Multiple State Fitting Example - The A-X transition in C₃

If you have trouble setting this up, the settings are saved in multc3fitinital.pgo.This is a plot of energy against J, but with 0.408J(J+1) subtracted from the energy, so simple states will give horizontal lines. The two excited states are clearly visible on this plot as the green and blue horizontal lines, and the circles give the observed energy levels calculated from the line list by adding the calculated lower state energy. The vertical lines join the calculated to the observed energy levels, and it is clear from this that about half some of the assignments are to the wrong energy level. This is because the line list constructed specifies states in terms of J, symmetry (+ or -) and eigenvalue number, and we have just added a state below the previous one, increasing the eigenvalue number. There are various ways of dealing with this and we work through several possibilities here, all of which work for this case, but may not in more complicated cases.

Different Manifolds

The first possibility is to put the new state in a different manifold from the initial Π state. The original line list will then work, as the manifold is specified for each state. The disadvantage of this is that perturbations between states can't be simulated, and states close together in energy as these are may well interact. The required set in is shown in the right hand side below, with the original set up on the left:

Setup with two states (Sigma, Pi) in a single excited state manifold (A)	Setup with each state in its own manifold (A for Pi, Asigma for Sigma)

Converting between the two is straightforward:

• Left to right: right click on A, select "Copy with Linked Items", right click on A again and select "Paste" to duplicate the upper manifold. Compete the process by deleting the one state from each of the excited state manifolds and renaming as required.
• Right to left: right click on Sigma, select "Copy", right click on A and select "Paste" to copy the Sigma state to the A manifold. Then right click on <Sigma|T(1)|v=0>, select "Copy", right click on <A|mu|X> and select "Paste" to copy the transition moment. The Asigma manifold can then be deleted.

Note that a Transition Moment from the ground state is required for both states (here <Pi|T(1)|v=0> and <Sigma|T(1)|v=0>), and a separate Transition Moments object (here <A|mu|X>and <Asigma|mu|X>) is required for each connected pair of manifolds. When the name of a manifold is changed the line list will need adjustment using one of the methods described below.

Using a separate line list file

For the most flexible handling, the line list should be put in a separate text file and edited with a separate text editor, the The Slave Editor included with PGOPHER or other text editor of your choice. Use of a separate editor is more flexible than editing in the line list window (undo/redo, search and replace), and it is easy to combine separate sources of data into a combined fit. To generate a file use "More, Save to File" in the Line List Window, and then use the Log Window to select the file and fit.

State Renumbering

The first option in fixing the line list is simply to change the eigenvalue number that are wrong in the new simulation set up. In the line list window this will be the #' column, and in this case the values for the odd J levels need increasing by 1. This change could be done manually, but a further change may need making if a state is added or removed, so we use the indexoffsets directive. To use this we put the line list in a separate file. Select "More, Save to File, In Standard Format" in the Line List Window window and choose a suitable filename, say APi.lin. Now go to the Log Window, check than the file you have just created is shown and press "Edit" to edit the file. To add the required offsets add:

upperindexoffsets 1 0

at the start of the file and then save the changed file. (If you use the the Slave Editor included with PGOPHER this step is not necessary as it a save is forced on each PGOPHER fit.) You can check this is right by hitting the reload button in the Energy Level Plot Window and the upper state should now show no long vertical lines, indicating a correct assignment:

State Searching

As an alternative the quantum numbers given in the comments at the end of each line can also be used to work out the eigenvalue number . In the Line List Window this can be forced for an individual line by clearing the eigenvalue number (#' and #") columns or in a line list file by setting the eigenvalue number to 0. The symmetry (S', S") can also be similarly left unset, though a negative value, rather than 0 must be used in the line list file to flag this. To force this for the entire file use:

upperindexoffsets search

to work out the eigenvalue numbers or

upperindexoffsets searchall

to work out both symmetry and eigenvalue number. Alternatively:

UpperState Pi

can be used to indicate that the eigenvalue numbers are to be taken within the state. Any of these methods will work fine in many cases, though where the quantum numbers are ambiguous, typically in the presence of strong state mixing, the quantum numbers assigned can change as the parameters change. In fitting the quantum number assignments for any given state can flip between successive fit cycles.

