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 *T*_{m }, 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: *f*_{1,1},
*f*_{1,2}, . . . , *f* _{K,1}, *f*_{K,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 (*f*_{1} + *f*_{2}) or average* f *can be reliably determined and
practically any can reproduce
almost linear temperature dependence.

#### Example: *f*_{1} = *f*_{2}

The general formula for a chemical shift

can be rewritten as

for *f*_{k, 1} = *f*_{k, 2}
= *f*_{k} it reduces to

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

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