A systematic approach using the SFCHECK software

Wodak, S. J.; Vagin, A. A.; Richelle, J.; Das, U.; Pontius, J.; Berman, H. M.

doi:10.1107/97809553602060000708

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 21.2, pp. 510-517 | 1 | 2 |

Section 21.2.3.1. A systematic approach using the SFCHECK software

S. J. Wodak,^a ^* A. A. Vagin,^b J. Richelle,^b U. Das,^b J. Pontius^b and H. M. Berman^c

^aUnité de Conformation de Macromolécules Biologiques, Université Libre de Bruxelles, avenue F. D. Roosevelt 50, CP160/16, B-1050 Bruxelles, Belgium, and EMBL–EBI, Wellcome Trust Genome Campus, Hinxton, Cambridge CB10 1SD, England, ^bUnité de Conformation de Macromolécules Biologiques, Université Libre de Bruxelles, avenue F. D. Roosevelt 50, CP160/16, B-1050 Bruxelles, Belgium, and ^cDepartment of Chemistry, Rutgers University, 610 Taylor Road, Piscataway, NJ 08854-8087, USA
Correspondence e-mail: shosh@ucmb.ulb.ac.be

21.2.3.1. A systematic approach using the SFCHECK software

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21.2.3.1.1. Tasks performed by SFCHECK

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21.2.3.1.1.1. Treatment of structure-factor data and scaling

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SFCHECK reads in the structure-factor data written in mmCIF format. It then performs the following operations: Reflections are excluded if they are systematically absent, negative, or have flagged σ values (99.9). Equivalent reflections are merged. The amplitudes of missing reflections are approximated by taking the average value for the corresponding resolution shell.

From the model coordinates read from the PDB (or mmCIF) atomic coordinates file, SFCHECK calculates structure factors and scales them to the observed structure factors. The scaling factor, S, is computed using a smooth cutoff for low-resolution data (Vaguine et al., 1999) (Table 21.2.3.1). This involves the calculation of the observed and calculated overall B factors from the standard deviations of the Gaussian fitted to the Patterson origin peaks [see Table 21.2.3.1 and Vaguine et al. (1999)]. In addition, SFCHECK also estimates the overall anisotropy of the data, following the approach of Sheriff & Hendrickson (1987), and applies the anisotropic scaling after the Patterson scaling is performed (Murshudov et al., 1998).

Table 21.2.3.1| top | pdf |
Parameters computed for the analysis of the structure-factor data

The first column lists the parameter, the second column gives the formula or definition of the parameter and the third column contains a short description of the meaning of the parameters when warranted.

Parameter	Formula/definition	Meaning
Completeness (%)	Percentage of the expected number of reflections for the given crystal space group and resolution
B_overall (Patterson)	$[8\pi^{2} \sigma_{\rm Patt}/(2)^{1/2}]$ ^†	Overall B factor
R_stand(F)	$[\langle \sigma (F)\rangle/\langle F \rangle]$ ^‡	Uncertainty of the structure-factor amplitudes
Optical resolution	$[(\sigma_{\rm Patt}^{2} + \sigma_{\rm sph}^{2})^{1/2}]$ ^† ^§	Expected minimum distance between two resolved atomic peaks
Expected optical resolution	Optical resolution computed considering all reflections
$[\hbox{CC}_{F}]$	$[\displaystyle{\langle F_{\rm obs} F_{\rm calc}\rangle - \langle F_{\rm obs}\rangle\langle F_{\rm calc}\rangle \over \left[(\langle F_{\rm obs}^{2} \rangle - \langle F_{\rm obs}\rangle^{2}) (\langle F_{\rm calc}^{2}\rangle - \langle F_{\rm calc}\rangle^{2})\right]^{1/2}}]$	Correlation coefficient between the observed and calculated structure-factor amplitudes
S	$[\left\{{\textstyle\sum\displaystyle (F_{\rm obs} f_{\rm cutoff})^{2} \over \textstyle\sum\displaystyle \left[F_{\rm calc} \exp (- B_{\rm diff}^{\rm overall} s^{2}) f_{\rm cutoff}\right]^{2}}\right\}^{1/2}]$ ^¶	Factor applied to scale $[F_{\rm calc}]$ to $[F_{\rm obs}]$
$[f_{\rm cutoff}]$	$[1 - \exp (- B_{\rm off} s^{2})]$ ^††	Function applied to obtain a smooth cutoff for low-resolution data

