Volume 69 Received 5 February 2013  Use of intensity quotients and differences in absolute structure refinement^{a}EaStCHEM School of Chemistry and Centre for Science at Extreme Conditions, The University of Edinburgh, King's Buildings, West Mains Road, Edinburgh EH9 3JJ, Scotland,^{b}Chimie minérale, analytique et appliquée, University of Geneva, Geneva, Switzerland, and ^{c}Novartis Institutes for BioMedical Research, 4002 Basel, Switzerland Several methods for absolute structure refinement were tested using singlecrystal Xray diffraction data collected using Cu K radiation for 23 crystals with no element heavier than oxygen: conventional refinement using an inversion twin model, estimation using intensity quotients in SHELXL2012, estimation using Bayesian methods in PLATON, estimation using restraints consisting of numerical intensity differences in CRYSTALS and estimation using differences and quotients in TOPASAcademic where both quantities were coded in terms of other structural parameters and implemented as restraints. The conventional refinement approach yielded accurate values of the Flack parameter, but with standard uncertainties ranging from 0.15 to 0.77. The other methods also yielded accurate values of the Flack parameter, but with much higher precision. Absolute structure was established in all cases, even for a hydrocarbon. The procedures in which restraints are coded explicitly in terms of other structural parameters enable the Flack parameter to correlate with these other parameters, so that it is determined along with those parameters during refinement. Keywords: intensity quotients; absolute structure refinement. 
When applied in crystallography the term absolute structure refers to the spatial arrangement of the atoms of a physically identified noncentrosymmetric crystal and its description by way of unitcell dimensions, space group and representative coordinates of all atoms (Flack & Bernardinelli, 1999, 2008b). Since inverted images of a noncentrosymmetric crystal structure are different, the question of absolute structure arises during analysis of any noncentrosymmetric crystal structure. The most important practical application of absolute structure refinement is, however, in the crystallographic determination of the absolute configuration of chiral molecules.
The fact that absolute structure can be obtained at all in a crystal structure determination is the result of resonant scattering, also known as anomalous scattering or anomalous dispersion, which introduces small differences in intensity between reflections h and which carry the information on absolute structure. Methods for absolute structure determination most commonly used today are based on a formulation first described by Flack (1983), in which the crystal under investigation is considered to be an inversion twin in which the reference domain has the absolute structure of the current refinement model, and the other domain is inverted. Measured intensities are then modelled according to
where F_{single}(h)^{2} and F_{single}()^{2} are model quantities based on a single crystal comprised of the reference domain. The two alternative absolute structures can be refined competitively against one another by refining the twin scale factor, x, which in this application is referred to as the Flack parameter.
The Flack parameter has a physically meaningful value in the range of 01, and represents the fraction of the inverted structure present in the crystal. A value of x = 0 implies that none of the crystal is in the inverted form and the model has the correct absolute structure; if x = 1 then all of the crystal is in the inverted form. Intermediate values of x point to inversion twinning.
It is important to interpret the value of the Flack parameter in the context of its standard uncertainty. From a statistical point of view, a value of 0.2 (8) has such a large standard uncertainty (0.8) that one neither knows whether the crystal is twinned by inversion or not, nor whether it is inverted or not. Further analysis shows that before any conclusions regarding absolute structure can be made, the standard uncertainty of the Flack parameter should be less than 0.1, even if a material is known to be enantiopure (Flack & Bernardinelli, 2000).
The ability to achieve a low standard uncertainty for the Flack parameter depends in part on the resonant scattering effects having sufficient magnitude to lead to measurably different intensities for Friedel pairs of reflections with indices h and . This depends on the chemical elements present in the crystal and the wavelength of the Xrays used to collect the diffraction data. The magnitude of resonant scattering effects in a given experiment can be conveniently quantified by the Friedif_{stat} parameter (Flack & Shmueli, 2007).
If Friedif_{stat} has a value of 80 or more, absolute structure determination presents little problem (Flack & Bernardinelli, 2008a). However, resonant scattering effects for elements such as C, N and O are small for commonly available Xray energies making it difficult to determine the Flack parameter with sufficient precision to establish absolute structure for many organic compounds. For example, the value of Friedif_{stat} for the amino acid Lalanine with Cu K radiation is only 34. Accordingly, the value of the Flack parameter obtained from a conventional refinement of Lalanine was 0.04 (27). The dataset was of excellent quality, yielding low merging and refinement residuals, yet the precision of the Flack parameter is too low to enable a definitive statement to be made regarding the absolute structure (Flack & Bernardinelli, 2000).
