International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 1.6, pp. 114128
Section 1.6.5. Specialized methods of spacegroup determination
H. D. Flack^{b}

In smallmolecule crystallography, it has been customary in crystalstructure analysis to make no use of the contribution of resonant scattering (otherwise called anomalous scattering and in older literature anomalous dispersion) other than in the specific area of absolutestructure and absoluteconfiguration determination. One may trace the causes of this situation to the weakness of the resonantscattering contribution, to the high cost in time and labour of collecting intensity data sets containing measurements of all Friedel opposites and for a lack of any perceived or real need for the additional information that might be obtained from the effects of resonant scattering.
On the experimental side, the turning point came with the widespread distribution of area detectors for smallmolecule crystallography, giving the potential to measure, at no extra cost, fullsphere data sets leading to the intensity differences between Friedel opposites hkl and . In 2015, the new methods of data analysis briefly presented here are in the stage of development and have not yet enjoyed widespread distribution, use and acceptance by the community. Flack et al. (2011) and Parsons et al. (2012) give detailed information on these calculations.
The basic starting point in this analysis is the following linear transformation of and , applicable to both observed and model values, to give the average (A) and difference (D) intensities:In equation (1.6.2.1), was denoted by . The expression for corresponding to that for given in equation (1.6.2.1) and using the same nomenclature isIn general is small compared to . A compound with an appreciable resonantscattering contribution has , whereas a compound with a small resonantscattering contribution has . For centric reflections, , and so the values of of these are entirely due to random uncertainties and systematic errors in the intensity measurements. of acentric reflections contains contributions both from the random uncertainties and the systematic errors of the data measurements, and from the differences between and which arise through the effect of resonant scattering. A slight experimental limitation is that a data set of intensities needs to contain both reflections hkl and in order to obtain and .
The Bijvoet ratio, defined byis the ratio of the rootmeansquare value of D to the mean value of A. In a structure analysis, two independent estimates of the Bijvoet ratio are available and their comparison leads to useful information as to whether the crystal structure is centrosymmetric or not.
The first estimate arises from considerations of intensity statistics leading to the definition of the Bijvoet ratio as a value called Friedif_{stat}, whose functional form was derived by Flack & Shmueli (2007) and Shmueli & Flack (2009). One needs only to know the chemical composition of the compound and the wavelength of the Xradiation to calculate Friedif_{stat} using various available software.
The second estimate of the Bijvoet ratio, Friedif_{obs}, is obtained from the observed diffraction intensities. One problematic point in the evaluation of Friedif_{obs} arises because A and D do not have the same dependence on and it is necessary to eliminate this difference as far as possible. A second problematic point in the calculation is to make sure that only acentric reflections of any of the noncentrosymmetric point groups in the chosen Laue class are selected for the calculation of Friedif_{obs}. In this way one is sure that if the point group of the crystal is centrosymmetric, all of the chosen reflections are centric, and if the point group of the crystal is noncentrosymmetric, all of the chosen reflections are acentric. The necessary selection is achieved by taking only those reflections that are general in the Laue group. To date (2015), the calculation of Friedif_{obs} is not available in distributed software. On comparison of Friedif_{stat} with Friedif_{obs}, one is able to state with some confidence that:
Example 1
The crystal of compound Ex1 (Udupa & Krebs, 1979) is known to be centrosymmetric (space group ) and has a significant resonantscattering contribution, Friedif_{stat} = 498 and Friedif_{obs} = 164. The comparison of Friedif_{stat} and Friedif_{obs} indicates that the crystal structure is centrosymmetric.
Example 2
The crystal of compound Ex2, potassium hydrogen (2R,3R) tartrate, is known to be enantiomerically pure and appears in space group . The value of Friedif_{obs} is 217 compared to a Friedif_{stat} value of 174. The agreement is good and allows the deduction that the crystal is neither centrosymmetric, nor twinned by inversion in a proportion near to 50:50, nor that the data set is unsatisfactorily dominated by random uncertainty and systematic error.
Example 3
The crystals of compound Ex3 (Zhu & Jiang, 2007) occur in Laue group . One finds Friedif_{stat} = 70 and Friedif_{obs} = 499. The huge discrepency between the two shows that the observed values of D are dominated by random uncertainty and systematic error.
It was shown in Section 1.6.5.1.2 that under certain circumstances it is possible to determine whether or not the space group of the crystal investigated is centrosymmetric. Suppose that the space group was found to be noncentrosymmetric. In each Laue class, there is one centrosymmetric point group and one or more noncentrosymmetric point groups. For example, in the Laue class mmm we need to distinguish between the point groups 222, 2mm, m2m and mm2, and of course between the space groups based on them. We shall show that it is possible in practice to distinguish between these noncentrosymmetric point groups using intensity differences between Friedel opposites caused by resonant scattering.
