 1. Introduction
 2. Centrosymmetric structures treated as noncentrosymmetric
 3. Pseudocentrosymmetric structures
 4. Predicting the uncertainty of the Flack parameter
 5. Centrosymmetric structures giving a high standard uncertainty on x
 6. Bias of the Flack parameter
 7. Inconsistent chemical and crystallographic data
 8. Concluding remarks
 Supporting Information
 References
 1. Introduction
 2. Centrosymmetric structures treated as noncentrosymmetric
 3. Pseudocentrosymmetric structures
 4. Predicting the uncertainty of the Flack parameter
 5. Centrosymmetric structures giving a high standard uncertainty on x
 6. Bias of the Flack parameter
 7. Inconsistent chemical and crystallographic data
 8. Concluding remarks
 Supporting Information
 References
research papers
Centrosymmetric and pseudocentrosymmetric structures refined as noncentrosymmetric
^{a}Laboratoire de Cristallographie, University of Geneva, Switzerland, ^{b}Dipartimento dei Materiali e delle Risorse Naturali, Università di Trieste sede di Pordenone, Italy, ^{c}Institute of Organic Chemistry, University of Zurich, Switzerland, and ^{d}Department of Crystal and Structural Chemistry, Bijvoet Center for Biomolecular Research, Utrecht University, The Netherlands
^{*}Correspondence email: howard.flack@cryst.unige.ch
The behaviour of the a priori estimate of the of the based only on the chemical composition of the compound and the wavelength of the radiation. The paper concludes with preliminary presentations of bias in the and of inconsistent chemical and crystallographic data.
for centrosymmetric and pseudocentrosymmetric crystal structures based on crystal structures published as being noncentrosymmetric is presented. It is confirmed for centrosymmetric structures that the value obtained for the is critically dependent on the Friedel coverage of the intensity data, approaching 0.5 for a coverage of 100% and sticking near the starting value for a coverage of 0%. For pseudocentrosymmetric structures, even those very close to being centrosymmetric, it is found that it is often possible to obtain significant values of the A theoretical basis for this surprising result is established. It has also been possible to establish an1. Introduction
Flack & Bernardinelli (2006) have recently published a short analysis of values of the Flack (1983) parameter obtained from 23 centrosymmetric structures but published in Inorganica Chimica Acta as being noncentrosymmetric. This analysis resides on the identification of the appropriate centrosymmetric space groups of these structures by Clemente (2005). In the present paper we extend our study of the values of the to a far greater number of crystal structures, published as being noncentrosymmetric but having subsequently been identified in published and unpublished works as being centrosymmetric.
It has also been possible to investigate the behaviour of the
for pseudocentrosymmetric structures. Although there are no published lists of such structures, it has nevertheless been possible to identify a sufficiently large number from our own and other sources to make a statistically significant study.Finally we undertake a short description and discussion of a new method for predicting a priori the of the Flack (1983) parameter based on the chemical composition of the compound under study, of bias in the Flack (1983) parameter, and of troublesome reported cases of incompatibility between chemical and crystallographic evidence.
