levels". The clearest plots are typically obtained by plotting E − B_effJ(J+1) versus J, rather than simply E v. J. B_eff is chosen empirically to give the clearest plots, usually close to the actual rotational constant. This "reduced" style of energy level plot makes deviations from the overall trend clearer. It can also plot the "observed" energy levels on the plot, calculated for the upper levels by taking the calculated energy for the lower levels and adding the observed energy, and vice-versa for the lower levels. This can reveal trends in errors (suggesting additional parameters are needed) and common difference failures. A good diagnostic for the current fit is obtained by:

• Setting "Subtract" to 0.42 cm⁻¹ (or whatever your B value is).
• Selecting "Linear: Species - LinearMolecule - Excited - v=1" at the top to plot just the upper vibrational state.
• Clicking on the line style button, under plot type, to show calculated energy levels as lines, rather than symbols.
  
• Clicking on the "Observed" buttons,  and , to plot the "observed" energy levels.
• For the current fit the "observed" values are so close to the calculated lines that the residual differences cannot be appreciated. They can be magnified by setting "Error mult" - try a value of 200 instead of the default of 1.
  

Combining data from more than one source is invaluable in improving the quality of fits. For the current example microwave data is available, so we will make use of that. The first step is to add the microwave spectrum to the simulation. All that is required is to add the relevant dipole moment to the model – the file constructed above only contains the v=1 – v=0 transition moment. The key steps are:

The tree of objects in the constants window (on the left) should then look like this:
The Simulation object contains parameters controlling the simulation, such as linewidth and temperature. The Ground and Excited objects are Manifolds, which contain one or more States (here v=0 and v=1). Manifolds can be used to group states together, with the only constraint on grouping that interacting states must be in the same manifold. For the simulation here, both states could be in the same manifold, though there is a slight speed advantage to keeping them separate. Along with the manifolds, Transition Moments objects (here <Excited|mu|Ground> and <Ground|mu|Ground>) contain one or more Transition Moment objects that link specific states. These are all under the higher level Molecule object; additional molecule objects could be added for isotopically substituted species, such as ¹⁵N¹⁴NO in this case. Additional Species objects could be added for completely unrelated molecules. For all these items, duplicate names under any given object should be avoided; they can be used when setting up the calculation in the first place but should be renamed when possible.

Pressing Simulate and then All should reveal a new set of pure rotational transitions at low frequency. In fact the microwave transitions are rather weaker than the infra-red transitions so they will not be visible on the same plot as the infra-red transitions. Zooming the plot into the 0-50 cm^-1 range should show them. As microwave transitions are usually reported in frequency rather than wavenumber you may want to switch the plot units to MHz – use “Plot”, Units”, “MHz”. (Any simulation widths set will automatically be converted, but note the line assignments generated as above are not.)

A microwave line list is available from the JPL spectral line catalogue at http://spec.jpl.nasa.gov/. Follow the “Catalog Directory with links” and look for entry 44004. The file c044004.cat is the line list in text format, and the first three lines look like:

Frequency/MHz FreqErr log10(I)DR   Elower gUP  Tag Qfmt Upper Qno   Lower Qno
   25123.2460  0.0004 -6.7155 3    0.0000  3 -440041303 1 0 0       0 0 0       
   50246.3700  0.0130 -5.8150 3    0.8380  5 -440041303 2 0 0       1 0 0       
   75369.2240  0.0100 -5.2911 3    2.5141  7 -440041303 3 0 0       2 0 0       

PGOPHER can understand and make use of .par files in various ways. To use it as a simple line list (i.e. as a list of frequency, intensity pairs) load it as an overlay. (Tip: to load at this point, without removing the previous overlay, right click on the .cat file in a file explorer and select “Copy” and then select “Overlays”, “Paste Another” in PGOPHER. Alternatively turn on “Merge” in the file menu before loading the overlay, and then turn it off.) Note you might need to adjust Jmax to reproduce the entire spectrum. This can be set at the state, manifold, molecule or species level, and defaults to 50 at the species level.

The .cat files have absolute intensities, in units of nm² MHz. To force the experimental overlay to use these units, make sure the intensity units in the Overlays window ("View", "Overlays") are set to nm2MHzperMolecule at both the top level and the c044004edit.cat object. To make PGOPHER use the same units, set the same intensity units in the Simulation object in the constants window. For quantitative agreement on intensities, the dipole must be set correctly. Helpfully this is provided in the JPL documentation associated with the .cat file. The value (0.16083 D) should be entered as the strength parameter in <v=0|T(I)|v=0> in the constants window.