An excellent intensity data set from a crystal (Ex2 above) of potassium hydrogen (2R, 3R) tartrate, measured with a wavelength of 0.7469 Å at 100 K, was used. The Laue group was assumed to be mmm. The raw data set was initially merged and averaged in point group 1 and all special reflections of the Laue group mmm (i.e. 0kl, h0l, hk0, h00, 0k0, 00l) were set aside. The remaining data were organized into sets of reflections symmetryequivalent under the Laue group mmm, and only those sets (589 in all) containing all 8 of the mmmsymmetryequivalent reflections were retained. Each of these sets provides 4 and 4 values which can be used to calculate R_{merge} values appropriate to the five point groups in the Laue class mmm. The results are given in Table 1.6.5.1. The value of 100% for R_{merge} in a centrosymmetric point group, such as mmm or 2/m, arises by definition and not by coincidence. The of the true point group has the lowest value, which is noticeably different from the other choices of point group.

The crystal of Ex1 above (space group ) was treated in a similar manner. Table 1.6.5.2 shows that values display no preference between the three point groups in Laue class 2/m.

Intensity measurements comprising a full sphere of reflections are essential to the success of the R_{merge} tests described in this section.
There is an excellent way in which to evaluate both data measurement and treatment procedures, and the fit of the model to the data, including the spacegroup assignment, at the completion of structure refinement. This technique is applicable both to noncentrosymmetric and to centrosymmetric crystals. A scattergram of D_{obs} against D_{model}, and 2A_{obs} against 2A_{model} pairs are plotted on the same graph. All (D_{obs}, D_{model}) pairs are plotted together with those (2A_{obs}, 2A_{model}) pairs which have . The range of values on the axes of the model and of the observed values should be identical. For acentric reflections, for both A and D, a good fit of the observed to the model quantities shows itself as a straight line of slope 1 passing through the origin, with some scatter about this ideal straight line. For an individual reflection, 2A and D are, respectively, the sum and the difference of the same quantities and they have identical standard uncertainties. It is thus natural to select 2A and D to plot on the same graph. In practice one sees that the spread of the 2A plot increases with increasing value of 2A. Fig. 1.6.5.1 shows the 2AD plot for Ex2 of Example 2 in Section 1.6.5.1.2, which is most satisfactory and confirms the choice of point group from the use of R_{merge}. The conventional R value for all reflections is 3.1% and for those shown in Fig. 1.6.5.1 it is 10.4%. The R value for all D values is good at 51.1%. Fig. 1.6.5.2 shows the 2AD plot for Ex1 of Example 1 in Section 1.6.5.1.2. The structure model is centrosymmetric so all values are zero. The conventional R value on A for all reflections is 4.3% and for those shown in Fig. 1.6.5.2 it is 9.1%. The R value on all the D values is 100%.

Dataevaluation plot for crystal Ex2. The plot shows a scattergram of all () pairs and those () pairs in the same intensity range as the D values. 
For macromolecular crystallography, succinct descriptions of spacegroup determination have been given by Kabsch (2010a,b, 2012) and Evans (2006, 2011). Two characteristics of macromolecular crystals give rise to variations on the smallmolecule procedures described above.
The first characteristic is the large size of the unit cell of macromolecular crystals and the variation of the cell dimensions from one crystal to another. This makes the determination of the Bravais lattice by cell reduction problematic, as small changes of cell dimensions give rise to differences in the assignment. Kabsch (2010a,b, 2012) uses a `quality index' from each of Niggli's 44 lattice characters to come to a best choice. GrosseKunstleve et al. (2004) and Sauter et al. (2004) have found that some commonly used methods to determine the Bravais lattice are susceptible to numerical instability, making it possible for highsymmetry Bravais lattice types to be improperly identified. Sauter et al. (2004, 2006) find from practical experience that a deviation δ as high as 1.4° from perfect alignment of direct and reciprocal lattice rows must be allowed to construct the highestsymmetry Bravais type consistent with the data. Evans (2006) uses a value of 3.0°. The large unitcell size also gives rise to a large number of reflections in the asymmetric region of reciprocal space, and taken with the tendency of macromolecular crystals to decompose in the Xray beam, fullsphere data sets are uncommon. This means that confirmation of the Laue class by means of values of R_{int} (R_{merge}) are rarer than with smallmolecule crystallography, although Kabsch (2010b) does use a `redundancyindependent R factor'. Evans (2006, 2011) describes methods very similar to those given as the second stage in Section 1.6.2.1. The conclusion of Sauter et al. (2006) and Evans (2006) is that R_{int} values as high as 25% must be permitted in order to assemble an optimal set of operations to describe the diffraction symmetry. Another interesting procedure, accompanied by experimental proof, has been devised by Sauter et al. (2006). They show that it is clearer to calculate R_{merge} values individually for each potential symmetry operation of a target point group rather than comparing R_{merge} values for target point groups globally. According to Sauter et al. (2006) the reason for this improvement lies in the lack of intensity data relating some target symmetry operations.