2. Centrosymmetric structures treated as noncentrosymmetric
From the publications of Clemente (2005, 2006), Clemente & Marzotto (2003, 2004), Herbstein et al. (2002), Herbstein & Marsh (1998), Marsh (1999, 2002, 2004, 2005), Marsh et al. (2002) and Marsh & Spek (2001), crystal structures were extracted corresponding to the following three criteria: (i) at least one atom should be of an element heavier than Ar, (ii) the paper should be published after 1995; and (iii) a centre of symmetry needs adding to the Criterion (i) was chosen to try to ensure small standard uncertainties on the Flack parameters determined and hence significant values of the parameter itself. Criterion (ii) was chosen to correspond to the widespread use of the Flack (1983) parameter in published crystalstructure determinations. Criterion (iii) allows us to study crystal structures which were published as being noncentrosymmetric but after careful examination by the authors cited above were shown to be centrosymmetric. This reassignment of is based on a search, often automated, in the set of atomic coordinates of the determined structure for supplementary symmetries which are not part of the chosen Often deviations of interatomic distances and angles between chemically equivalent bonds in the same or equivalent molecules can be ascribed to instabilities in the leastsquares Aberrant shapes in the ellipsoid representation of atomic displacement parameters also provide an indication of an incorrect choice of Frequently it is not possible to carry out a leastsquares in the higher symmetry due to the diffraction intensity data being unavailable as supplementary material to the published structure. As sources of data, the published papers and any supplementary material freely available for download from the journal's online site, the archive of the Cambridge Crystallographic Data Centre (CCDC) or sometimes from the Fachsinformationszentrum (FIZ), Karlsruhe, were used. Authors were never contacted for supplementary material. Each paper available in searchable digital form was scanned for the following strings of characters: BASF, Flack, absolute, enantio, CD, optical, chiral, racem. It is of particular importance in the current study to determine whether pairs of Friedel opposites h k l and −h −k −l (or symmetry equivalent) have been measured and used separately in the leastsquares For this purpose we define the Friedel coverage as a measure of the completeness of the diffraction intensity data with regard to inversion in the origin of If for each value of h k l the intensity of the Friedel opposite −h −k −l (or one symmetryequivalent to it) has not been measured, then the Friedel coverage is 0%. However, if for each value of h k l, both the reflection h k l and its Friedel opposite −h −k −l (or one symmetryequivalent to it) have been measured and used separately in the leastsquares then the Friedel coverage is 100%. For some special values of the indices h k l (e.g. h 0 l in P2_{1}) the reflections h k l and −h −k −l are symmetry equivalent and are generally called centrosymmetric reflections. The centrosymmetric reflections are not included in the calculation of the Friedel coverage. From the information provided in the original paper and supplementary publication it is not always possible to determine unequivocally the Friedel coverage.
A table containing the pertinent aspects of the set of 231 centrosymmetric crystalstructure analyses refined as noncentrosymmetric is presented in the supplementary material.^{1} As concerns the reporting of the absolutestructure determination, 89 of these have no report and eight of them have a dismissive textual report. 24 of them report `x = 0.00' for the absolutestructure determination, implying that due to some data manipulation error (Clegg, 2003) the of the was not reported or not carried out. Thus, overall 52% of the structure reports are inadequate in their reporting of the absolutestructure determination. In five of the remaining crystalstructure analyses the on the is too high for the value itself to be interpretable. For the remaining crystalstructure determinations, 28 analyses have been carried out with a Friedel coverage of 100%, 67 with a coverage of 0% and 10 with an intermediate coverage.
Table 1 gives values of the reported in the 28 structure analyses based on a Friedel coverage of 100%. One can see that the Flack parameters tend to take a value of 0.5 although the spread is considerable. With a Friedel coverage of 100%, both observed intensities I_{obs}(h k l) and I_{obs}(−h −k −l) are present in the leastsquares and in the absence of random and systematic effects these would have the same intensity since the structure is centrosymmetric. In the noncentrosymmetric equality of the model intensities I_{model}(h k l) and I_{model}(−h −k −l) is achieved when the takes a value of 0.5. Other details of this argument are given by Flack & Bernardinelli (2006), and Table 1 confirms this analysis.

Table 2 gives values of the reported in the 67 structure analyses based on a Friedel coverage of 0%. The values are arranged around the value of zero but with a tendency to be positive. With a Friedel coverage of 0% only the observed intensities I_{obs}(h k l) are present in the and I_{obs}(−h −k −l) are absent. Consequently there is no specific influence operating on the model intensities I_{model}(h k l) and I_{model}(−h −k −l) in the leastsquares forcing them to become equal by way of a close to 0.5. As a result the is not driven to any particular value and will tend to stay close to or `stick at' its starting value. Other details of this argument are given by Flack & Bernardinelli (2006). All but one of the structure analyses contributing to Table 2 used software that uses a starting value of 0.0 for the The single exception is the value of 0.57 for which the starting value of the was 0.5. It is our contention that, were the crystal structures contributing to Table 2 refined in the noncentrosymmetric model but with data corresponding to a Friedel coverage of 100%, then values of the close to 0.5 would be obtained.