The simulation and overlay of the microwave should look fairly similar at this point, but can be improved by using the line list for as input to a fit. This is (in principle) easy for .cat files, as PGOPHER can (in principle) take the assignments from the file without modification. To fit:

Fits should now work, though if you have been following the complete instructions above you will need to fix some now irrelevant parameters for sensible results. (The “Undo Fit” button might be needed here and, in the constants window, "Fix", "Fix All Parameters...", "Mixture".) The residuals window is, as before, essential in tracking the progress of the fit (“View”, “Residuals” if not already visible). Note that c440004.cat contains a wide range of observation weights (= standard deviation of the observations), so you may want to set the plot to “Unweighted Obs-Calc” to understand the fit in the early stages. It also contains a mixture of calculated and observed line positions, and PGOPHER will use only the observed line positions for fitting by default. The file provided seems sufficient to determine B, D, H and L for the ground state. In addition to make the text displayed in the log window useful you will need to increase the precision of the displayed values - the Precision setting at the top level should be increased to 7 or 8. (Click on the N2Ostage.pgo at the top of the tree to set this.)

The file at this stage, after fitting ground state rotational constants and dipole moment is available at N2Ostage4.pgo.

6. Fitting using multiple sources of data

To produce a best overall fit, using both the infra-red and microwave line positions, the lines are best placed in separate text files. The line list window is convenient for getting started in fitting, but for anything other than the simplest fits a setup along the lines described below is much more flexible. See List Input File Format for full details of the format of the observation files.

From the instructions above, we have two files, N2OIRv1.obs. with the infra-red data and c044004edit.cat. To combine these we need a control file with the following contents:

Colour Red
include N2OIRv1.obs
Colour Blue
include c044004edit.cat

Save this as N2Ocombined.obs and:

• Bring up the log window: “View”, “Log”.
• Drag and drop the N2Ocombined.obs file to the log window, or use the file open button () to select the file via the standard dialog. Either will set the fit file name in the top centre of the log window.
• Make sure all the parameters are fixed ("Fix", "Fix All Parameters...", "Mixture")
• Make sure the fit type in the top left is set to “Line”.
• In the residuals window, press the reload button ()

The Colour directives in the control file govern the colour of the points in the residuals window, and this makes it very clear that the relative estimated errors of the two data sets need adjustment:

The microwave data has weighted residuals, (Observed-Calculated)/Estimated Error, in the range of -3 to +3, which is as expected if the estimated errors correspond to one standard deviation, and indeed the .cat file has an individual uncertainty assigned to each line. The infra-red data has a much smaller range of residuals, suggesting estimated error is much too large, and indeed the default of 1 that we used above is much larger than the resolution of 0.001cm^-1. (Note that only the relative magnitudes of the estimated errors are important in fitting, so the default of one is normally adequate if the data is all from the same source.) We could edit the estimated errors in N2OIRv1.obs but it is easier to add an override in the control file, as in:

Colour Red
ScaleStdDev 0.001
include N2OIRv1.obs
ScaleStdDev
Colour Blue
include c044004edit.cat

The ScaleStdDev directive scales all subsequent estimated errors by the value given, or 1 if no value given. In fact just scaling by the linewidth still looks like too large an error:
and a better guide is the average error in the infra-red fits, 5×10^-5. Using ScaleStdDev 5e-5 gives:
The relative errors now look good, but there is clearly a systematic error in the infra-red data. We can now do a combined fit, floating everything we have floated before - B, D, H and L in the ground state, and the Origin, B, D, q and q_D in the excited state. A few fitting cycles gives:
suggesting a good overall fit. The files at this stage are N2Ostage5.pgo and N2Ocombined.obs.

7. More Advanced Fitting - Hot Bands

If you look carefully at the infra-red spectrum you can see two additional weak Q branches in the spectrum, at 579.36 cm^-1 and 588.98 cm^-1. (The expand vertical scale button, , is helpful in seeing the latter.) These are due to hot bands, i.e. transitions where the lower state is a vibrationally excited state, and in fact for these two branches the lower state is actually the upper state of the infra-red transition we have been working on so far. We expect P and R branches associated with these Q branches to be present also, but are going to be rather difficult to spot.

The v₂=2 state of N₂O will actually have two components with bending quantum number l = 0 and 2 corresponding to  Σ+ and Δ symmetry respectively. To set up the data file to include these perform the following steps in the constants window; we will start by just adding a single state.