^† $[\sigma_{\rm Patt}]$ is the standard deviation of the Gaussian fitted to the Patterson origin peak.
^‡F is the structure-factor amplitude, and $[\sigma({F})]$ is the structure-factor standard deviation. The brackets denote averages.
^§ $[\sigma _{\rm sph}]$ is the standard deviation of the spherical interference function, which is the Fourier transform of a sphere of radius $[1/d_{\min}]$ , with $[d_{\rm min}]$ being the minimum d spacing.
^¶ $[B_{\rm diff}^{\rm overall} = B_{\rm obs}^{\rm overall} - B_{\rm calc}^{\rm overall}]$ is added to the calculated overall B factor, $[B_{\rm overall}]$ , so as to make the width of the calculated Patterson origin peak equal to the observed one; s is the magnitude of reciprocal-lattice vector.
^†† $[B_{\rm off} = 4 d_{\rm max}^{2}]$ , where s and $[d_{\rm max}]$ , respectively, are the magnitude of the reciprocal-lattice vector and the maximum d spacing.

To assess the quality of the structure-factor data, the program computes four additional quantities (see Table 21.2.3.1 for details): the completeness of the data, the uncertainty of the structure-factor amplitudes, the optical resolution and the expected optical resolution. The latter two quantities represent the expected minimum distance between two resolved atomic peaks in the electron-density map when the latter is computed with the set of reflections specified by the authors and with all the reflections, respectively.

21.2.3.1.1.2. Global agreement between the model and experimental data

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To evaluate the global agreement between the atomic model and the experimental data, the program computes three classical quality indicators: the R factor, $[R_{\rm free}]$ (Brünger, 1992b) and the correlation coefficient $[\hbox{CC}_{F}]$ between the calculated and observed structure-factor amplitudes (Table 21.2.3.1). The R factor is computed using all the reflections considered (except those approximated by their average value in the corresponding resolution shell) and applying the same resolution and σ cutoff as those reported by the authors. $[R_{\rm free}]$ is computed using the subset of reflections specified by the authors. In addition, the R factor is evaluated using the `non-free' subset of reflections (those not used to compute $[R_{\rm free}]$ ). The correlation coefficient is computed using all reflections from the reported high-resolution limit, applying the smooth low-resolution cutoff (see Table 21.2.3.1) but no σ cutoff.

21.2.3.1.1.3. Estimations of errors in atomic positions

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The errors associated with the atomic positions are expressed as standard deviations (σ) of these positions. SFCHECK computes three different error measures. One is the original error measure of Cruickshank (1949). The second is a modified version of this error measure, in which the difference between the observed and calculated structure factors is replaced by the error in the experimental structure factors. The first two error measures are the expected maximal and minimal errors, respectively, and the third measure is the diffraction-component precision indicator (DPI). The mathematical expressions for these error measures are given in Table 21.2.3.2, and further details can be found in Vaguine et al. (1999).