The ability to determine absolute structure precisely also depends on low levels of random and systematic errors in intensity measurements. Analysis of noncentrosymmetric crystal structures published in Acta Cryst. C in 2007, 2011 and 2012 has shown that even the intensity data of structures with large Friedif_{stat} values may be entirely dominated by random uncertainties and systematic errors (Flack, 2012; Flack et al., 2011).
There is a longstanding interest in finding ways to improve the precision of the Flack parameter in lightatom structures. A postrefinement Bayesian statistical procedure has been described by Hooft et al. (2008, 2010), which can be used either to define a probability that a refined absolute structure is correct or to obtain an estimate of the Flack parameter.^{1} Methods in which refinement weights are modified for data in proportion to their sensitivity to the Flack parameter have also been described (Bernardinelli & Flack, 1985; Parsons, Wagner et al., 2012). It has further been shown that precision may be improved by the use of aspherical scattering factors (Dittrich et al., 2006).
While each of the methods described has been shown to yield lower standard uncertainties on Flack parameters than conventional refinement, all are open to potential criticisms. The Hooft method, being a postrefinement method, does not formally allow x to correlate with other parameters during refinement, and this, theoretically at least, may compromise values of the standard uncertainty obtained (Hooft et al., 2008). Reweighting methods, involving the selection of data for upweighting, can magnify errors in the intensity measurements, so that values of x can be precise but inaccurate. Parsons, Wagner et al. (2012), for example, refer to one structure where the elimination of just two poorly measured data points shifted the Flack parameter from 0.35 (12) to 0.02 (14). Use of aspherical scattering factors yields improvements in precision, but in most of the examples tested the change was too small to enable sufficiently precise absolute structure determination for lightatom compounds (see Table 3 in Dittrich et al., 2006). This said, the use of invariom models in combination with other methods, described above and herein, merits further investigation.
In this paper we will describe methods based on intensity differences and quotients that enable x to be refined along with all other parameters. The purpose of this paper is to demonstrate that this leads to more precise estimates of x than conventional refinement methods while avoiding the potential criticisms discussed above.
Differences between the observed intensities of Friedel pairs of reflections
can be modelled following equation (1) with
where . The quantities D, which are also referred to as Friedel or Bijvoet differences, have been used in strategies for absolute structure determination described by Hooft et al. (2008, 2010) and Le Page et al. (1990), in a procedure available in the DIRDIF suite of programs (Beurskens et al., 1996) and in the procedure described by Thompson & Watkin (2011).
In principle, on a fourcircle diffractometer equipped with a point detector it is possible to measure the intensities of reflections h and at setting angles (2, , and ) and (2, , and ) (Le Page et al., 1990). The first set of setting angles is appropriate for reflection hkl and the second set for . In the second set, both the incident and reflected beam directions are reversals of those of the first set. If a crystal has a centrosymmetric habit then the beam paths through the crystal of these two measurements are identical. Consequently, their absorption and extinction corrections are identical, and the quotient I_{obs}(h)/I_{obs}() is absorption and extinction (and scale) free.
The corresponding model quotient can be written in terms of F_{single}(h)^{2} and F_{single}()^{2}
While this type of formulation has been used in the structure refinement of the kinase inhibitor roscovitine (Wang et al., 2001), it is nonlinear in x, and the standard uncertainties of quotients defined for h and are not the same. These problems are removed, and the independence from absorption, extinction and scale maintained, by reformulating the quotients in terms of sums and differences of Friedelpair intensities, so that the observed quotients
are modelled with
where
All of the test datasets used in this study were collected using modern diffractometers equipped with area detectors. These do not in general perform reversed beampath measurements, and so our initial contention that quotients can be measured in such a way that errors cancel does not hold.^{2} For such data we follow Parsons, Pattison & Flack (2012) in writing
where s(h) and s(h) are the average and halfdifference of the systematic errors in reflections I_{obs}(h) and I_{obs}() remaining after the application of a multiscan correction. If the term is small enough that is a good approximation for , this leads to
In practice Q_{model}(h) is usually considerably less than 1, and so provided the difference in systematic errors in I_{obs}(h) and I_{obs}() is small relative to the overall systematic error, the assumption that Q_{model}(h) = Q_{obs}(h) should still hold approximately.
