An alternative to the goodness of fit

Progress in detection of and correction for low-energy contamination

Contamination with low-energy radiation leads to an increased number of weighted residuals being larger in absolute terms than three standard uncertainties. For a Gaussian distribution, these rare events occur only in 0.27% of all cases, which is a small number for small- to medium-sized data sets. The correct detection of rare events - and an adequate correction procedure - thus relies crucially on correct standard uncertainties, which are often not available [Henn (2019), Crystallogr. Rev. 25, 83-156]. It is therefore advisable to use additional, more robust, metrics to complement the established ones. These metrics are developed here and applied to reference data sets from two different publications about low-energy contamination. Other systematic errors were found in the reference data sets. These errors compromise the correction procedures and may lead to under- or overcompensation. This can be demonstrated clearly with the new metrics. Empirical correction procedures generally may be compromised or bound to fail in the presence of other systematic errors. The following systematic errors, which were found in the reference data sets, need to be corrected for prior to application of the low-energy contamination correction procedure: signals of 2λ contamination, extinction, disorder, twinning, and too-large or too-low standard uncertainties (this list may not be complete). All five reference data sets of one publication show a common resolution-dependent systematic error of unknown origin. How this affects the correction procedure can be stated only after elimination of this error. The methodological improvements are verified with data published by other authors.

X-ray constrained wavefunctions based on Hirshfeld atoms. I. Method and review

The X-ray constrained wavefunction (XCW) procedure for obtaining an experimentally reconstructed wavefunction from X-ray diffraction data is reviewed. The two-center probability distribution model used to perform nuclear-position averaging in the original paper [Grimwood & Jayatilaka (2001). Acta Cryst. A57, 87-100] is carefully distinguished from the newer one-center probability distribution model. In the one-center model, Hirshfeld atoms are used, and the Hirshfeld atom based X-ray constrained wavefunction (HA-XCW) procedure is described for the first time, as well as its efficient implementation. In this context, the definition of the related X-ray wavefunction refinement (XWR) method is refined. The key halting problem for the XCW method - the procedure by which one determines when overfitting has occurred - is named and work on it reviewed.

On predicted R factors for dynamic structure crystallography

For the evaluation of data sets from dynamic structure crystallography, it may be helpful to predict expected R = I O N I O F F $R = {{{I_{ON}}} \over {{I_{OFF}}}}$ -based agreement factors from the observed intensities and their corresponding standard uncertainties with laser ON and with laser OFF. The predicted R factors serve three purposes: (i) they indicate, which data sets are suitable and promising for further evaluation, (ii) they give a reference R value for the case of absence of systematic errors in the data and (iii) they can be compared to the corresponding predicted F 2-based R factors. For point (ii) it is inevitable, that the standard uncertainties from the experiment are adequate, i.e. they should adequately describe the noise in the observed intensities and must not be systematically over- or under estimated for a part of the data or the whole data set. It may be this requirement, which is currently the largest obstacle for further progress in the field of dynamic structure crystallography.

Experimental charge-density studies: data reduction and model quality: the more the better?

In this review, recent developments concerning data and model quality in experimental charge-density investigations from a personal view-point are described. Data quality is not only achieved by the high resolution, high I/σ(I) values, low merging R values and high multiplicity. The quality of the innermost reflections especially is crucial for mapping the density distribution of the outermost valence electrons and can be monitored by (I/σ)^asymptotic. New detector technologies seem to be promising improvements. Empirical corrections to correct for low-energy contamination of mirror-focused X-ray data and for resolution- and temperature-dependent errors caused by factors such as thermal diffuse scattering are described. Shashlik-like residual density patterns can indicate the need for an anharmonic description of the thermal motion of individual atoms. The physical reliability of the derived model must be thoroughly analysed. The derived probability density functions for the mean-squared atomic vibrational displacements especially should have only small negative values. The treatment of H atoms has been improved by methods to estimate anisotropic thermal motion. For very high resolution data, the polarization of the core density cannot be neglected. Several tools to detect systematic errors are described. A validation tool is presented that easily detects when the refinement of additional parameters yields a real improvement in the model or simply overfits the given data. In all investigated structures, it is proved that the multipole parameters of atoms with a comparable chemical environment should be constrained to be identical. The use of restraints could be a promising alternative.