Anharmonic motion in experimental charge density investigations.

In the charge density study of 9-diphenylthiophosphinoylanthracene the thermal motion of several atoms needed an anharmonic description via Gram-Charlier coefficients even for data collected at 15 K. As several data sets at different temperatures were measured, this anharmonic model could be proved to be superior to a disorder model. Refinements against theoretical data showed the resemblance of an anharmonic model and a disorder model with two positions very close to each other (~0.2 Å), whereas these two models could be clearly distinguished if the second position is 0.5 Å apart. The refined multipole parameters were distorted when the anharmonic motion was not properly refined. Therefore, this study reveals the importance of detecting and properly handling anharmonic motion. Unrefined anharmonic motion leads to typical shashlik-like residual density patterns. Therefore, careful analysis of the residual density and the derived probability density function after the refinement of the Gram-Charlier coefficients proved to be the most useful tools to indicate the presence of anharmonic motion.

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.

High yield access to silylene RSiCl (R = PhC(NtBu)2) and its reactivity toward alkyne: synthesis of stable disilacyclobutene.

Two new approaches for synthesizing RSiCl, (R = PhC(NtBu)(2)) are reported by the reaction of RSiHCl(2) with bis-trimethyl silyl lithium amide and N-heterocyclic carbene respectively. In the former method silylene is produced in 90% yield. The silylene was treated with biphenyl alkyne to afford the disilacyclobutene system. This is a rare example of two five-coordinate silicon centers arranged adjacent to each other in a four-membered ring. Furthermore, we fluorinated the four-membered ring by trimethyltin fluoride to obtain the fluoro substituted disilacyclobutene.

Lewis-base-stabilized dichlorosilylene: a two-electron sigma-donor ligand.

The first structurally described cobalt(I) Lewis-base-stabilized silylene complex [Co(CO)(3){SiCl(2)(IPr)}(2)](+)[CoCl(3)(THF)](-) [1; IPr = 1,3-bis(2,6-diisopropylphenyl)imidazol-2-ylidene] was prepared by applying the two-electron sigma-donor ligand SiCl(2)(IPr) through coordination with Co(2)(CO)(8). The bonding situation between ligand SiCl(2)(IPr) and the cobalt(I) metal center in [Co(CO)(3){SiCl(2)(IPr)}(2)](+) of 1 was investigated by (1)H NMR and IR spectroscopy, single-crystal X-ray structural analysis, and density functional theoretical calculations.

Electron densities of three B12 vitamins.

The electron densities of the three natural B(12)-vitamins, two of them being essential cofactors for animal life, were determined in a procedure combining high-order X-ray data collection at low to very low temperatures with high-level density functional calculations. In a series of extensive experimental attempts, a high-order data set of adenosylcobalamin (AdoCbl) could be collected to a resolution of sin theta/lambda = 1.00 A(-1) at 25 K. This modification contains only minor disorder at the solvent bulk. For methylcobalamin (MeCbl), only a severely disordered modification was found (sin theta/lambda = 1.00 A(-1), 100 K, measured with synchrotron radiation). The already published data set of cyanocobalamin (CNCbl) (sin theta/lambda = 1.25 A(-1), 100 K) was reintegrated to guarantee similar treatment of the three compounds and cut to sin theta/lambda = 1.11 A(-1) to obtain a higher degree of completeness and redundancy. On the basis of these accurate experimental geometries of AdoCbl, MeCbl, and CNCbl, state-of-the-art density functional calculations, single-point calculations, and geometry optimizations were performed on model compounds at the BP86/TZVP level of theory to evaluate the electronic differences of the three compounds. AdoCbl and MeCbl are known to undergo different reaction paths in the body. Thus, the focus was directed toward the characterization of the dative Co-C(ax) and Co-N(ax) bonds, which were quantifed by topological parameters, including energy densities; the source function including local source; and the electron localizability indicator (ELI-D), respectively. The source function reveals the existence of delocalized interactions between the corrin macrocycle and the axial ligands. The ELI-D indicates unsaturated Co-C(ax) bonding basins for the two biochemically active cofactors, but not for CNCbl, where a population of 2.2e is found. This may be related to significant pi-backbonding, which is supported by the delocalization index, delta, of 0.15 between the Co atom and the N atom of the cyano ligand. Considering all results, the inherent electronic differences between AdoCbl and MeCbl are found to be small thus, supporting earlier findings that the interaction with the protein site mainly controls the type of Co-C(ax) bond cleavage.

Experimental charge density studies of disordered N-phenylpyrrole and N-(4-fluorophenyl)pyrrole.

The static electron densities of the title compounds were extracted from high-resolution X-ray diffraction data using the nucleus-centered finite multipole expansion technique. The interpretation of the data collected for the N-phenylpyrrole crystal revealed a static disorder that could be successfully resolved within the aspherical-atom formalism. The local and integrated topological properties of the density obtained via a constrained multipole refinement are in statistical agreement with those calculated at the B3LYP/cc-pVTZ level of theory for the isolated molecule and for those derived from the experimental density of the para-fluorinated derivative N-(4-fluorophenyl)pyrrole. The topological analysis of the densities indicates neither pyramidal character of the pyrrole N-atom nor a quinoidal structure of the phenyl rings in either molecule. The fluorine substitution appears to have only a minor effect on the density of the remaining constituents but it results in markedly different features of the electrostatic potential of the two compounds. The consistency of the multipole refinement is validated by residual density analysis.