Equation (6) can also be interpreted as being equivalent to equation (3) with an additional weighting term, 1/2A_{model}(h). The incorporation of this term can be justified on the basis of the leverage analysis presented in Parsons, Wagner et al. (2012), in which it was shown that the data with the greatest influence on the precision of the Flack parameter were those with weak to moderate intensities. The factor 1/2A_{model}(h) upweights these data. Very weak data, which have little leverage on the Flack parameter but high values of Q_{obs}(h) on account of a small value of A_{obs}(h), should be omitted from an analysis based on quotients.
A series of test datasets was used in this study; selected crystal data are given in Table 1. The compounds selected for study contain no element heavier than oxygen, and all have Friedif_{stat} of 36 or less for Cu K radiation. RMandelic acid and Lalanine were obtained from SigmaAldrich and were used as received; for other samples solutionphase optical rotation measurements or chiral separation established enantiomeric excesses of > 98%. Resonant scattering factors f' and f'' are independent of resolution, and the contribution of resonant relative to nonresonant scattering is therefore greatest at high values of sin /. For this reason the test datasets were collected at low temperature. Datasets were highly redundant, with average multiplicities of observations between 8 and 35.

Datasets for the samples listed in Table 1 carrying the superscript A were collected with Cu K radiation on an Agilent Technologies SuperNova incorporating a microsource generator. The temperature of data collection was 150 K except for Rmandelic acid. This material crystallizes as plates, and cooling to 150 K was found to cause strainbroadening, and so data were collected at 220 K. Processing, including integration and a multiscan absorption correction (Blessing, 1995), was accomplished with CrysAlis PRO (Oxford Diffraction Ltd, 2010).
Datasets carrying the superscript B were collected using Cu K radiation at 100 K using a Bruker Microstar finefocus rotating anode generator with a SMART 6000 CCD detector or a Bruker D8 microsource, also equipped with a SMART 6000 detector. Data were processed with SAINT (BrukerNonius, 2006) and corrected for absorption and other systematic errors using the multiscan procedure SADABS (BrukerNonius, 2006; Sheldrick, 2008a).
Data were merged using SORTAV with unit weights and robustresistant downweighting of outliers (Blessing, 1997).^{3}
Structures were solved using direct methods (SHELXS; Sheldrick, 2008b) or charge flipping (SUPERFLIP; Palatinus & Chapuis, 2007) and refined against F^{2} in SHELXL2012 (beta test version 2012/9) using all data (Sheldrick, 2012). Data were merged in point groups 2, 222 or 3 for the monoclinic, orthorhombic and trigonal structures, respectively. Isopropyl groups in structures R and SCYCLO are disordered over two orientations (Wang et al., 2001). The disorder components were restrained to have similar bond distances and angles. The water of crystallization in structure A0034a is disordered about a crystallographic twofold axis, which also induces disorder in two hydroxyl Hatom positions. All fullweight nonH atoms were refined with anisotropic displacement parameters. Full weight H atoms were refined freely; those part of disordered groups were either refined with restraints (A0034a) or placed in ideal positions (R and SCYCLO). The Flack parameter was refined either by fullmatrix least squares (i.e. using the TWIN/BASF commands in the SHELXL .ins file) or postrefinement based on quotients defined in §2.2. The results of the first of these refinements were used to calculate the Hooft parameter via the BIJVOET routine in PLATON. In all cases a Gaussian prior was used (Hooft et al., 2008; Spek, 2003).
The same models as described above were refined in CRYSTALS (Version 14.40; Betteridge et al., 2003) and the Flack parameter estimated via the procedure described by Thompson & Watkin (2011).
TOPASAcademic, Version 5 (Coelho, 2012), allows userequations to be written in the form of a function, in a similar way to the definition of a function or subroutine in a programming language such as Fortran or C++ (Coelho et al., 2011). These equations can then be used in restraints or to define other parameters. This option enables equation (6) to be incorporated into the refinement in the form of a set of restraints where the equation is coded in terms of atomic positional, displacement and occupancy parameters for each quotient.