1. Duplicate the upper (v₂=1) state of the infra-red transition by right clicking on "Excited" and selecting "Copy With Linked Items" then right click again and select "Paste".
2. Some renaming is called for - the names are arbitrary, but to reduce confusion right click on "Copy of Excited", select "Rename" and enter "Second". Right click on "v=1" below Second, select "Rename" and enter "v=2".
3. This has copied the constants from v=1 to v=2 which is a reasonable starting point for B but not the origin, which will be about twice that for v=1. Try 1176 for this - click on v=2 and enter this at Origin.
4. Pressing the simulate button and then "All" will show that the 2-0 overtone band is now visible in the spectrum. This is not the required hot band, which we fix by right clicking on <Second|mu|Ground>, selecting "Rename" and changing "Ground" to "Excited" in the resulting dialogue box.
   
5. This will not (yet) produce any transitions in the simulation. To make PGOPHER consider a manifold (such as Ground or Excited here) as an initial state in a transition the Initial flag must be set. Click on "Excited" and change Initial from False to True.
6. There should now be some hot bands in the simulation, but they are likely to be difficult to see as they are so weak. As a quick fix, set the vibrational temperature, Tvib, to 3000 K. (Click on the Simulation object in the constants window for this.) This allows a different effective temperature for the distribution over rotational and vibrational energy levels, which you may need to simulate spectra in some circumstances.
   
7. To distinguish spectra from multiple sources colour coding can help. Click on v=2 and change Colour to Navy. Clicking on the "Individual Spectra" button, , will then show the hot bands in blue. You will need to turn off the display of the overall spectrum with the "Show Sum" button, . Also try cycling through the various plot styles with the "Next Plot Style" button, .

Your plot should now look something like this (the vibrational temperature, Tvib has been reset to the default (-1) for this, implying the rotational and vibrational temperatures are the same).

The data file at this stage is available as N2Ostage6.pgo. The object tree at this stage looks like:
Note there are several possible transition moments that could be added to this picture, including <v=1|T(1)|v=1> under a <Excited|mu|Excited> transition moments object (which would give microwave transitions within v₂=1) and <v=2|T(1)|v=0> under a <Second|mu|Ground> object which, as we saw above, would simulate an overtone transition.

To fit the Q branch at 579.36 cm^-1:

1. Adjust the v=2 Origin so the Q branch hot band is in about the right place.
2. The state type will not be right - try both Σ+ and Δ for v=2. For the Δ state both q and q_D should be set to 0, as l doubling is typically too small to be observed. (To avoid confusion they should also be cleared for the Σ state, but only because they are ignored for a Σ state.) This is a good application for the mouse adjust function (F3) as only the upper state constants (v=2) need adjusting. Good values for the increments are 0.001 for the Origin and 1e-5 for B. In addition the way lines merge is important for the appearance of the spectrum here, so a width should be added to the simulated spectrum. Try setting the Gaussian width of the spectrum to 0.001 in the Gau box at the top of the main window.
3. The difference between Σ+ and Δ is fairly subtle - the band selected is actually Σ+,  and visually the difference is that the transition to the Δ has two branches, corresponding to the two different parities. Compare (Σ+, Origin=1168.1305, B=.41992 available as N2Ostage7.pgo):
   
   with (Δ, Origin=1168.1305, B=.41915):
   

Even given the assignment of the state symmetry, the assignment of individual lines is awkward as the numbering of the of the peaks is not easy to establish. This is made more difficult in this case, as the centrifugal distortion constant, D, is important in determining the appearance of the spectrum, and also the Q branch only nature of the spectrum means that common differences are not available. A possible approach is to assign the band head as the lowest J transition (Q(1) in this case), and then try a few different assignments.  The possible assignments will only differ in J, and this is easiest to change if the line list is in a text file. Try the following:

1. Bring up the line list window ("View", "LineList") and use "Clear" to empty it. (The assignments currently in this window have been saved above.)
2. Assign as many lines as possible, floating the upper state Origin, B and D to help this. It should be possible to assign 20 or so lines with relative J values correct even if you are not sure as to the absolute numbering, so the band head won't necessarily look right
3. Save the assignments in Branch format to a text file. (In the line list window use "More", "Save To File", "In Branch Format..." and save it to N2OIRv2SigmaBr.obs).
4. This file requires a little editing before it can be used for fitting; the upper and lower states must now be specified as the edits above means there is now a choice. Edit the file so it starts with:
   

   UpperManifold Second
   UpperState v=2
   LowerManifold Excited
   LowerState v=1
       Q(7)  579.3595            1 5.8582  : N2O_0.001....dpt - N2Onu2.ovr
       Q(8)  579.358             1 6.4287  : N2O_0.001....dpt - N2Onu2.ovr
       Q(9)  579.35695           1 6.9292  : N2O_0.001....dpt - N2Onu2.ovr
   