The topic of a suitable or necessary datameasurement region was treated early on by Bernardinelli & Flack (1987) in a study using potassium hydrogen (2R,3R) tartrate. It was found that of the was stable and unbiased when using a Friedel coverage of either 100% or 0%, in the latter case with several different choices of data region, e.g. h k l all positive or all negative. The of the obtained with a Friedel coverage of 0% was higher than when using a coverage of 100% due to the diminished number of observations. The results of the Bernardinelli & Flack (1987) and the present study are not in contradiction. The early study concerned a which was undoubtedly noncentrosymmetric with significant The current study applies to centrosymmetric structures mistakenly analysed as being noncentrosymmetric.
All in all, of the 231 centrosymmetric structure analyses treated as noncentrosymmetric, 86% of them (= 198/231; 198 = 231 − 28 − 5) are to be considered as unsatisfactory from the point of view of measurement, data reduction,
or reporting. Of course, 100% of these structure solutions are unsatisfactory due to an incorrect choice of space group.It must be stressed that the real crystal structures contributing to Tables 1 and 2 are centrosymmetric. Consequently, the is undefined and has no physical meaning for them as a description of One perceives in this situation that the value obtained for the of the noncentrosymmetric model is a junk parameter as concerns an absolutestructure specification of the real Its refined value is dictated by other incidental factors as described in this section. In accordance with the view of Wilson (1975), `Do the right calculation and get the right answer…all that really needs to be said is' that the authors of these papers should have had available a Friedel coverage of 100% in the discovery or structuredetermination phase of the analysis and should have realised for themselves that the was centrosymmetric.
3. Pseudocentrosymmetric structures
There are no published lists of pseudocentrosymmetric structures. Nevertheless, those authors who study the literature to detect crystal structures which are faulty in one way or another all have their own unpublished material for structures which appeared erroneous at first glance but on further examination turned out to be correct. This material includes those structures which are nearly centrosymmetric but are definitely noncentrosymmetric. A table containing the pertinent aspects of the set of 180 crystalstructure analyses is presented in the supplementary material.^{1} Many, but not all, of these structures are deemed to be noncentrosymmetric because they contain an molecule or group. The data selection and extraction procedures were very similar to those used for the centrosymmetric structures described in §2. We will just mention the differences here. The only selection criterion was that the paper had been published in or after 1993, a relaxation of the previous 1995 limit which was too restrictive. An additional keyword disorder was used on the searchable electronic documents. In a few instances there were direct contacts with the principal author of the paper. Such instances are marked in the notes to the compound. The measure of the pseudocentrosymmetry of each structure was obtained from PLATON (Spek, 2003). In fact, PLATON produces one fit parameter for each to be added. Often, but not always, the values of these fit parameters are identical. When this is not the case, we adopt the practice of reporting the lowest value. A measure of the amount of for each data set was obtained by use of the formula (Girard et al., 2003)
Of the 180 crystal structures, 78 had insufficient treatment and reporting of absolutestructure determination; for eight we were unsure whether the structures are centrosymmetric or noncentrosymmetric; another 14 were judged to be definitely centrosymmetric and included in the analysis in §2; and 48 had absolutestructure information but had been measured with zero or low Friedel coverage. The final set of 32 structures had absolutestructure information and had been measured with a Friedel coverage of 100%. So 82% of these structure determinations and reports were deficient in one way or another. In view of the results of §2, it is only on this final set of structures measured with a high Friedel coverage that we have concentrated our attention.
Table 3 contains the most important data harvested from the pseudocentrosymmetric structures measured with a Friedel coverage of 100%. It is immediately apparent from Table 3 that with a significant resonantscattering contribution (medium to large value of Rescat) and a Friedel coverage of 100% it is frequently possible to obtain an absolutestructure determination by way of the despite the structure being very nearly centrosymmetric as witnessed by the PLATON fit parameter. It is also very clear that as the amount of increases, so the on the diminishes. At low values of Rescat, large standard uncertainties and large deviations from zero are observed for the For compounds with Rescat > 160 (see §4) the average value of the of −0.0019 is close to the expected value of zero. For detailed information on the two compounds [DOJYAC, x = 0.50 (10), and MAKDAE, x = 0.27 (7)] for which Rescat > 80 and the is significantly different from zero, the notes in the supplementary publication should be consulted. Counter to our own intuition and to that of others (Jones, 1984; Iwasaki, 1974; Cianci et al., 2005), the works well in situations where the is definitely noncentrosymmetric but strongly pseudocentrosymmetric. Why?