The quotients Q_{obs}(h) were calculated from the integrated datasets using reflections for which both I_{obs}(h) and I_{obs}() were greater than three times their respective uncertainties. This cutoff condition eliminates quotients which are large because the term A_{obs}(h) is small (see §2.2). Q_{obs}(h) may also be calculated using XPREP (Sheldrick, 2001). Outlier data for which D_{obs}(h) was greater than twice the maximum value of D_{single} were also excluded. This condition mirrors the procedure used for outlier detection in the Bijvoet routine in PLATON (Spek, 2003). For surviving data the standard uncertainty u[Q_{obs}(h)] was propagated from the values of u [I_{obs}(h)] and u [I_{obs}()] obtained from merging
The observations used in TOPAS for refinement took the form of Friedelaveraged intensity data, A_{obs}(h), and quotients, Q_{obs}(h), the latter in the form of restraints.
The values of A_{obs}(h) and their uncertainties were obtained by merging the centric and paired acentric data in the relevant Laue group (e.g. mmm for an orthorhombic structure). Any unpaired acentric data were omitted. The structure was first refined in CRYSTALS against A_{obs}(h) only, and the weighting scheme optimized. This weighting scheme was then held fixed in subsequent refinement in TOPAS. The A_{obs}(h) values were modelled both in CRYSTALS and TOPAS using a Flack parameter equal to 0.5 in order to correctly account for the averaging of Friedelpair intensities.
The quotient data, Q_{obs}(h), were modelled with equation (6) using a second Flack parameter; it is this parameter which characterizes the absolute structure. The quotient restraints were initially given a weight, w_{restraint}(h) = 1/u^{2} [Q_{obs}(h)]. After initial cycles of refinement the values of the deviates were used to calculate a reduced ^{2} statistic. The structure was then rerefined with the quotient restraint weights scaled by 1/^{2}. A normal probability plot (Abrahams & Keve, 1971) of deviates was also inspected to detect further outliers and for validation purposes.
The same procedure was applied to the refinement using difference restraints based on equation (3). No cutoff condition based on values of I_{obs}(h)/u(I_{obs}(h)) was applied, but outliers were detected as described above. The initial restraint weights were w_{restraint}(h) = 1/u^{2} [D_{obs}(h)] = 1/(u^{2} [I_{obs}(h)] + u^{2}[I_{obs}()]), but these were rescaled after initial cycles of refinement, also as described above.
Clearly it is only possible to calculate Q_{obs}(h) and D_{obs}(h) for acentric data where both I_{obs}(h) and I_{obs}() have been measured; coverage statistics along with numbers of data omitted as outliers are given in Table 2. Input files for quotient refinement of Lalanine^{B} in TOPASAcademic are available in the supplementary material .

Programs written to calculate Q_{obs}(h) and D_{obs}(h), detect outliers and write files of symbolic restraints made use of subroutines available in the CrysFML Fortran library (RodríguezCarvajal & González Platas, 2009).
The results of `conventional' refinement of x, that is as a twin factor in fullmatrix least squares, are listed in Table 1 in the column x(twin). They are completely consistent with the results of Flack & Bernardinelli (2008a), which indicates that compounds with Friedif_{stat} in the range 936 would be expected to yield standard uncertainties of the Flack parameter of between 0.8 and 0.2. None of the conventional refinements yields a Flack parameter with sufficient precision to enable a conclusion to be made about the absolute structures of the crystals being studied.
It is, however, remarkable that the values of the Flack parameter in Table 1 cluster around zero much more tightly than would be anticipated on the basis of their high standard uncertainties. The reduced ^{2} calculated from the data in the x(twin) column (assuming the true value of x is zero in each case) is only 0.031, suggesting that the uncertainties are overestimated by a factor of 5.5, meaning that more information about absolute structure is present in the data than implied by the uncertainties calculated by least squares. While this finding runs counter to the general underestimation of standard uncertainties in crystallographic least squares (Hamilton & Abrahams, 1970), it is consistent with Thompson & Watkin's (2011) histogram of Flack parameters obtained from refinements of 150 structures with Friedif_{stat} in the range 3.410.8. The histogram was centred at zero; had the data been devoid of absolute structure information, the histogram should have been centred about 0.5.
It appears that conventional refinement of x yields rather pessimistic uncertainty estimates. Although the data quoted in Table 1 were derived using SHELXL2012, similar results were obtained with CRYSTALS and TOPAS. The methods published elsewhere by Hooft et al. (2008, 2010), Thompson & Watkin (2011) and Parsons, Wagner et al. (2012) and those described below attempt to obtain more realistic uncertainty estimates.
