by Nikolai V. Shokhirev

Prof. F. Ann Walker Research Group, Department of Chemistry, University of Arizona, Tucson, Arizona 85721, USA

Temperature dependence fitting

Least Squares Method

Suppose the NMR spectrum consists of K lines and the temperature dependences are measured at Tm , m = 1,..., M . The number of experimental points is K * M . The system can be described in terms of the above theory with (2 K + 1) parameters: f1,1, f1,2, . . . , f K,1, fK,2, . If K * M > 2 K then the parameters in principle can be extracted. 

A common approach is the Least Squares Method. The  sum of squares of deviations of experimental values from theoretical values is minimized with respect to (2 K + 1) parameters: 

The optimal parameters when the sum reaches its minimum are used for the description of the molecular properties.

(see  Optimization Algorithm for details. The program TDFw implements this algorithm).

Accuracy and Resolution

In this approach the K additional experimental values are implicitly used: the diamagnetic chemical shifts (or chemical shifts at 1/T = 0). The uncertainty of diamagnetic shifts is difficult to to estimate. Moreover, these values are subtracted from all other experimental shifts and increase the errors. This is the disadvantage of the method. The advantage is the extension of the interval of measurements to 1/T = 0 or T = . Here we trade the accuracy for resolution: we can distinguish the dependences that could coincide (within the experimental accuracy) at a narrow interval. The low resolution in temperature dependencies leads to the low accuracy in   (The accuracy and resolution in indirect measurements is discussed here ).

The alternative variant is to include the diamagnetic shifts into the set of fitting parameters. It would be a better solution if the temperature interval is wide enough and the accuracy is high. Unfortunately it is not the case.

Regardless the solution method, both the accuracy and resolution of the reconstruction of parameters depends on the numerical values of the parameters. For example, if the values of the Curie factors are close then only the sum (f1 + f2) or average f can be reliably determined and practically any can reproduce almost linear temperature dependence. 

Example:  f1f2

The general formula for a chemical shift


can be rewritten as


for  fk, 1fk, 2fk  it reduces to

 The above formula displays pure Curie dependence and the energy gap cannot be determined.

©Nikolai V. Shokhirev, F. Ann Walker, 2002

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