A straightforward analysis will enable us to understand why reliable values of the
may be obtained from pseudocentrosymmetric crystal structures. A suitable model consists of a structure with two components:(i) (C), a centrosymmetric part formed of resonant atoms giving a contribution, F_{c}(h), to the complete F(h), which may be written as F_{c}(h) = s(C + iC′′), where s = +1 or −1 according to the sign of the factor F_{c}(h), and C and C′′ are its nonresonant and resonant atomic scattering amplitudes, F_{c}(h) being obtained by a sum over all atoms in the F_{c}(h) may be written in this form even if the atoms in the are of various chemical elements.
(ii) (N), a noncentrosymmetric part formed of resonant atoms all of the same F(h) of F_{n}(h) which may be written as
This part gives a contribution towhere N and N′′ are the nonresonant and resonant atomic scattering amplitudes and φ_{n} is the phase angle of the F_{n}(h) being obtained by a sum over all atoms in the unit cell.
The total F(h) = F_{c}(h) + F_{n}(h) and F(−h) may be expressed in terms of the same quantities by reversing the sign of sin φ_{n}. In a crystal twinned by inversion the intensities of the reflection h and its Friedel opposite −h are I(h) = (1 − x)F(h)^{2} + xF(−h)^{2} and I(−h) = xF(h)^{2} + (1 − x)F(−h)^{2}. The critical quantity Δ(h) is the difference in intensity between reflections h and −h,
The latter equation tells us much of what we need to know. For a φ_{n} to be near 90° or 270°. An Argand diagram of the complex plane indicating F_{c}(h), F_{n}(h) and F_{n}(−h) confirms this result. One way to make the term C′′N − CN′′ large is to set C′′ large and N′′ to zero. Indeed this is probably the case for many of the pseudocentrosymmetric structures studied here. These structures consist of an inner centrosymmetric part often with a heavy metal atom of large C′′ at the centre and an outer noncentrosymmetric part of light atoms with N′′ ≃ 0. These conditions systematically maximize Δ(h). One should note especially that for the best contrast Δ(h), curiously the resonant atoms are in a centrosymmetric arrangement and the noncentrosymmetric part of the structure (possibly chiral) is composed of nonresonant atoms.
of reflections to have a large absolute difference it is best for4. Predicting the uncertainty of the Flack parameter
Table 3 presented the results on pseudocentrosymmetric crystal structures reporting values of the determined with a high Friedel coverage. We already pointed out in §3 that as the amount of (Rescat) increases so the (u) on the diminishes. It is possible to quantify this observation to some extent. We restrict our attention to data with high Friedel coverage for this is the sure way to undertake absolutestructure determination. In Fig. 1, log_{10}(u) is plotted against log_{10}(Rescat) for the values in Table 3. Although there is a fair amount of scatter of the points in this plot, the general tendency shows a linear variation of log(u) with log(Rescat) and may be represented by a best straight line: log_{10}(u) = −1.01 log_{10}(Rescat) + 0.873. On this line, a value of u = 0.04 corresponds to a value of Rescat = 160, and a value of u = 0.10 corresponds to a value of Rescat = 80. These two values of u are the limiting values chosen by Flack & Bernardinelli (2000) as upper limits in the general case (u = 0.04) and with a compound known to be (u = 0.10) for absolutestructure determination. Consequently, the corresponding values of Rescat (160, general, and 80, enantiopure) are lower limits for absolutestructure determination calculable from a knowledge of the elemental composition of the compound alone. These two values are thus of practical use in the choice of compound and radiation wavelength for absoluteconfiguration determination prior to experimentation and in the evaluation of the value of u obtained. Of course, Fig. 1 has been derived from the pseudocentrosymmetric structures that we had available and it is difficult to know how the relation between log(u) and log(Rescat) might look for a more general class of compounds measured with a Friedel coverage of 100%.
