Once the NOESY spectrum and its gradient are calculated, the relaxation energy can be expressed as a function of the difference between (functions of) observed and calculated intensities, and analytic derivatives with respect to atomic coordinates can be readily obtained by using the chain rule
where is the energy constant for the relaxation term, and are respectively the calculated and observed intensities, is an error estimate for , is a weight factor, is the calibration factor for each spectrum, and is the number of cross peaks in each spectrum. The function is defined as the absolute value of the difference between the nth powers of a and b, where b has an error estimate :
The individual error estimates reflect the errors in the peak volumes, usually subjective estimates, especially due to noise and spectral overlap. At present, the error estimates are also used to ascertain if a measured intensity is to be used in the determination of the overall calibration factor.
Values for the exponents of and m=2 (Eqs. 21.9 and 21.10) correspond to the refinement of the residual in X-ray crystallography. These values tend to put a high weight on the large intensities, resulting in a bad fit of intensities for which the calculated value is too small. Following a suggestion by James et al. (1991), use m=2 and . A value of m=1 results in the refinement of the R value directly, instead of the residual. The discontinuity of the gradient may lead to instabilities during the refinement.
In addition to the overall weight , individual weights can be applied to each term in the sum in Eqs. 21.9 and 21.12, e.g., in order to increase the relative weight of the small intensities. (It should be noted that this is achieved already by setting in Eq. 21.9.) The scheme corresponds to a common weighting scheme used in crystallography if experimental values are unreliable or unavailable. In the NMR case, however, there is no theoretical justification for this weighting scheme. (In crystallography, the statistical error of an intensity measurement is .) The weights are scaled such that .
The calibration factor between observed and back-calculated intensities is determined simply as the ratio of the sums of all calculated and observed intensities:
This ratio can be determined separately for each spectrum, or overall for all data points. Volumes that are not very reliable (i.e., they have a large error estimate) can be excluded from the ratio. The calibration factor can be updated automatically at every step. For technical reasons, this does not allow the calculation of the exact derivatives at present, so the calibration should not be updated automatically at every step during conjugate gradient minimization. During annealing, the effects due to the error in the gradient are negligible.