Section 15.2.6. Estimation of overall coordinate error

In principle, since the distribution of observed and calculated amplitudes is determined largely by the coordinate errors of the model, one can determine whether a particular coordinate-error distribution is consistent with the amplitudes. Unfortunately, it turns out that the coordinate errors cannot be deduced unambiguously, because many distributions of coordinate errors are consistent with a particular distribution of amplitudes (Read, 1990).

If the simplifying assumption is made that all the atoms are subject to a single error distribution, then the parameter D (and thus the related parameter ) varies with resolution as the Fourier transform of the error distribution, as discussed above. Two related methods to estimate overall coordinate error are based on the even more specific assumption that the coordinate-error distribution is Gaussian: the Luzzati plot (Luzzati, 1952) and the plot (Read, 1986). Unfortunately, the central assumption is not justified; atoms that scatter more strongly (heavier atoms or atoms with lower B factors) tend to have smaller coordinate errors than weakly scattering atoms. The proportion of the structure factor contributed by well ordered atoms increases at high resolution, so that the structure factors agree better at high resolution than if there were a single error distribution.

It is often stated, optimistically, that the Luzzati plot provides an upper bound to the coordinate error, because the observation errors in have been ignored. This is misleading, because there are other effects that cause the Luzzati and plots to give underestimates (Read, 1990). Chief among these are the correlation of errors and scattering power and the overfitting of the amplitudes in structure refinement (discussed below). These estimates of overall coordinate error should not be interpreted too literally; at best, they provide a comparative measure.