5. In the branch format it is easy to slide all the assignments along by one - simply change Q(7) to Q(8), Q(8) to Q(9) and so on. Alternatively the BranchExpr format format can be used by editing the file to look as follows:

   UpperManifold Second
   UpperState v=2
   LowerManifold Excited
   LowerState v=1
   BranchExpr Q(J)
     7 579.3595            1 5.8582  : N2O_0.001....dpt - N2Onu2.ovr
     8 579.358             1 6.4287  : N2O_0.001....dpt - N2Onu2.ovr
     9 579.35695           1 6.9292  : N2O_0.001....dpt - N2Onu2.ovr
   

   (Versions of PGOPHER after 9.0.116 can also save directly to this format from the line list window). To offset the assignment replace BranchExpr Q(J) with BranchExpr Q(J-1) or BranchExpr Q(J+1).
   
6. Using either of the methods above try a few different assignments, and look at the overall quality of the fit, and how well the band head is reproduced. An offset of 2 from the assignments given above gives the best fit to the band head:

8. Band Contour Fits

The spectrum above shows only partially resolved rotational structure, and it may be worth trying a contour fit in these circumstances. This fits directly to the observed spectrum, rather than to line positions. While it sounds easier (in that assignments are not required) a good starting point is required as peaks must typically be within half a linewidth of their true positions for the fit to succeed. For a spectrum like the one used here contour fits only make sense as a follow up to a standard line position fit. See Contour Fitting for a full description.

For the current spectrum we use a slice of the original spectrum at full resolution without averaging; this is available at N2Onu2Sigmaonly.ovr. (Fitting to the entire spectrum would not make sense.) Try the following steps:

1. In the overlays window (View, Overlays), click on the topmost item and ensure that "IntensityUnits" are set to "Normalized" in both objects.
2. In the constants window (View, Constants) and click on "Fix", "Fix All Parameters" and select "Mixture"  to ensure that no molecular constants are floated to start with.
3. The first parameters to be floated must be the scale and baseline of the overlayed spectrum. In the overlays window (View, Overlays), float Scale and Baseline (click in the float column so it reads “yes”).
4. In the log window (View, Log) choose "Contour" as the fit type in the top left. On pressing the "Fit" button the main window will change to show two plots. The upper plot shows the residual (the difference between the experimental spectrum and the simulation) is shown in the main window (see below for an example) and the lower plot the experimental and observed spectra simultaneously. After a few cycles it should look like:
   
   
5. This is actually quite good already, but try floating the linewidth (Gaussian under the Simulation object in the constants window), and Origin, B and D for the upper state. You will find that the fit requires some help to converge - the "Undo Fit" button, , is essential here. A specific problem here is that the default increments used for calculating derivatives for the least squares fit are too large. This is a fairly common problem, which is why the constants window allows setting an increment for each parameter. Derivatives are calculated by repeating the simulation with the parameter displaced by the increment, and taking the difference from the initial simulation. Try floating one extra parameter at a time, and if the fit diverges use undo fit to step backwards to sensible values and then experiment with the increment for the parameter. (Try 1e-8 for the Origin and 1e-13 for B). In this case a minor improvement in the simulation is possible:

The final file is available as N2Ostage9.pgo. The overlay values used were Baseline=0.02664, Scale=91699.

9. Adding Custom Parameters - The Other Hot Band

The Δ - Π hot band at 588.98 cm^-1 is left as mainly as an exercise for the reader, but some tips are in order:

1. There are two different branches (corresponding to e and f parities, showing as Qe(J) and Qf(J)) - turn on the Fortrat plot to see this. There is significant overlap between the two, so right clicking will often select more than one transition. If this occurs, consider deleting one or more of the lines before measurement. For initial fitting at low J the f transitions are stronger, so try fitting the f transitions only to start with.
2. There are several possible ways to do this; in fact I picked out a few f transitions and used the BranchExpr technique described above to shift the assignment. To add more transitions I then used the line list window as follows:
   1. Clear the line list window.
      
   2. Assign a few lines (Right click on the simulation, right click and drag on the experimental spectrum.)
   3. Add to the line list file by pressing "All", "More", "Copy Branch Table" in the line list window and using paste in the editor with the line list file open.
   4. Fit with the new data, and then repeat from step 1.
   The final line list file, N2OIRv2DeltaBrEx.obs then contains a mixture of formats for the observations.
3. Despite the earlier comment about l doubling being typically too small to see for Δ state the resolution here is sufficiently high to observe it, so q must be floated for the Δ state.
4. Even more unusually the centrifugal distortion of q is also determined. This is not included in the standard PGOPHER terms, but illustrates nicely how perturbations can be used to add non-standard terms in a general way. See Λ Doubling and perturbation objects for more details.