Table 21.2.3.2| top | pdf |
Estimation of errors in atomic coordinates

Parameter	Formula/definition	Meaning
$[\sigma(x)]$	$[\displaystyle{\sigma\hbox{(slope)} \over \hbox{curvature}}]$ ^†	Standard deviation of the atomic coordinates following Cruickshank (1949) for the minimal and maximal errors (Vaguine et al., 1999)
σ(slope) for maximal error	$[\displaystyle{2\pi \left\{\textstyle\sum\displaystyle \left[h^{2} (F_{\rm obs} - F_{\rm calc})^{2}\right]\right\}^{1/2} \over V_{\rm unit \ cell} a}]$ ^‡	Expression for σ(slope) in the expected maximal error following Cruickshank (1949)
Curvature	$[\displaystyle{2\pi \textstyle\sum\displaystyle (h^{2} F_{\rm obs}) \over V_{\rm unit \ cell} a^{2}}]$	Expression for the curvature following Murshudov et al. (1997)
σ(slope) for minimal error	$[\displaystyle{2\pi^2 \left\{\textstyle\sum\displaystyle \left[h^{2} \sigma (F_{\rm obs})^{2}\right]\right\}^{1/2} \over V_{\rm unit \ cell} a}]$ ^§	Expression for σ(slope) in the expected minimal error, following Cruickshank (1949)
DPI	$[\displaystyle{\sigma (x) = \left({N_{\rm atoms} \over N_{\rm obs} - 4 N_{\rm atoms}}\right)^{1/2} c^{-1/3} d_{\min} R}]$ ^¶	Atomic coordinate error estimate following Cruickshank (1996)

^†σ(slope) and curvature are the slope and curvature of the electron-density map at the atomic centre, in the x direction, for spherically symmetric peaks; $[\sigma (x)\simeq \sigma(y)\simeq \sigma(z)]$ .
^‡a is the crystal unit-cell length, h is the Miller index and V _{unit cell} the unit-cell volume.
^§ $[\sigma(F_{\rm obs})]$ is the standard deviation of the structure-factor amplitude.
^¶c is the structure-factor data completeness expressed as a fraction (0–1), R is the conventional R factor, $[N_{\rm atoms}]$ is the total number of atoms in the unit cell, $[N_{\rm obs}]$ is the total number of observed reflections and $[d_{\rm min}]$ is the minimum d spacing.

21.2.3.1.1.4. Local agreement between the model and the experimental data

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In addition to the global structure quality measures, SFCHECK also determines the quality of the model in specific regions. Several quality estimators can be calculated for each residue in the macromolecule and, whenever appropriate, for solvent molecules and groups of atoms in ligand molecules. These estimators are the normalized atomic displacement (Shift), the correlation coefficient between the calculated and observed electron densities (Density correlation), the local electron-density level (Density index), the average B factor (B-factor) and the connectivity index (Connect), which measures the local electron-density level along the molecular backbone. These quantities are computed for individual atoms and averaged over those composing each residue or group of atoms [see Table 21.2.3.3 and Vaguine et al. (1999) for details].

Table 21.2.3.3| top | pdf |
Parameters computed by SFCHECK to assess the quality of the model in specific regions

Parameter	Formula/definition	Meaning
Shift	$[(1/N\sigma)\textstyle\sum\limits_{i}^{N}\displaystyle \Delta_{i},\hbox{ with } \Delta_{i} = (\hbox{gradient}_{i}/\hbox{curvature}_{i})]$ ^†	Normalized average atomic displacement computed over a group of atoms or residue; reflects the tendency of the group of atoms to move from their current position
Density correlation	$[\displaystyle{\textstyle\sum\displaystyle \rho_{\rm calc}(x_{i})[2\rho_{\rm obs}(x_{i}) - \rho_{\rm calc}(x_{i})] \over \left(\left[\textstyle\sum\displaystyle \rho_{\rm calc}^{2} (x_{i})\right]\left\{\textstyle\sum\displaystyle \left[2\rho_{\rm obs}(x_{i}) - \rho_{\rm calc}(x_{i})\right]^{2}\right\}\right)^{1/2}}]$ ^‡	Electron density correlation coefficient computed over a group of atoms or residue; reflects the local agreement of the model with the electron density
Density index	$[\left[\textstyle\prod\displaystyle \rho(x_{i})\right]^{1/N}/\langle \rho \rangle_{\rm all \ atoms}]$ ^§	Reflects the level of the electron density for a group of atoms; is a local measure of the density level
Connect		Same as Density index, but considering only backbone atoms.^¶

^†Gradient _i is the gradient of the $[F_{\rm obs} - F_{\rm calc}]$ map with respect to the atomic coordinates, curvature _i is the curvature of the model map computed at the atomic centre (see Agarwal, 1978