The second characteristic of macromolecular crystals is that the compound is known, or presumed, to be chiral and enantiomerically pure, so that the crystal structure is chiral. This limits the choice of space group to the 65 Sohncke space groups containing only translations, pure rotations or screw rotations. For ease of use, these have been typeset in bold in Tables 1.6.4.2–1.6.4.30.
For the evaluation of protein structures, Poon et al. (2010) apply similar techniques to those described in Section 1.6.2.3. The major tactical objective is to identify pairs of αhelices that have been declared to be symmetryindependent in the structure solution but which may well be related by a rotational symmetry of the crystal structure. Poon et al. (2010) have been careful to test their methodology against generated structural data before proceeding to tests on real data. Their results indicate that some 2% of Xray structures in the Protein Data Bank potentially fit in a highersymmetry space group. Zwart et al. (2008) have studied the problems of underassigned translational symmetry operations, suspected incorrect symmetry and twinned data with ambiguous spacegroup choices, and give illustrations of the uses of group–subgroup relations.
In powder diffraction, the reciprocal lattice is projected onto a single dimension. This projection gives rise to the major difficulty in interpreting powderdiffraction patterns. Reflections overlap each other either exactly, owing to the symmetry of the lattice metric, or approximately. This makes the extraction of the integrated intensities of individual Bragg reflections liable to error. Experimentally, the use of synchrotron radiation with its exceedingly fine and highly monochromatic beam has enabled considerable progress to be made over recent years. Other obstacles to the interpretation of powderdiffraction patterns, which occur at all stages of the analysis, are background interpretation, preferred orientation, pseudotranslational symmetry and impurity phases. These are general powderdiffraction problems and will not be treated at all in the current chapter. The reader should consult David et al. (2002) and David & Shankland (2008) or the forthcoming new volume of International Tables for Crystallography (Volume H, Powder Diffraction) for further information.
It goes without saying that the main use of the powder method is in structural studies of compounds for which single crystals cannot be grown.
Let us start by running through the three stages of extraction of symmetry information from the diffraction pattern described in Section 1.6.2.1 to see how they apply to powder diffraction.
There has been considerable progress since 2000 in the automated extraction by software of the set of conditions for reflections from a powderdiffraction pattern for undertaking stage 3 above. Once the conditions have been identified, Tables 1.6.4.2–1.6.4.30 are used to identify the corresponding space groups. The output of such software consists of a ranked list of complete sets of conditions for reflections (i.e. the horizontal rows of conditions given in Tables 1.6.4.2–1.6.4.30). Accordingly, the bestranked set of conditions is at the top of the list followed by others in decreasing order of appropriateness. The list thus is answering the question: Which is the most probable set of reflection conditions for the data to hand? Such software uses integrated intensities of Bragg reflections extracted from the powder pattern and, as mentioned above, the results are sensitive to the particular profile integration procedure used. Moreover, only ideal Wilson (1949) p.d.f.'s for space groups P1 and are implemented. The art of such techniques is to find appropriate criteria such that the most likely set of reflection conditions is clearly discriminated from any others. Altomare et al. (Altomare, Caliandro, Camalli, Cuocci, da Silva et al., 2004; Altomare, Caliandro, Camalli, Cuocci, Giacovazzo et al., 2004; Altomare et al., 2005, 2007, 2009) have used a probabilistic approach combining the probabilities of individual symmetry operations of candidate space groups. The approach is pragmatic and has evolved over several versions of the software. Experience has accumulated through use of the procedure and the discrimination of the software has consequently improved. Markvardsen et al. (2001, 2012) commence with an indepth probabilistic analysis using the concepts of Bayesian statistics which was demonstrated on a few test structures. Later, Markvardsen et al. (2008) made software generally available for their approach. Vallcorba et al. (2012) have also produced software for spacegroup determination, but give little information on their algorithm.
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