 1. Introduction
 2. General information
 3. Twofold pseudoorthorhombic twinning, a straightforward example
 4. Threefold pseudohexagonal twinning using a nonconventional space group setting, B21
 5. Fourfold pseudotetragonal twinning via an Icentred supercell
 6. Conclusions
 Supporting information
 References
 1. Introduction
 2. General information
 3. Twofold pseudoorthorhombic twinning, a straightforward example
 4. Threefold pseudohexagonal twinning using a nonconventional space group setting, B21
 5. Fourfold pseudotetragonal twinning via an Icentred supercell
 6. Conclusions
 Supporting information
 References
research communications
Practical hints and tips for solution of pseudomerohedric twins: three case studies
^{a}Department of Chemistry, University of Kentucky, Lexington, KY 40506, USA
^{*}Correspondence email: s.parkin@uky.edu
by pseudomerohedry is a common phenomenon in smallmolecule crystallography. In cases where twincomponent volume fractions are markedly different, structure solution is often no more difficult than for nontwinned structures of similar complexity. When twincomponent volume fractions are similar, however, structure solution can be much more of a problem. This paper presents hints and tips for such cases by means of three worked examples. The first example presents the most common (and simplest) case of a twocomponent pseudoorthorhombic twin. The second example describes structure solution of a reticular threefold pseudohexagonal twin that benefits from use of an unconventional spacegroup setting. The third example covers structure solution of a reticular fourfold pseudotetragonal twin. All three structures are ultimately shown to be monoclinic crystals that twin as a consequence of unitcell metrics that mimic those of higher symmetry crystal systems.
Keywords: pseudomerohedry; pseudoorthorhombic; pseudotetragonal; pseudohexagonal; reticular; twinning; HKLF 5 format.
1. Introduction
e.g. Hahn & Klapper, 2006 and references therein), nomenclature and classification (Donnay & Donnay, 1959; Nespolo & Ferraris, 2003; Nespolo, 2015, 2019), identification and structure (HerbstIrmer & Sheldrick, 1998, 2002; Parsons, 2003; Petríček et al., 2016; Sevvana et al., 2019) for molecular crystals have been extensively covered in the literature, and need not be repeated here. Nevertheless, a brief introduction is warranted. Within a twin, the component parts are mapped onto each other via twin operations (inversion, rotation, reflection) that occur with respect to a twin element (point, axis, plane). The family of symmetrically equivalent twin operations, i.e. those that result in the same mapping of component parts onto each other, constitute the twin law. This definition distinguishes a true twin from a mere aggregate (i.e. a random conglomerate of two or more pieces). The term `twin' refers to the whole crystalline entity, which is composed of individuals or components (Nespolo, 2015) related by the Although sometimes used interchangeably, the terms twin domain and individual are not synonymous. An individual might comprise a single domain or many domains. Domains having identical orientations comprise a single domain state. For a macroscopic twin, `domain state' and `individual' are interchangeable (Nespolo, 2019). Twins have been classified in different ways (e.g. Donnay & Donnay, 1974; Nespolo & Ferraris, 2000; Hahn & Klapper, 2006; Petríček et al., 2016). Commonly used terms include twinlatticesymmetry (TLS), twinlatticequasisymmetry (TLQS), twinning by merohedry, pseudomerohedry, nonmerohedry etc., with further subdivisions possible. Definitions of each are given in the Online Dictionary of Crystallography (IUCr ODC, 2021).
