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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:

### 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.

- 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.

The following shows an example appropriate for a linear molecule:

CDstartNotes:

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

- 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.)

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`".

# 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