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About Curve Fitting

A processed NMR spectrum is nothing but an array of points. We prefer interpreting it as a sum of peaks (belonging either to the main compound or to impurities), baseline distortions, artifacts and noise. A few of these components are interesting, the other ones are just causing troubles. In particular, we need to extract two parameters from the good peaks: the frequency and the area (in studies of relaxation the line width is important too).
When the spectrum is crowded with peaks, or affected by excessive noise, curve-fitting may be the only way to measure the above quantities. Beware that it makes no miracles, it is not a substitute for higher magnetic fields, more abundant samples or prolonged acquisitions. The name “deconvolution” means (more or less): “removing the shape”.

theoretical shapes

The theoretical shape of an NMR peak is Lorentzian. Field inhomogeneity and weighting functions can however yield partial Gaussian line-shapes. From a merely graphical (and practical) point of view, they are two complementary shapes, in the sense that most symmetrical peaks can be approximated with suitable combinations of Lorentzian and Gaussian components. For your convenience, iNMR defines a mixed function, for which you can vary at will the percentages of Lorentzian and Gaussian character.

requisites

Not only iNMR makes the assumption that all lines can be described as Lorentzian (or Gaussian, or anything in between), it also assumes that the peaks are perfectly symmetric (perfectly phase-corrected) and the baseline is flat (completely removed). The principle in action is that, for processing to be effective, you must factorize it in many steps. Curve-fitting arrives as the last step, assuming that everything before has been performed correctly. Everything also includes properly shimming the magnet!

battlefield

To start a deconvolution, first select, with the mouse, a small region of a 1D, processed, spectrum. Ideally the region should contain an isolated peak, or an isolated multiplet, or an isolated ensemble of overlapping peaks. In all cases, the region must contain at least 5 points (for each peak to calculate). The command Simulate > Deconvolution creates a new window, where the selected region has been copied. It is drawn in black. Superimposed are some components (peak shapes) proposed by iNMR, drawn in green. They constitute a mere starting point; you can add/remove components and change their parameters, either directly or using the automatic facility. The deconvolution window can be expanded and can be as large as the screen. You cannot, however, use the usual iNMR tools and commands. Think at this module like another application, running in its own window, with its own commands. You can create how many deconvolution windows you will and alternate from them to document windows.

plot area

The experimental spectrum is drawn in black, while the synthetic spectrum is green. The latter can be displayed in two modes: either decomposed into the individual peaks or as their sum. The toolbar command “toggle” lets you switch between the two modes. In both cases, a single peak is always selected. You can see four square handles around the selected peak. The bottom handle defines the position of the peak along the frequency scale, the top handle defines the height, the side handles define the width. If you drag the handles, the curve changes. With the bottom handle you can also move the plot, vertically, inside the window. To select another peak, click near to it.
When the individual components are on display, the selected peak is in red. When the total is on display, the red line represents instead the difference between the experimental and the calculated spectrum. The number called “residual error” is a synthetic measure of this difference spectrum. You can tell which mode you are in by the presence or the absence of the “residual error“ label. It comes together with the total/difference mode, it disappears with the individual components mode.

table

On top of the plot you can see the list of parameters. Each line of the table corresponds to a single peak. To select a peak, click on its line or use the arrow keys (up and down) of the keyboard. When the list becomes long, a scroll bar appears at its right. Note that the table contains the “area” parameter, not the height of the peak. What it means becomes clear when you try changing the width: the height will simultaneously change, and the area will remain constant. Compare this behavior to what happens when you drag the width handles. In this case, the height remains constant and the area changes.
You can change the numerical values either with the keyboard or with the small arrows under the list. Note that not all values are accepted. For example, you cannot enter more than 100 in the percentage field! (Each field has its own limits). The check buttons determine which parameters can change when you click the “FIT” icon. If the parameter is not checked, its value is kept constant. If no parameter is checked, the FIT command is disabled.
Peaks are constantly sorted by their frequencies. You can invert the sorting order by clicking the header of the frequency column. The index that appears at the left has no importance. When you issue the “copy” or the “export” commands, peaks are sorted again and the indices may change.

numerical values for the intensities and the residual error

The intensity values shown by the deconvolution window are comparable with the integrals of the experimental spectrum and both are normalized in the same way. In practice, when you cut an integral in the experimental spectrum, if there aren't others already defined, that first integral corresponds to the value of 1 (the unit of integration). You can redefine the unit value with a double click. When a document is created, or when all the integral regions are removed, the normalization factor does not exist. When you create a deconvolution window, it copies the normalization factor, if it exists. If not, the unit is calculated as 1/100 of the area of the selected region. Afterwards there is no connection at all between the two windows. If you redefine the normalization factor in the document, the integrals are not comparable.
You might have realized that the best practice is: first complete the standard integration, then start with curve-fitting!

The residual error is expressed in the same unit.
It is calculated as the square root of the sum of the squares of the points in the difference spectrum:
		residual error = √ Σ (Experimentali - Calculatedi)2.

automatic constrains

At the end of each fitting run (not during it), iNMR checks if the recalculated values are within the expected limits. For example, all intensities must be positive and all percentages within 0 and 100. If they are not, the value is corrected and the check mark removed. It is advisable that the user clicks another time the “FIT” icon. The residual error can decrease further.
At the beginning of each run, the height of each peak is also looked at. If it is too low, that peak is kept constant and non recalculated. All check marks are removed from the corresponding line in the table.
Even during manual fitting, iNMR sets limits to all parameters.

strategies

If you don't know the shape of your peaks, you can let iNMR calculate the Lorentzian percentage. After the first run, if you notice that all percentages are above 90, you'd better set them all at 100, and keep them constant.

Generally speaking, all parameters should be recalculated simultaneously. In this way, however, a dilution effect is created. Let's say that all parameters are OK but one. If you adjust that wrong parameter, you can get a 10% improvement in the reconstruction of the corresponding peak but only a 1% improvement on the whole spectrum. iNMR can conclude that it's not worth to further refine 80 parameters only to get a 1% improvement, and simply stops the fitting at the current stage. In such cases, you should leave a single check mark, on the single wrong parameter, and start a new fitting run.

For exactly the same reason, avoid selecting a wide region if it can be dissected in smaller ones.

The fitting algorithm implemented is good at finding the nearest minimum, (and it must be really near!), not the absolute minimum. Spend some time in a preliminary manual fit!

To simulate a shoulder, you can define an additional (weak) peak. By adding peaks, you might eventually simulate any positive shape. The big problem is that the fitting algorithm often removes these small peaks.

Every now and then, copy the list of parameters and save it somewhere. If you get lost, you can always paste the list back into the table.

Related Topics

Curve Fitting Commands

Measuring the Area by Fitting to a Model

 

Web Tutorial

Line Fitting Primer