The final file is available at N2Ostage10.pgo includes this perturbation. The object tree looks like:
The two v₂=2 states, v=2 (the Σ+ state) and v=2Delta (the Δ state), have been put in the same manifold (Second). These states, which are close together in energy, are a typical case where one state might perturb the other so putting then in the same manifold makes sense. This would allow the addition of a Perturbation object to the manifold to model the interaction. No interactions are observed here, so the two states could equally be put in separate manifolds.

Left as an exercise for the reader is adding P and R branch transitions from the hot bands to the observed line list - there are some visible.

10. Using HITRAN data

Line lists from the HITRAN database (or line listings with the same format) are understood by PGOPHER and can be used in several different ways. At its simplest, the line positions and intensities can be overlaid as an "experimental" spectrum, but the files can also be used as input to line position or intensity fits (PGOPHER can use the quantum numbers in the file directly) or imported as an external source allowing more general application of many of the features of PGOPHER, such as energy level plots, in addition to standard simulations. See Using HITRAN files for further details.

The file N2OHITRAN2012v2edit.par was obtained from HITRAN, in this case by using HITRAN on the web at http://hitran.iao.ru/. Browse to “Molecules”, select Nitrous Oxide (molecule 4), select the ν2 band (0110 - 0000), hit “Simulate Spectrum” and then select “Numerical Representation” and save the resulting text. To trigger PGOPHER’s processing of HITRAN files the extension should be .par.

Alternatively, use the newer HITRAN online at http://hitran.org/. Select “Data Access”, “Line-by-Line”, “N2O”. In the “Select isotopologues” page select ¹⁴N¹⁶O only. The frequency ranges can be left at the default, but for the current exercise “Select vibrational bands” is required. This option is only available if you are logged in to the site, and “Select by band” is turned on in your profile. In this case select “Upper state” = 0 1¹ 0 (1), “Lower state” = 0 0⁰ 0 (1) and make sure the output format is selected as “.par (160 chars)”. Use the link to download the generated .par file.

This .par file can be used as an overlay - drag and drop the the .par file onto the PGOPHER window of use "File", “Load Overlay...”. The .par files have absolute intensities, in units of cm² cm^-1 Molecule^-1 at 296 K. To force the experimental overlay to use these units, make sure the intensity units in the Overlays window ("View", "Overlays") are set to cm2WavenumberperMolecule at both the top level and the N2OHITRAN2012v2edit.par object. To make PGOPHER use the same units, set the same intensity units in the Simulation object in the constants window.For the simulated intensities to be correct the transition dipole must be set correctly; this is the Strength parameter of <v=0|T(I)|v=0> in the constants window. Manual adjustment suggests a value of 0.04 D is required.

The .par file can also be used as a line list for fitting, in the same way as the .cat files described above. If you try that for the supplied files position fits suggests a systematic error of 0.0002 cm-1 at low J (about 1/3 of the linewidth, probably the result of a small calibration offset) increasing to higher J (suggesting HITRAN is derived from a larger range of J than here). You will also see a systematic difference in intensities, suggesting a Custom Transition Moment is required.

HITRAN files can also be imported into PGOPHER using “File, Import, HITRAN File....” The resulting file will produce simulations in the normal way, allowing the temperature and linewidth to be adjusted and right clicking on any transition will give the underlying quantum numbers. Note however, that two extra steps will normally be required to produce complete results:

1. If more than one isotopologue is present in the file the abundances for each will need to be entered manually. Importing the complete HITRAN file for N₂O  will produce several HITRANMolecule objects named 41, 42, 43, ... corresponding to the HITRAN numbers. 41 corresponds to the main isotopologue, and the others can be deleted or the Abundance set to the correct value in the Molecule object. (Strictly the abundance should be set to 0.990333 rather than the default of 1)
2. By default, the partition function is calculated by summing over the energy levels deduced from .par file. This can give a good starting point, but may not be correct. This is because relevant levels may have been excluded from the HITRAN line list and also the process of deducing the effective energy level pattern from the line list is not foolproof. (Any error this introduces will not affect relative intensities at a given temperature.) To correct this, specify the partition function as a table of values using an Interpolated Partition Function. To add this, right click on the HITRANMolecule object and select "Add New...", "Interpolated Partition Function" and then right click on the Interpolated Partition Function object, select "Load..." and then select a file with the required values. HITRAN online makes the requiered file available under "Documentation", "Isotopologues". Look for the link to q21.txt.

An exercise for the reader is to use this import process to verify the assignments above.