), N is the number of atoms in the group considered and σ is the standard deviation of the $[\Delta_{i}]$ values computed in the structure.
^‡ $[\rho_{\rm calc}(x_{i})]$ and $[\rho_{\rm obs}(x_{i})]$ are, respectively, the electron density computed from calculated and observed structure-factor amplitudes at the atomic centre. The summation is performed over all the atoms in the group considered. For polymer residues, D_corr is computed separately for backbone and side-chain atoms. For the calculation of the electron density at the atomic centre, see Vaguine et al. (1999)

.
^§ $[[\prod{\rho (x_{i})}]^{1/N}]$ is the geometric mean of the $[2F_{\rm obs} - F_{\rm calc}]$ electron density of the atom subset considered and $[\langle \rho \rangle_{\rm all \ atoms}]$ is the average electron density of the atoms in the structure. For water molecules or ions which are represented by a unique atom, the above expression reduces to the ratio $[\rho(x_i)/\langle \rho \rangle_{\rm all \ atoms}]$ .
^¶Backbone atoms are N, C, C^α for proteins and P, O5′, C5′, C3′, O3′ for nucleic acids.

21.2.3.1.2. Evaluation of individual structures

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Figs. 21.2.3.1–21.2.3.3 summarize the analysis carried out by SFCHECK on the protein rusticyanin from Thiobacillus ferrooxidans (1RCY) (Walter et al., 1996). Fig. 21.2.3.1 displays the numerical results from the analysis of the structure-factor data and from the evaluation of the global agreement between the model and the data. The R-factor and $[R_{\rm free}]$ values, computed by SFCHECK (Model vs. Structure Factors panel) using the identical reflection subset to that reported by the authors (Refinement panel), show negligible differences with the reported values. These differences are 0.175 versus 0.172 for the R factor and 0.25 versus 0.243 for $[R_{\rm free}]$ . The small R-factor difference may stem from the fact that SFCHECK considers a somewhat different number of reflections (9144) than the authors (9098), although it uses the same d-spacing range and σ cutoff as those reported.

Figure 21.2.3.1| top | pdf |

Typical SFCHECK output in PostScript format, illustrated for the protein rusticyanin from Thiobacillus ferrooxidans (1RCY) (Walter et al., 1996). Summary panels displaying the numerical results from the analysis of the deposited structure-factor data and from the evaluation of the global agreement between the model and these data. The top elongated panel lists the PDB title record, deposition date and PDB code. The Crystal panel summarizes the crystal parameters, provided by the authors, as read from the model input files. The Model and Refinement panels list the information provided by the authors on the model and the refinement procedure, respectively. This information is read from the PDB coordinates entry. The Structure Factors panel summarizes the information on the deposited structure-factor data (Input section) and on the data used and criteria computed by SFCHECK (SFCHECK section). The numbers given under `Anisotropic distribution of Structure Factors' are the ratios of the eigenvalues of the symmetric anisotropic thermal tensor to the maximum eigenvalues. The Model vs. Structure Factors panel summarizes the results of the verifications made by SFCHECK. The values listed under `Anisothermal Scaling (Beta)' are those of the overall anisotropic thermal tensor ( $[b_{11}, b_{12}, b_{13}, b_{22}, b_{23}, b_{33}]$ ). The meanings of other listed quantities are either self-explanatory or are described in the text.

Figure 21.2.3.2| top | pdf |

Graphical output from the SFCHECK analysis of global characteristics of the structure-factor data and the model agreement with those data for the same structure as in Fig. 21.2.3.1. From left to right and top to bottom: the Wilson plot; the behaviour of the optical resolution as a function of the nominal resolution (d spacing); the data completeness and structure-factor standard error as a function of the d spacing; the maximal and minimal coordinate error dependence on d spacing; a stereographic projection of the averaged radial structure-factor data completeness; and, finally, the R-factor dependence and Luzzati plots for a given atomic error.

Figure 21.2.3.3| top | pdf |

SFCHECK evaluation summary of the local agreement between the model and the electron density for the same structure as in Fig. 21.2.3.1. Five criteria are plotted for each residue of the macromolecule (designated by its one-letter code), as well as for each solvent molecule (w), or hetero group. These criteria are: (1) Shift, (2) Density correlation, (3) Density index, (4) B factor, (5) Connect. The definitions of these criteria are given in the text. Note that the values of the Connect parameter are truncated to a maximum of 1. The SFCHECK output shown in Figs. 21.2.3.1–21.2.3.3 was generated using routines from PROCHECK kindly provided by R. Laskowski.