Values of Q_{obs}(h) [equation (5)] can be calculated using the observed intensity data, while values of [equation (6)] can be calculated from the refinement model. A `Q plot' of Q_{obs}(h) against should be a straight line passing through the origin with a gradient of (1  2x). Similarly, a `D plot' of D_{obs}(h) [equation (2)] against [equation (3)] should also be linear with a gradient of (1  2x).
The method can be illustrated with reference to the data collected for Lalanine. The gradient of the weighted leastsquares best straight line in the Q plot shown in Fig. 1(a) is 0.984 (68), which yields a value of the Flack parameter of 0.01 (3). This estimate is very much more precise than that obtained with the same data in a conventional refinement. Some of the error bars in Fig. 1 are huge, but these points have little influence on the (weighted) fit. Much more important is the very clear unit slope in the bulk of the data points.
 Figure 1 Plot of Q_{obs}(h) against Q_{single}(h) [as defined in equations (5) and (6)] for (a) Lalanine and (b) cholestane. (c) as (b), but only the top 200 most influential points are shown; the axes are chosen to be the same as (b). 
As the effects of resonant scattering become smaller, the trend in Qplots becomes harder to discern by eye, as illustrated in the plot for cholestane (Friedif_{stat} = 9) in Fig. 1(b). The leastsquares fit for the data shown nevertheless yields a Flack parameter of 0.02 (12). Part of the problem in Fig. 1(b) is that the majority of points have little influence on the fit. Fig. 1(c) shows the 200 most influential points in Fig. 1(b) (as measured by their leverages; Parsons, Wagner et al., 2012; Prince, 2004; Merli et al., 2000). The noticeable gap in the middle of the plot occurs because for a oneparameter linear fit the leverages are proportional to ; other missing points in Fig. 1(c) have a low leverage on account of their high error bars. Although not exactly obvious, the trend in Fig. 1(c) is a little clearer than it is in Fig. 1(b). More significantly the value of x obtained for these points is 0.09 (15): even though we are only working with 5% of the data, the precision is only marginally affected.
This linear Qfitting method has been incorporated into SHELXL2012, and the results for the other compounds studied are the top values listed in the column labelled `x(quotient)' in Table 1. Also included in Table 1 [in column y(Hooft)] are the estimates of Hooft's Bayesian method. The results of the two procedures are in excellent agreement, both showing consistently higher precision than conventional methods [x(twin)].
Although the method described in §4.2 yields precise and accurate values of the Flack parameter, like other postrefinement methods, it has the disadvantage that x is not allowed to correlate with other parameters. A way around this difficulty is to incorporate equation (6) into the structure refinement (Murphy et al., 2010; Parsons & Flack, 2004; Wang et al., 2001). Values of Q_{obs}(h) defined in equation (5) are calculated from the intensity data, as before, but facilities available in TOPAS enable Q_{single}(h) to be written in terms of the parameters of the refinement model (atomic coordinates, displacement parameters and occupancies) and built into the refinement as an equation of restraint; an example is provided in the supplementary material . The Flack parameter can now correlate with other parameters during refinement.
A quotient can be defined for each Friedel pair of intensities in the dataset so that the observations in the refinement now take the form of Friedelaveraged intensity measurements, A_{obs}(h), and the quotients, Q_{obs}(h). The values of A_{obs}(h) are obtained by merging the centric and paired acentric data in the centrosymmetric Laue group, and all absolute structure information is contained in the quotients. The number of quotients used in the test examples lay between a few hundred to several thousand depending on the size of the structure (details are in Table 2).
The bottom values listed in the x(quotient) column of Table 1 show the results. They are very similar to those obtained with the postrefinement procedure described in §4.2, and much more precise than those obtained with the conventional refinement technique. Acceptable precision has been obtained for datasets with Friedif_{stat} as low as 12.
The Flack parameter is sensitive to outliers in the data, and it is important that these are detected and eliminated. The sensitivity to outliers can be illustrated using the dataset TWA16a. A refinement performed with no outlier elimination at all yielded a Flack parameter equal to 0.18 (8). A normal probability plot calculated for this refinement (Fig. 2) was linear, but with one point (at the bottom left in Fig. 2) deviating substantially from the straight line. Elimination of this one observation changed the value of x to 0.08 (8).