5. Centrosymmetric structures giving a high on x
For the five centrosymmetric crystal structures in §2 which gave large standard uncertainties on the we have examined various causes for this effect. The following effects are not responsible for the high uncertainty of the in these structure determinations:
(i) Inadequate
contribution. The parameter Rescat for these compounds takes values between 143 and 297 which is clearly adequate.(ii) Overall bad fit of the model. The conventional R values lie between 0.047 and 0.085 so, although these structure determinations are not of the highest quality, the overall fit is adequate.
(iii) Inadequate data. The Friedel coverage is 100% in all five cases.
It is possible, however, to explain the high standard uncertainties on x by supposing that the refined noncentrosymmetric model (remember the itself is centrosymmetric) is very close to being centrosymmetric. Indeed the PLATON fit parameter for four of these structures is 100%, the remaining structure having a value of 97%. Expressing the scattering density in terms of a centrosymmetric (C, C′′) and an antisymmetric (A, A′′) component, we may use Δ(h) = 4(1 − 2x)(C′′A − CA′′), similar to the expression obtained in §3. The approach to a centrosymmetric arrangement is characterized by the values of A and A′′ vanishing simultaneously, resulting in tiny values of Δ(h) and ∂Δ(h)/∂x, and consequently large values of the of the A high value of the of the for a reasonable value of Rescat may be used as another sign that the is centrosymmetric.
6. Bias of the Flack parameter
Preliminary statistical tests on crystal structures determined by three separate groups of structure analysts in Geneva (Bernardinelli), Utrecht (Spek) and Zurich (Linden) seem to indicate that there may be a small systematic negative bias in the values of the x and the number of structures used in the survey are as follows: Geneva, −0.0050 for 109 structures; Utrecht, −0.0097 for 160 structures; Zurich, −0.014 for 80 structures. Although the bias is small, its significance seems to be confirmed by the fact that the three groups of structure analysts have different equipment, operate at different wavelengths, use various software and follow their own procedures. Moreover the choice of compounds studied is varied and independent.
The average values ofA brainstorming session over the possible cause(s) of such a bias brought forward many suggestions which we now list. Certain apparatus might overestimate or underestimate strong or weak intensities. Weak reflections may be recorded with toohigh an intensity owing to a contribution from multiple reflection. Background estimation may be systematically in error. f ′′ values may be systematically biased. Weighting schemes may play a role. Overestimated atomic displacement parameters due to the lack of an extinction or absorption correction would lead to a negative bias of the No definite statement can be made at this stage.
We recall that Tables 1 and 2 present values of the obtained from centrosymmetric crystals refined with a noncentrosymmetric model. It is not appropriate to talk of or analyse bias for such data as the expected value of the of a centrosymmetric crystal is undefined. Tables 1 and 2 contain values of a junk parameter of the real crystal structure.
7. Inconsistent chemical and crystallographic data
Crystalstructure analyses arise in which the crystallographic and chemical evidence are in conflict. Over 20 cases have been identified by Djukic et al. (2006) in a review of planarchiral metallacycles. Often the compound has been declared to be but the structure analysis results in a significantly close to 0.5 or even in the of a racemate. The twinningbyinversion model of the implies, for a value close to 0.5, that the compound is a racemate, in contradiction with the chemical evidence or hypothesis. We shall mention a few effects which may be at the origin of inconsistencies of this type:
(i) Although Friedel opposites are measured, their intensities are averaged before structure refinement.
(ii) A semiempirical absorption correction has been applied which attempts to reduce intensity differences between Friedel opposites.