The information in Figs. 21.2.3.1 and 21.2.3.2 allows one to make some judgement about the quality of the structure-factor data for this protein. The relatively high resolution of this structure (1.9 Å) is accompanied by limited data completeness (82.1%). The Rstand(F) plot on the same graph shows, furthermore, a decrease in quality of the high-resolution data (2.2–1.9 Å). The average radial completeness plot (bottom left-hand plot of Fig. 21.2.3.2) allows one to identify the regions in reciprocal space with incomplete data.

Fig. 21.2.3.3 presents the SFCHECK analysis of the local agreement of the model with the electron density for 1RCY. The shift plot shows that both backbone and side-chain shifts are of comparable size, with several residues (1, 2, 16, 25) displaying shifts as high as 0.16 Å. The density correlation is excellent throughout the entire molecule, except for residues 2, 16 and 29, which display poorer correlation. In particular, the side chains of these residues seem to be more poorly defined in the electron-density map. The backbone density index plunges in a few regions, notably at the N-terminus (residues 5–7) and in the segments comprising residues 25–30 and 68–70. The side chains display, in general, a poorer density index than the backbone, with some regions (for example, residues 5–7, 23–30, 58–60) displaying rather low density indices. The same segments also display higher backbone and side-chain B factors. The backbone Connect parameter is, on the other hand, quite good throughout, except for residues 5–7 and 28–29 (Fig. 21.2.3.3).

Water molecules (labeled w in the SFCHECK output) are also evaluated. The relevant plots for these molecules are those of the Shift, Density index and B factor parameters. The first 50 or so water molecules in the list (appearing sequentially along the plot from left to right) tend to display a higher density index and lower B factors (< 30 Å²) than the following molecules in the list. They thus seem to be more reliably positioned than subsequent molecules, whose density indices sometimes drop perilously. A steady climb of the B factors is also apparent as one goes down the list of water molecules. The analysis of the density indices and B factors of individual water molecules performed by SFCHECK could be a very useful guide in investigations of the properties of crystallographic water molecules and their interactions with protein atoms.

21.2.3.1.3. Quality assessment based on surveys across structures

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21.2.3.1.3.1. Assessing the quality of a structure as a whole

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As for the evaluation of the geometric and stereochemical parameters of the model, surveying the same quality indicators across many structures is crucial. It allows one to establish the ranges of expected values for each indicator and to identify structures with unexpected features – those for which the values of one or more quality indicators are outside their standard range.

The global quality indicators computed by SFCHECK are the nominal resolution (d spacing), the R factor, $[R_{\rm free}]$ , the minimal and maximal errors in atomic positions, the DPI, and the correlation coefficient $[\hbox{CC}_{F}]$ . Another type of global quality indicator can be obtained by computing the average values of local quality measures across a given structure. This can be done for the per-residue (or per-group) atomic displacement and the Density correlation and B factor parameters as well as for the Density index and Connect parameters.

Many of the geometric and stererochemical quality indicators vary as a function of resolution – some linearly and some not (Laskowski et al., 1993). This is also the case for most of the global quality indicators described here. Examples of this dependence are given in Fig. 21.2.3.4, which shows how the correlation coefficient, the maximal error, the average atomic displacement and average density index vary as a function of resolution in the 104 nucleic acid structures surveyed. This variation is approximately linear for all four parameters. The density correlation and average density index decrease, whereas the maximal error and average atomic displacements increase, as the resolution gets poorer. In all four plots of Fig. 21.2.3.4, the points tend to display significant scatter as the d spacing increases, and at least three points, corresponding to the same three structures, appear as outliers in all plots. These structures also appear as outliers in the analysis of other parameters. A closer examination revealed that in the vast majority of the cases, the abnormal behaviour of these structures could be traced back to problems with data formats or errors that occurred during data deposition and entry processing.