 Figure 2 Outlier detection. Normal probability plot calculated using observed quotient restraint deviates for TWA16a. The extreme point at the bottom left was omitted as an outlier. This plot was calculated prior to rescaling of weights. 
Normal probability plots are a powerful means for detecting outliers, although in this work we follow Spek's procedure in PLATON in eliminating Friedel pairs with D_{obs}(h) more than twice the maximum calculated absolute difference for the entire dataset. This is a more objective procedure, although it means that more data are omitted the lower the value of Friedif_{stat}. The numbers of outliers omitted are given in Table 2.
An alternative procedure is to base the restraints on differences rather than quotients. The target value for each restraint is D_{obs}(h) as defined in equation (2). The model value D_{model}(h) is defined in equation (3) and written in terms of the refinement parameters. The procedure was otherwise identical to that described in §4.3.
A similar procedure is available in CRYSTALS. Here refinement is carried out against F_{obs}(h)^{2} data merged in the appropriate crystal (as opposed to Laue) point group and D_{model}(h) is a numerical value calculated from the current model (see Fig. 2 in Thompson & Watkin, 2011).
The results of the two procedures are listed in the x(difference) column of Table 1; the top value is obtained from the CRYSTALS procedure, the bottom via explicit restraints. The results are in agreement with each other and those obtained for quotients.
The differences defined in equations (2) and (3) are used in the Bayesian and numerical restraint methods for estimation of the Hooft or Flack parameters available in PLATON and CRYSTALS. Our original idea of basing absolute structure analysis on quotients was conceived about a decade ago, when fourcircle instruments with point detectors were still in common use. The cancellation of absorption and extinction errors which occurs on taking intensity quotients using reversebeam measurements of Friedel pair intensities cannot in general be achieved with modern areadetector instruments (but see footnote 2, §2.2). The analysis presented in §2 suggests that some approximate cancellation of errors occurs provided the difference in systematic errors in I_{obs}(h) and I_{obs}() is small relative to the overall systematic error.
One means for assessing whether error cancellation is achieved in practice with areadetector data is to compare R factors based on the observed and model values of quotients or differences [equation (10)]. If errors are really cancelled R(Q) should be systematically lower than R(D).
Values of R(Q) and R(D) are listed for each dataset in Table 3. The differences are mostly quite marginal, a finding consistent with a similar listing in Table 10 of Parsons, Pattison & Flack (2012).

A second method for assessing the presence of systematic errors is to examine normal probability plots based on the weighted residuals and for the quotient and difference refinements, respectively. Systematic errors shift the intercept of the plot away from the origin and cause the plot itself to deviate from linearity. The intercepts, gradients and Pearson correlation coefficients (r^{2}) of normal probability plots for quotient and difference refinements are also listed in Table 3. There is generally rather little systematic difference between the intercepts calculated for quotients or differences, all falling very close to the origin. There is no systematic difference between the correlation coefficients for the quotient and difference plots. The gradients are all near the ideal value of unity, but this is a consequence of the weightscaling described in §3.4.
The Rfactor and normal probability calculations indicate that cancellation of systematic errors can occur on taking quotients of intensities collected with area detectors, but the improvement, if present, is usually small and the results of using either method essentially the same. This is possibly because the assumptions about relative systematic errors referred to above are not met or because absorption and extinction are not the principal systematic errors present.
The linearity and small intercepts of the normal probability plots listed in Table 3 indicate that the weights applied to the quotients and differences reflect the uncertainties in the data. The values of reduced ^{2} listed in Table 1 are near unity for both quotient and differencebased methods, indicating that the magnitudes of the standard uncertainties are realistic. Taken with the accuracy of the values of the Flack parameters listed in Table 1 this shows that quotient and difference methods are both appropriate for absolute structure determination.
Some insight into why the methods presented increase the precision in x can be gained by considering the relative influences of the observations A_{obs}(h) and Q_{obs}(h) or D_{obs}(h) on the structural and Flack parameters. Leverage analysis was carried out on the refinement of Lalanine^{B} using CRYSTALS (Parsons, Wagner et al., 2012).
Fig. 3(a) is a histogram of leverages for the F_{obs}(h)^{2} (orange) and the D_{obs}(h) (green) data. The D_{obs}(h) leverages cluster near zero showing that they have rather little effect on the overall datafitting. The insensitivity of the structural parameters to the Friedel difference intensities reflects a similar finding described in §4.1 of Flack et al. (2011).