Notes on Standard Line List Format

A few lines of the line list file are shown below; see Line List Input File Format for a detailed explanation of each field.

LinearMolecule   A   1  1  1   X   0  0  1    27179.774 1 .03172      0 :     R(0) : A Pi  1 e - X v=0  0 e : LDS751.dr b - C3AX.ovr
LinearMolecule   A   3  1  1   X   2  0  1    27181.277 1 .05494      0 :     R(2) : A Pi  3 e - X v=0  2 e : LDS751.dr b - C3AX.ovr
LinearMolecule   A   5  1  1   X   4  0  1    27182.625 1  .0589      0 :     R(4) : A Pi  5 e - X v=0  4 e : LDS751.dr b - C3AX.ovr
...

Notes on this file:

• Most of the values after the "Std Dev" column, the last column of all 1's, are not used in fitting but are just shown in the fit log. The column giving the conventional quantum numbers ("A Pi 1 e - X v=0 0 e" is only used if state searching is invoked as described above. The column headed .03172 contains the line intensities from the simulation for information, but is never used in the fit.
  
• The layout is not critical except where the simple quantum numbers on the end are used (e.g. "A Pi 1 e - X v=0 0 e")
• The molecule and manifold fields are optional; if it is not clear which should be used then defaultmolecule, uppermanifold or lowermanifold directives can be used:

UpperManifold A
1  1  1  0  0  1 27179.774 1
3  1  1  2  0  1 27181.277 1
5  1  1  4  0  1 27182.625 1
...

Branch Format

An alternative way of specifying transitions is to use traditional branch notation, such as P(3), implying J' = 2, J" = 3. Such a file can be generated from the Line List Window using "More, Save to File, In Branch Format". The resulting file (in APibranch.lin) looks like this:

UpperState Pi
R(0)    27179.774 1        
R(2)    27181.277 1        
R(4)    27182.625 1        
...

For clarity, the intensity and comment fields have been removed. The "UpperState Pi" is required to identify which of the two possible upper vibronic states is required. defaultmolecule, uppermanifold, lowermanifold or lowerstate directives might also be required if there is more than one possibility for each.

Simple Format

It is also possible to use the conventional quantum numbers to specify the transitions. Such a file can be generated from the Line List Window using "More, Save to File, In Simple Format". The resulting file (in APisimple.lin) looks like this:

NQN 2
UpperState Pi
 1 e  0 e    27179.774 1        
 3 e  2 e    27181.277 1        
 5 e  4 e    27182.625 1       
...

For clarity, the intensity fields have been removed. The "UpperState Pi" is required to identify which of the two possible upper vibronic states is required. defaultmolecule, uppermanifold, lowermanifold or lowerstate directives might also be required if there is more than one possibility for each. The quantum numbers here are J' parity' J" parity"; see  Simple Format for the possible formats of the quantum numbers. Note than NQN, the number of quantum numbers present for each state, must be specified.

Multiple State Fit

With the Π state line list saved to a separate file, the line list window can be cleared and an independent fit of the Σ-Σ transition can be performed using the standard method. If trying this for yourself, you will need to fix all the Π state parameters, and remove q from the Σ state. When right clicking on the simulation you may end up selecting both Σ and Π state lines; you will have to check for any Π state lines manually and delete them. The resulting file is available in multc3fitsigma.pgo. A fit to both states simultaneously can now be done by saving the line list for the Σ state to a separate file in, say simple format (ASigma.lin) looking like this:

NQN 2
 1 e  2 e    27164.765 1        
 3 e  4 e    27163.026 1        
...

To use this file with one of the previous Π state files set up a separate file, say Aboth.lin, with the following format:

Colour Green
Include APisimple.lin

Colour Navy
UpperState Sigma
Include ASigma.lin

The include statements simply read the contents of the given file. All the information could be put in one file, or split in any other way as convenient. The colour directives set the colour of the points in the Residuals Window and the indicating marks at the top of the main window. The final fit () gives an excellent overall simulation: