Combination Difference Fitting

Observations containing an upper or lower level in common can be combined into a single observation corresponding to the difference in energy between the two levels in the other state. Fitting in this way can be helpful if one of the states is perturbed or unassigned. If using the linelist window fitting using this mode can be invoked by turning on advanced mode ("More" , "Advanced") and selecting "Upper State CD Fit" or "Lower State CD Fit" from the fit type drop down. When fitting in this way the observations are internally sorted by upper or lower state as appropriate, and the resulting list is then scanned, combing observations where possible. Any observations that cannot be combined are discarded.

For more control, put the observations in a separate file, rather than using the linelist window, and it is then probably easiest to work using directives in the input file as shown below, though CDlower or CDupper can be selected as the default mode for a file in the log window. It is also possible to manually construct the appropriate equivalent transitions within the state of interest, even if they do not correspond to allowed transitions; PGOPHER does not require fitted transitions to be allowed.

The CDupper or CDlower directive can be used for observations that are already sorted so that those with common levels are adjacent. The directive applies to all following observations until a directive changes the observation type. For example, using lines in branch format, the pair of observations input as follows:

CDlower
P(6) 992.1870 1
R(4) 1012.0070 1

will be transformed into a single observation of 1012.0070-992.1870 = 19.82 corresponding to the difference between J" = 6 and J" = 4 in the lower state. Any observations that cannot be combined causes an error.

If the observations are not sorted the AutoCDupper or AutoCDlower directives can be used to automatically sort the observations before looking for common differences. In this mode a CDstart directive marks the start of the observations to be considered for conversion to combination differences, and the conversion happens when the AutoCDupper or AutoCDlower is encountered in the file. There is an implicit CDstart directive before the first observation, and after any observation type directive that changes the observation type. The AutoCDupper or AutoCDlower directive can be followed by a flag that controls the handling of observations that are not part of an upper state combination difference:

• ignore - discard the observations. (This is the mode used if CD fit is invoked from the linelist window.)
  
• force (the default if the flag is not given) - report an error for each observation not included in a combination difference.
  
• include - include the observation in the fit as a normal transition, including the "other" state.

This mode offers less control than the CDupper or CDlower directives as the mapping between the observations input and those actually fit can be less direct.. For example, any duplicate observations (those with identical upper and lower states) are combined into a single observation before looking for combination differences.

The following shows an example appropriate for a linear molecule:

CDstart
Excited  8 e 1 Ground  7 e 1 3035.9776 1  R(7) : Excited v=1  8 e - Ground v=0  7 e
Excited  8 e 1 Ground  8 e 1 2871.5452 1  Q(8) : Excited v=1  8 f - Ground v=0  8 e
Excited  9 e 1 Ground  8 e 1 3052.5730 1  R(8) : Excited v=1  9 e - Ground v=0  8 e
Excited  9 e 1 Ground 10 e 1 2663.1901 1 P(10) : Excited v=1  9 e - Ground v=0 10 e
Excited 10 e 1 Ground 10 e 1 2863.9385 1 Q(10) : Excited v=1 10 f - Ground v=0 10 e
Excited 11 e 1 Ground 10 e 1 3084.2618 1 R(10) : Excited v=1 11 e - Ground v=0 10 e
AutoCDupper include

Notes:

• Upper state combination differences are used, requiring the lower state quantum numbers to be identical. The sample here has been sorted by lower state quantum number for clarity, but this is not required.
• The first line with an observation, R(7), is the only observation with J"=7, so cannot be combined into a combination difference. The include flag on the last line means that the observation will be included in the fit as a normal line position observation. Use AutoCDupper ignore to discard it.
  
• The next two lines, Q(8) and R(8) will be combined into a single observation.
  
• The next three lines, P(10), Q(10) and R(10) will be combined into two observations corresponding to J'=11 - J'=10 and  J'=10 - J'=9. (Note the Q(10) transition should be forbidden, but this does not affect the fitting.)

The output from the fit will be:

1 observation(s) not converted to common differences
Last unconverted observation at line 2 of test.lin

Residuals before fit for LinearMolecule
    J'S'  #'            J"S"  #"     Observed    Calculated      Obs-Calc StdDev  Correl
    8 e   1  Ground     7 e   1     3035.9776     3035.8171        0.1605 1.00000        R(7) : Excited v=1  8 e - Ground v=0  7 e  :   2:test.lin
    9 e   1 Excited     8 e   1      181.0278      180.5544        0.4734 1.41421        R(8)/Q(8) : Excited v=1  9 e - Excited v=1  8 f :  3:test.lin :   4:test.lin
   10 e   1 Excited     9 e   1      200.7484      200.6160        0.1324 1.41421 -.5000 Q(10)/P(10) : Excited v=1 10 f - Excited v=1  9 e :  5:test.lin :   6:test.lin
   11 e   1 Excited    10 e   1      220.3233      220.6776       -0.3543 1.41421        R(10)/Q(10) : Excited v=1 11 e - Excited v=1 10 f :  6:test.lin :   7:test.lin

4 Observations,  0 Parameters (scaled)
Average Error: 0.224679208483449  (Unweighted: 0.313426155635298)

The text on the end of each line shows which lines have been combined; 2:test.lin should be read as "line 2 of test.lin". Note that the upper manifold is not listed in the output, as it is always "Excited".

Estimated errors and correlations in the observations

When two observations are combined, the estimated standard deviation of the observation is derived from the standard deviations of the two observations, using standard error propagation methods. For two observations with standard deviation 1, the standard deviation of the difference will be sqrt(2). If three observations are combined into two common differences, as for P(10), Q(10) and R(10) in the example above, then the correlation between the observations must be considered to obtain the correct statistical analysis, and results in the -0.5 entry in the "Correl" column. See J. Tellinghuisen, J. Molec. Spectrosc. 221, 244 (2003) for a discussion of this point. (Note this paper also points out that only two differences should be used for fitting in this case; including the third difference will also produced incorrect statistics.)

Overall example

The sample input file below shows how several of these possibilities can be combined in a single file. The data has been generated for a simple linear molecule, using the parameters given in the paper mentioned above.

# Normal lines
Line
Excited  5 e 1 Ground  5 e 1 2879.9890 1 Q(5) : Excited v=1  5 f - Ground v=0  5 e
Excited  9 e 1 Ground  9 e 1 2867.9519 1 Q(9) : Excited v=1  9 f - Ground v=0  9 e

# Explicit common differences in correct order
CDupper # Applies to following lines, which reduce to three observations
Excited  1 e 1 Ground  2 e 1 2844.2656 1 P(2) : Excited v=1  1 e - Ground v=0  2 e
Excited  2 e 1 Ground  2 e 1 2884.8006 1 Q(2) : Excited v=1  2 f - Ground v=0  2 e
Excited  3 e 1 Ground  2 e 1 2945.5659 1 R(2) : Excited v=1  3 e - Ground v=0  2 e

Excited  8 e 1 Ground  9 e 1 2686.9224 1 P(9) : Excited v=1  8 e - Ground v=0  9 e
Excited 10 e 1 Ground  9 e 1 3068.6981 1 R(9) : Excited v=1 10 e - Ground v=0  9 e

CDlower # Applies to following lines, which reduce to four observations
Excited  4 e 1 Ground  3 e 1 2964.5797 1 R(3) : Excited v=1  4 e - Ground v=0  3 e
Excited  4 e 1 Ground  4 e 1 2881.9914 1 Q(4) : Excited v=1  4 f - Ground v=0  4 e
Excited  4 e 1 Ground  5 e 1 2778.8584 1 P(5) : Excited v=1  4 e - Ground v=0  5 e

Excited  7 e 1 Ground  6 e 1 3018.8444 1 R(6) : Excited v=1  7 e - Ground v=0  6 e
Excited  7 e 1 Ground  8 e 1 2710.3938 1 P(8) : Excited v=1  7 e - Ground v=0  8 e
Excited  7 e 1 Ground  7 e 1 2874.7888 1 Q(7) : Excited v=1  7 f - Ground v=0  7 e

# Automatic common differences
CDstart # Start of block
# Note first line not a common difference
Excited  1 e 1 Ground  1 e 1 2885.6067 1 Q(1) : Excited v=1  1 f - Ground v=0  1 e
Excited  2 e 1 Ground  1 e 1 2926.1472 1 R(1) : Excited v=1  2 e - Ground v=0  1 e
Excited  2 e 1 Ground  3 e 1 2822.8136 1 P(3) : Excited v=1  2 e - Ground v=0  3 e
Excited  3 e 1 Ground  3 e 1 2883.5772 1 Q(3) : Excited v=1  3 f - Ground v=0  3 e
Excited  3 e 1 Ground  4 e 1 2801.0182 1 P(4) : Excited v=1  3 e - Ground v=0  4 e
Excited  5 e 1 Ground  4 e 1 2983.1304 1 R(4) : Excited v=1  5 e - Ground v=0  4 e
Excited  6 e 1 Ground  5 e 1 3001.2369 1 R(5) : Excited v=1  6 e - Ground v=0  5 e
Excited  5 e 1 Ground  6 e 1 2756.3627 1 P(6) : Excited v=1  5 e - Ground v=0  6 e
Excited  6 e 1 Ground  6 e 1 2877.5749 1 Q(6) : Excited v=1  6 f - Ground v=0  6 e
Excited  6 e 1 Ground  7 e 1 2733.5398 1 P(7) : Excited v=1  6 e - Ground v=0  7 e
AutoCDlower ignore # Applies to preceding lines; implicit block start

CDstart # Optional here, as preceding AutoCDlower forced block start
# Note first line not a common difference
Excited  8 e 1 Ground  7 e 1 3035.9776 1  R(7) : Excited v=1  8 e - Ground v=0  7 e
Excited  8 e 1 Ground  8 e 1 2871.5452 1  Q(8) : Excited v=1  8 f - Ground v=0  8 e
Excited  9 e 1 Ground  8 e 1 3052.5730 1  R(8) : Excited v=1  9 e - Ground v=0  8 e
Excited  9 e 1 Ground 10 e 1 2663.1901 1 P(10) : Excited v=1  9 e - Ground v=0 10 e
Excited 10 e 1 Ground 10 e 1 2863.9385 1 Q(10) : Excited v=1 10 f - Ground v=0 10 e
Excited 11 e 1 Ground 10 e 1 3084.2618 1 R(10) : Excited v=1 11 e - Ground v=0 10 e
Excited 10 e 1 Ground 11 e 1 2639.1611 1 P(11) : Excited v=1 10 e - Ground v=0 11 e
Excited 11 e 1 Ground 11 e 1 2859.4863 1 Q(11) : Excited v=1 11 f - Ground v=0 11 e
Excited 12 e 1 Ground 11 e 1 3099.2950 1 R(11) : Excited v=1 12 e - Ground v=0 11 e
AutoCDupper include

Line
Excited  1 e 1 Ground  0 e 1 2906.2757 1 R(0) : Excited v=1  1 e - Ground v=0  0 e