 Figure 3 Leverage analysis for Lalanine^{B}. (a) Leverages of F_{obs}(h)^{2} and difference data, D_{obs}(h). (b) Relative influences of F_{obs}(h)^{2} and D_{obs}(h) on the precision of the Flack parameter (as expressed by the quantity T). In each case F_{obs}(h)^{2} data are shown in orange and D_{obs}(h) data in green. 
Fig. 3(b) shows a histogram of the quantity T, which measures the influence of observations on the precision of the Flack parameter (David, 2004; David et al., 1993; Parsons, Wagner et al., 2012; Prince, 2004). Here the situation seen in the leverage plot is reversed, the orange F_{obs}(h)^{2} data cluster about zero, whereas the green D_{obs}(h) data span the range ±100.
The improvement in the precision of the Flack parameter which is gained by using differences or quotients is the result of transforming the observations into one set [A_{obs}(h)] which is sensitive to the structure but independent of the Flack parameter, and another [D_{obs}(h) or Q_{obs}(h)] which is sensitive to the Flack parameter, but very insensitive to the atomic parameters. The transformation means that correlation between the Flack parameter and the other refined parameters is essentially absent, and this explains why the results of the postrefinement methods are so similar to those obtained with the method outlined above.
Absolute structure refinements have been carried out for a series of 23 lightatom crystal structures with Friedif_{stat} values for Cu K radiation of between 9 and 36. Accurate values of the Flack parameter and its standard uncertainty were obtained, but with a precision higher and more realistic than conventional refinement. The results of the methods implemented in SHELXL2012, CRYSTALS and PLATON are essentially the same as those obtained when quotients or differences are explicitly coded into the refinement in TOPAS. We conclude that the potential problems discussed in §1 associated with a lack of correlation between the Flack parameter and the other structural parameters are not significant in absolute structure determinations of lightatom compounds. This justifies the use of postrefinement algorithms for absolute structure determination provided a complete set of Friedel pair intensity measurements is available.
On the basis of the results presented in the current paper, and those cited herein, it is possible to provide an outline of the treatment of diffraction data which leads to a reliable value of the Flack parameter with as low and realistic standard uncertainty as possible. The steps are described below in terms of intensity differences (D), although they could equally well be applied to quotients (Q)

The transformation of the data described in (ii) yields one set of observations [the centric data and A_{obs}(h)] which is sensitive to the structure but independent of the Flack parameter, and another [D_{obs}(h) or Q_{obs}(h)] which is sensitive to the Flack parameter, but highly insensitive to the atomic parameters. The agreement between A_{obs}(h) and A_{model}(h) is usually much better than between D_{obs}(h) and D_{model}(h) or their equivalents based on quotients (e.g. Flack et al., 2011; Parsons, Pattison et al., 2012); one advantage of the transformation of data into A and D (or Q) is that different schemes for the selection of outliers and weights can be applied to each. A conventional refinement is compatible only with a single `onesizefitsall' weighting scheme. Use of the transformed data also removes correlation between the Flack parameter and the other structural parameters.
All data in this work were collected with Cu K radiation, although there is no reason why the methods described could not be applied to data from more than one source, for example, Cu K radiation for the Friedelaveraged intensity data A_{obs}(h) and Cr K radiation for the difference or quotient data D_{obs}(h) or Q_{obs}(h). This will be investigated in due course.
We thank Dr Alan Coelho for making the substantial enhancements to TOPASAcademic which made the work reported in this paper possible, and Mr Philippe Piechon, Mr Tom Heathcote, Mr Paul McGovern and Dr Oliver Presly for the collection of data used in this work. We would also like to acknowledge the continued interest and advice given by Dr Richard Cooper, Professor George Sheldrick, Dr Devinder Sivia and Dr David Watkin. We thank Dr Robert Westwood (Cyclacel Ltd) for samples of R and SCYCLO, and the anonymous referees for their comments on the manuscript of this paper.
Abrahams, S. C. & Keve, E. T. (1971). Acta Cryst. A27, 157165.
Bernardinelli, G. & Flack, H. D. (1985). Acta Cryst. A41, 500511.
Betteridge, P. W., Carruthers, J. R., Cooper, R. I., Prout, K. & Watkin, D. J. (2003). J. Appl. Cryst. 36, 1487.