Figure 21.2.3.4| top | pdf |

Variation of global quality indicators with the nominal resolution (d spacing) of the crystallographic data. The following quality indicators were computed by SFCHECK for each of the 104 nucleic acid crystal structures considered in the study of Das et al. (2001): (a) correlation coefficient, (b) maximal error, (c) average atomic displacement and (d) average density index. For the meaning of the various quantities see Table 21.2.3.2. The three structures for which the reported and re-computed R factors differ by more than 10% are highlighted as black circles. The NDB (PDB) codes for these structures are ADFB72 (256D), ADF073 (257D) and ADJ081 (320D).

As the number of structures with deposited structure-factor data becomes large enough, plots such as those of Fig. 21.2.3.4 could be used to define the expected range of values for a quality indicator in a structure determined at a given resolution or refined under given conditions. Structures yielding quality indicators outside this range could then be identified as unusual on a more solid statistical basis.

21.2.3.1.3.2. Assessing the quality in specific regions of a model

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The main purpose for computing the four local quality measures, the B factor, the Density index, the atomic displacement (Shift) and the Density correlation (Table 21.2.3.3), is to identify problem regions in a model. In order to do this effectively, it is necessary to evaluate the degree of redundancy between these measures and to establish the standard ranges for their values. The latter task, in particular, is not straightforward since it depends crucially on the quality of the experimental data and biases introduced by the scaling procedure and refinement protocol. In this regard, several issues are presently still under investigation.

A preliminary investigation of the mutual relations between the above-mentioned local measures has been performed in several protein and nucleic acid structures taken individually. This shows that that the B factor is strongly correlated with the density index, as illustrated in Fig. 21.2.3.5(a), and to a lesser extent with the atomic displacement (Fig. 21.2.3.5b). A weaker correlation was detected between the latter three measures and the residue density correlation (data not shown).

Figure 21.2.3.5| top | pdf |

Pairwise correlations between the various local quality indicators computed by SFCHECK. (a) Correlation between the average residue B factor and the density index, and (b) between the B factor and the atomic displacement. The values displayed were computed for residues in the crystal structure of carboxypeptidase (1YME). The meaning of the parameters displayed is given in Table 21.2.3.3.

Analyses across structures could, in principle, be carried out for all four local measures computed by SFCHECK, provided these measures are not subject to systematic biases due to differences in scaling procedures and refinement practices. Such biases are, however, well known for the B factors of individual atoms or residues. This is illustrated in Fig. 21.2.3.6(a). This figure plots, side-by-side, the average residue B factors in 21 protein structures determined at different d spacings. It shows that for proteins determined at poorer resolution (d spacing above 2 Å), the B factors of different structures are systematically shifted relative to one another. Such systematic shifts are much smaller for structures determined at 2 Å resolution or better (Fig. 21.2.3.6a). This is not surprising, since in lower-resolution structures, $[N_{\rm refl}/N_{\rm atoms}]$ is often too low (< 4) to yield meaningful values for the B factors.

Figure 21.2.3.6| top | pdf |

B factors and density indices for residues across different structures. (a) Average B values in residues of 21 protein structures; (b) average density indices of the same set of residues and structures. The 21 protein structures analysed are from the following PDB entries: 1YME, 1MCT, 1PDO, 1VHH, 1WBA, 1CNS, 1RG7, 1UCO, 1BRO, 1EMB, 1FXI, 1KBA, 1XSM, 1HIB, 1IVF, 1QRS, 1AGX, 1NSN, 1ZOO, 1TGK, 1JCK.

Interestingly, the residue Density index, a very different parameter from the B factor, which measures the level of electron density at the atomic positions, does not display the systematic shifts observed for the B factors (Fig. 21.2.3.6b), despite the fact that the two measures are rather strongly correlated in individual structures. An indicator such as this one, and ultimately the atomic s.u.'s themselves, should be better suited for analysing and comparing the trends in the quality of specific regions of the model across different structures.

References

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International Tables for Crystallography (2006). Vol. F. ch. 21.2, pp. 510-517