About systematic errors in charge-density studies

Abstract

The formerly introduced theoretical R values [Henn & Schönleber (2013). Acta Cryst. A69, 549-558] are used to develop a relative indicator of systematic errors in model refinements, R(meta), and applied to published charge-density data. The counter of R(meta) gives an absolute measure of systematic errors in percentage points. The residuals (Io - Ic)/σ(Io) of published data are examined. It is found that most published models correspond to residual distributions that are not consistent with the assumption of a Gaussian distribution. The consistency with a Gaussian distribution, however, is important, as the model parameter estimates and their standard uncertainties from a least-squares procedure are valid only under this assumption. The effect of correlations introduced by the structure model is briefly discussed with the help of artificial data and discarded as a source of serious correlations in the examined example. Intensity and significance cutoffs applied in the refinement procedure are found to be mechanisms preventing residual distributions from becoming Gaussian. Model refinements against artificial data yield zero or close-to-zero values for R(meta) when the data are not truncated and small negative values in the case of application of a moderate cutoff Io > 0. It is well known from the literature that the application of cutoff values leads to model bias [Hirshfeld & Rabinovich (1973). Acta Cryst. A29, 510-513].

An alternative to the goodness of fit

“…We chose data by Zhurov et al (2011), because in our previous studies these proved to be the most accurate data [see data sets 8-13 in Henn & Meindl (2014a), in Henn & Meindl (2015b) and in Henn & Meindl (2014b) as well as paragraph 4.4 and Fig. 4 in Henn & Meindl (2015a) and the supporting information].…”

“…The parameter values a, b, f and the total number of parameters in the refinement, N par , are also given. The data set numbering scheme from Henn & Meindl (2014a) was used in column one to identify the data sets consistently with Henn & Meindl (2014b, 2015a Remaining errors -if present -originate as a consequence only from the experimental standard uncertainties. Table 2 shows the corresponding charge-density data.…”

More about systematic errors in charge-density studies

“…In this section the meta residual value is discussed for the experimental data sets (1-23) that were introduced with references in Henn & Meindl (2014) together with the artifi-cial data sets (24-29) that correspond to refinements with (set Nos. 25,27,29) and without (set Nos.…”

“…In two preceding publications, the theoretical R value (Henn & Schö nleber, 2013) and a meta residual factor R meta (Henn & Meindl, 2014), which employs the theoretical R value, were developed. The concept of a theoretical R value and a meta residual factor is completely general and applicable to all fields where least-squares fits are conducted.…”

“…1 Application to crystallographic standard structures and high-resolution charge-density studies revealed that residual distributions reminiscent of a Gaussian only rarely appear. The application of the meta residual factor in Henn & Meindl (2014) focused on the experimental standard uncertainties (s.u. 's) of the crystallographic data, i.e.…”