Fitting of the correction function

The program allows to correct the predicted wavenumbers WnCalc by adding a correction function n_corr, which can be formally regarded as a contribution to the upper state energy

Because all quantum numbers except J are fixed in this expression, the correction function depends only on the upper state J’ quantum number.  It is assumed to have the form

where the terms with half-integer powers can account for resonance effects that cannot be described well by perturbation theory, to which the part of n_corr with the a_i coefficients corresponds. All the coefficients of the correction function can be adjusted in a least-squares fit to minimize the residuals dn(J’). One correction function is common to all branches, which share the same series of upper state levels, i.e. those with the same values of quantum number that are kept fixed. These are also branches whose Loomis-Wood diagrams are related by common LSCD’s and which can be opened simultaneously. If the correction function is not yet optimized, the residuals dn (J’) in all related branches should be more or less the same and thus it is possible to correct them in all these branches by fitting the common correction function.

The fit of coefficients is performed on the assigned transitions of the active branch. It should be noted that for fitting, at least one Loomis-Wood window has to be opened. Because the LW program blocks opening Loomis-Wood windows for branches not belonging to the common series of upper state levels, this feature prevents fitting of correction function to branches that are not related to the common series of upper state levels.

The fitting procedure can be called in several ways

• Clicking the FIT button in the Branches window
• Selecting the Fit Correction Function item in the Branches menu
• Pressing the hot key Ctrl+F

This opens the Fit Window. The selection of parameters to be fitted is done by clicking on the check-boxes beside the parameter labels. The fit itself is done by clicking on the Fit parameters button. Clicking on the Undo button returns the parameters to values from before the fit. Fitting of the correction function minimizes the residuals, which graphically corresponds to aligning the triangle symbols of peaks in the Loomis-Wood window along the central vertical line. The fit reflects the weights of assigned transitions, which are calculated as inverse of squares of the corresponding accuracies. The standard deviation of the fit in individual branches is displayed for all branches, for which the LW Windows are opened, but the fit itself is performed only on the data from the active branch. It is obvious that if the branches are correctly related by LSCD’s, the fit performed on any of them should minimize standard deviations of all branches in a consistent manner. The errors of parameters are displayed in the yellow cells right from the Value cells.

It is sometimes convenient, especially when the patterns in the LW diagrams are not apparent, to change the values of parameters manually and search for emerging patterns. Besides editing the numbers in corresponding Value cells of the Fit window, a stepwise change function can be invoked by clicking on the toggle button Step> which changes the meaning (and color) of the Error cell to cell setting the Step. The value of step for each parameter can be then edited in each Step cell and the parameter changed by this step by clicking the Up/Down buttons between the Value and Step cells. After any such change of parameter(s) the new correction function is calculated by clicking the Calculate button.