Beurskens, P. T., Beurskens, G., Bosman, W. P., de Gelder, R., GarciaGranda, S., Gould, R. O., Israel, R. & Smits, J. M. M. (1996). DIRDIF96. University of Nijmegen, The Netherlands.
Blessing, R. H. (1995). Acta Cryst. A51, 3338.
Blessing, R. H. (1997). J. Appl. Cryst. 30, 421426.
BrukerNonius (2006). SAINT, Version 7. BrukerAXS, Madison, Wisconsin, USA.
Coelho, A. A. (2012). TOPASAcademic, Version 5. Bruker AXS, Karlsruhe, Germany.
Coelho, A. A., Evans, J., Evans, I., Kern, A. & Parsons, S. (2011). Powder Diffr. 26, s22S25.
David, W. (2004). J. Res. Natl Inst. Stand. Technol. 109, 107.
David, W. I. F., Ibberson, R. M. & Matsuo, T. (1993). Proc. R. Soc. London Ser. A, 442, 129146.
Dittrich, B., Strumpel, M., Schäfer, M., Spackman, M. A. & Koritsánszky, T. (2006). Acta Cryst. A62, 217223.
Flack, H. D. (1983). Acta Cryst. A39, 876881.
Flack, H. D. (2012). Acta Cryst. C68, e13e14.
Flack, H. D. & Bernardinelli, G. (1999). Acta Cryst. A55, 908915.
Flack, H. D. & Bernardinelli, G. (2000). J. Appl. Cryst. 33, 11431148.
Flack, H. D. & Bernardinelli, G. (2008a). Acta Cryst. A64, 484493.
Flack, H. D. & Bernardinelli, G. (2008b). Chirality, 20, 681690.
Flack, H. D., Sadki, M., Thompson, A. L. & Watkin, D. J. (2011). Acta Cryst. A67, 2134.
Flack, H. D. & Shmueli, U. (2007). Acta Cryst. A63, 257265.
Hamilton, W. C. & Abrahams, S. C. (1970). Acta Cryst. A26, 1824.
Hooft, R. W. W., Straver, L. H. & Spek, A. L. (2008). J. Appl. Cryst. 41, 96103.
Hooft, R. W. W., Straver, L. H. & Spek, A. L. (2010). J. Appl. Cryst. 43, 665668.
Le Page, Y., Gabe, E. J. & Gainsford, G. J. (1990). J. Appl. Cryst. 23, 406411.
Merli, M., Ungaretti, L. & Oberti, R. (2000). Am. Mineral. 85, 532542.
Murphy, P. J., Thomas, D. A., Sievert, N., Caulket, P. W. R. & Parsons, S. (2010). Lett. Org. Chem. 7, 508510.
Oxford Diffraction (2010). CrysAlis PRO, Version 1.171.33.55. Oxford Diffraction Ltd, Yarnton, Oxfordshire, England.
Palatinus, L. & Chapuis, G. (2007). J. Appl. Cryst. 40, 786790.
Parsons, S. & Flack, H. (2004). Acta Cryst. A60, s61.
Parsons, S., Pattison, P. & Flack, H. D. (2012). Acta Cryst. A68, 736749.
Parsons, S., Wagner, T., Presly, O., Wood, P. A. & Cooper, R. I. (2012). J. Appl. Cryst. 45, 417429.
Prince, E. (2004). Mathematical Techniques in Crystallography and Materials Science, 2nd ed. Berlin: Springer.
RodríguezCarvajal, J. & González Platas, J. (2009). CrysFML. Institut Laue Langevin, Grenoble, France, Universidad de La Laguna, La Launa, Spain.
Sheldrick, G. M. (2001). XPREP. University of Göttingen, Germany.
Sheldrick, G. M. (2008a). SADABS, Version 20081. University of Göttingen, Germany.
Sheldrick, G. M. (2008b). Acta Cryst. A64, 112122.
Sheldrick, G. M. (2012). SHELXL2012. University of Göttingen, Germany.
Spek, A. L. (2003). J. Appl. Cryst. 36, 713.
Thompson, A. L. & Watkin, D. J. (2011). J. Appl. Cryst. 44, 10171022.
Wang, S., McClue, S. J., Ferguson, J. R., Hull, J. D., Stokes, S., Parsons, S., Westwood, R. & Fischer, P. M. (2001). Tetrahedron Asymmetry, 12, 28912894.