Carbanion or amide? First charge density study of parent 2-picolyllithium.

The negative charge originating from deprotonation of the methyl group is distributed over the 2-picolyl ring. Bonding properties derived from the electron density distribution support the enamide character of picolyllithium (PicLi; the picture shows the deformation density of [2-PicLi x PicH](2)), but electrophilic attack occurs at the deprotonated C atom. This reactivity is rationalized by the electrostatic potential, which guides electrophiles towards the nucleophilic C atom.

Foundations of residual-density analysis.

New and concise descriptors of the residual density are presented, namely the gross residual electrons, the net residual electrons and the fractal dimension distribution. These descriptors indicate how much residual density is present and in what way it is distributed, i.e. the extent to which the distribution is featureless. The amount of residual density present accounts for noise in the experimental data as well as for modeling inadequacies. Therefore, the minimization of the gross residual electrons during refinement serves as a quality criterion. In the case where only Gaussian noise is present in the residual density, the fractal distribution is parabolic in shape. Deviations from this shape therefore serve as an indicator for systematic errors. The new measures have been applied to simulated and experimental data in order to study the effects of noise, model inadequacies and truncation in the experimental resolution. These measures, although designed and examined with particular regard to applications of space residual density, are very general and can in principle also be applied to space and momentum residual densities in a one-, two-, three- or higher-dimensional Euclidean space.

The P(bth)2- anion as a Janus head staple between lithium and manganese (bth = benzothiazol-2-yl, C7H4NS).

Deprotonation of HP(bth)2 (1) affords the lithium phosphanide [(Et2O)2Li(bth)2P], (2) with both nitrogen atoms coordinated to the lithium atom while in the heterobimetallic complex [Li(bth)2P{Mn(CO)2Cp}2](n) (3) additionally the phosphorus atom micro-bridges two Mn(CO)2Cp residues.

An alternative to the goodness of fit.

An alternative measure to the goodness of fit (GoF) is developed and applied to experimental data. The alternative goodness of fit squared (aGoFs) demonstrates that the GoF regularly fails to provide evidence for the presence of systematic errors, because certain requirements are not met. These requirements are briefly discussed. It is shown that in many experimental data sets a correlation between the squared residuals and the variance of observed intensities exists. These correlations corrupt the GoF and lead to artificially reduced values in the GoF and in the numerical value of the wR(F2). Remaining systematic errors in the data sets are veiled by this mechanism. In data sets where these correlations do not appear for the entire data set, they often appear for the decile of largest variances of observed intensities. Additionally, statistical errors for the squared goodness of fit, GoFs, and the aGoFs are developed and applied to experimental data. This measure shows how significantly the GoFs and aGoFs deviate from the ideal value one.

Statistical tests against systematic errors in data sets based on the equality of residual means and variances from control samples: theory and applications.

Statistical tests are applied for the detection of systematic errors in data sets from least-squares refinements or other residual-based reconstruction processes. Samples of the residuals of the data are tested against the hypothesis that they belong to the same distribution. For this it is necessary that they show the same mean values and variances within the limits given by statistical fluctuations. When the samples differ significantly from each other, they are not from the same distribution within the limits set by the significance level. Therefore they cannot originate from a single Gaussian function in this case. It is shown that a significance cutoff results in exactly this case. Significance cutoffs are still frequently used in charge-density studies. The tests are applied to artificial data with and without systematic errors and to experimental data from the literature.

More about systematic errors in charge-density studies.

In order to detect and graphically visualize the absence or presence of systematic errors in fit data, conditional probabilities are employed to analyze the statistical independence or dependence of fit residuals. This concept is completely general and applicable to all scientific fields in which model parameters are fitted to experimental data. The applications presented in this work refer to published charge-density data.

About systematic errors in charge-density studies.

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].

More about residual values.

The usual residual values are complemented by expectation values based solely on the experimental data and the number of model parameters. These theoretical R values serve as benchmark values when all of the basic assumptions for a least-squares refinement, i.e. no systematic errors and a fully adequate model capable of describing the data, are fulfilled. The prediction of R values as presented here is applicable to any field where model parameters are fitted to data with known precision. For crystallographic applications, F(2)-based residual benchmark values are given. They depend on the first and second moments of variance, intensity and significance distributions, <σ(2)>, , . Possible applications of the theoretical R values are, for example, as a data-quality measure or the detection of systematic deviations between experimental data and model predicted data, although the theoretical R values cannot identify the origin of these systematic deviations. The change in R values due to application of a weighting scheme is quantified with the theoretical R values.

Expectation values for integer powers of a Poisson-distributed random number.

It is shown that expectation values of Poisson-distributed random numbers exist not only for the well known positive integer powers but also for negative integer powers. A recursion formula for the calculation of expectation values of powers differing by one is given. This recursion formula helps to find an analytical representation for both positive and negative integer powers in terms of the hypergeometric function.