(iii) An unconverged leastsquares
has been undertaken due to inappropriate procedures.(iv) The bulk compound is a racemate (in contradiction to the chemical evidence presented) and crystallizes by
but giving crystals that are twinned by inversion.(v) The enantiopurity of the bulk compound is high but not exactly 100%. The crystal chosen for the diffraction experiment may then be that of the racemate. Cases where the temperature of fusion of the crystalline racemate is much higher than that of the ee's is: (100%) > eutetic > bulk > racemate (0%), the eutetic being closer to enantiopurity than the bulk. In such cases the first crystals to appear are those of the racemate.
compound lead to phase diagrams in which the order of theIn general, resolving such ambiguities requires more information than is available in the publication. In such cases characterization of the bulk and the individual crystal used for the diffraction experiment is crucial. The most suitable measurements are those of CD spectra in solution or solid, and enantioselective
On the diffraction side it is advisable to check the diffraction apparatus and all related software by measurement of a well characterized standard compound containing significant resonant scattering.8. Concluding remarks
Cianci et al. (2005) have recently published a very detailed review of the application of to problems in structural chemistry and biology. This text misses no opportunity to highlight the considerable number of past achievements and future potential of synchrotron radiation to structure analysis. However, Cianci et al. (2005) make no mention of how future developments in synchrotronradiation technology and measurement techniques could be of help in the common problem of the determination of from lightatom smallmolecule crystal structures. Doubtless, Cianci et al. (2005) have in mind that the K absorption edges of light atoms occur at long wavelengths, making diffraction data very sparse, owing to a small and experimentation difficult, owing to high absorption (see e.g. Biou et al., 2005).
It is very clear indeed from the results presented in §2 and §3 that provided the value of Rescat is larger than a threshold value of perhaps 40, it must be considered mandatory to measure and use a Friedel coverage of 100% for a new presented as being noncentrosymmetric. If this is done, for centrosymmetric structures, values of the close to zero due to `sticking' will be avoided and values close to 0.5 will be obtained. Thus, as long as a Friedel coverage of 100% has been used, a value of the significantly close to 0.5 may be taken as one indication that the may in fact be centrosymmetric, and a value significantly close to zero that it is definitely noncentrosymmetric. It is not valid to justify the choice of a noncentrosymmetric by a value of the close to zero if the Friedel coverage is not close to 100%. It must be borne in mind that a value of the significantly close to 0.5 for a truly noncentrosymmetric structure indicates that the macroscopic crystal is twinned by inversion in a ratio of approximately 50:50. In the late stages of analysis, we thoroughly recommend the evaluation of the by procedures such as PLATON in order to detect missed symmetry.
We recall yet again that, for estimates of the ) is of the very greatest importance.
and its to be valid, they need to be determined by a converged fullmatrix leastsquares In this respect the advice of Clegg (2003We hope that many engaged in the endeavour of crystalstructure determination, evaluation, reporting and archiving will have enjoyed, or at least benefited from, reading this paper. Our study confirms that crystal structures are far too often described with incorrect space groups or that crucial information is missing from a paper and its associated supplementary publications. Thus, standards of authorship, refereeing, software writing, automated evaluation, editing and publication need to be further improved for all the journals that we have encountered. Although Acta Crystallographica Section B, Section C and Section E achieve a very high quality in their published material, one suspects from the following remark of Clegg & Watson (2006) that this is often not the case for submitted manuscripts! `Papers which are difficult to understand, which fall well short of the requirements of the Notes for Authors, or which require extensive text correction and editing will be returned to the author for revision without a detailed review; it is the responsibility of authors, not of Coeditors, to generate an acceptable text for the Abstract, Comment and Experimental sections and to ensure that other CIF items are correct and complete.'
Supporting information
List of wrong structures. DOI: 10.1107/S0108768106021884/gp5012sup1.pdf
Pseudosymmetric structures. DOI: 10.1107/S0108768106021884/gp5012sup2.pdf
Acknowledgements
Our heartfelt thanks go to Dr R. E. Marsh for having supplied us with his list of pseudocentrosymmetric structures which formed the basis of a very considerable part of this work. He certainly merits to be included as an author of this paper but declined this due recognition, feeling he had not contributed sufficiently. We appreciated the excellent service provided by the Cambridge Crystallographic Data Centre in making available supplementary material from their
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