in crystallography is the phenomenon by which a crystalline entity may be composed of two or more crystals that are mutually related by precise mathematical relationships. The theoretical aspects of (seeIn the diffraction pattern of a twin, the reciprocal lattices of domains comprising an individual superimpose exactly, resulting in the diffraction pattern of that individual. Diffraction patterns from each individual overlap, to an extent determined by the orientation matrix (Busing & Levy, 1967). Similar to nontwinned crystals, a (pseudo)merohedric twin requires just one orientation matrix. In such cases, all diffraction maxima receive a contribution from each individual present. The term reticular is used as a modifier for by (pseudo)merohedry in particular cases where only a welldefined fraction of individual points, and hence twinrelated diffraction maxima, overlap. The diffraction pattern of a reticular (pseudo)merohedric twin may also be indexed by just a single orientation matrix, but any individual only contributes to the aforementioned fraction, the reciprocal of the twin index (Donnay & Donnay, 1959), of the observed diffraction maxima. Any deviation from exact overlap for by pseudomerohedry is quantified by obliquity (Friedel, 1926; Donnay & Donnay, 1959; Wolten, 1966) or by twin misfit (Nespolo & Ferraris, 2007). by nonmerohedry results in a combination of full, partial, and nonoverlapping diffraction maxima. Indexing of such a twin requires a separate orientation matrix for each individual.
and weighted by their relative irradiated volume fractions, thereby producing the diffraction pattern of the (whole) twin. The degree of overlap from each individual thus readily provides, to a first approximation, a quick and convenient means of assessment. Exact superposition of individual reciprocal lattices occurs for Aside from the special case of is far more common in minerals and crystals of inorganic compounds than in organic or organometallic crystals. Close, but not symmetrically exact overlap, occurs for by pseudomerohedry, and is common in molecular crystals. Indexing of any crystal requires finding the link between the and the coordinate system of the diffractometer, which takes the form of a mathematical transformation, theFor novice or otherwise inexperienced crystallographers, solution and e.g. Parkin & Hope, 1998; Hou et al., 2019) or worse, e.g. incommensurate modulation (van Smaalen et al., 1995; Wagner & Schönleber, 2009), order–disorder phenomena (DornbergerSchiff, 1956) etc., once the has been accounted for, completion of pseudomerohedric twin structures is nowadays often no more difficult than nontwinned structures of similar complexity. Twin laws for by (pseudo)merohedry may be derived by decomposition of the (Flack, 1987). However, in many cases, plausible twin operations can be obtained simply by inspection of the metrics. In less obvious cases, computer programs (e.g. Aroyo et al., 2006; Boyle, 2014) have been written to derive twin laws using the algorithms described by Flack (1987). Details of structure solution itself, however, particularly tips and tricks for nontrivial cases, have received less attention. For many smallmolecule pseudomerohedric twins in which twin component fractions are notably different, diffraction from the major component is often sufficiently dominant that structure solution is quite straightforward. This paper presents, by way of three worked examples of differing complexity, practical tips and hints for the more problematic case of crystals twinned by pseudomerohedry in which the relative volume fractions of individuals are close to equal.
of twinned crystal structures can appear to be a daunting task. Nevertheless, in the absence of other problems such as extensive disorder (2. General information
HerbstIrmer & Sheldrick (1998, 2002) and others (e.g. Rees, 1980; Yeates, 1988) describe a number of general observations and statistics that have proven useful for the identification and diagnosis of In addition, the failure of conventional of structure solution has also been noted as a common consequence of (Parsons, 2003). Nowadays, most smallmolecule structures are solved by dualspace algorithms of one sort or another (e.g. Oszlányi & Sütő, 2004; Sheldrick, 2015a). Such methods have proven immensely successful, such that their failure, persists as a strong indicator of Even prior to data collection, evidence of is often apparent from optical microscopy (e.g. Fig. 1). Reentrant angles (e.g. Kitamura et al., 1979) between crystal faces, and optical extinction for transparent crystals viewed between crossed polarizers clearly indicate the presence of macroscopic twin domains (Hartshorne & Stuart, 1950; Nespolo & Ferraris, 2003 and references therein). Depending on the primary purpose of where possible, it is often advisable to perform microsurgery to extricate a singlecrystal fragment (Fig. 1a), though this is not always feasible (Fig. 1b). When modern autoindexing routines [e.g. in XAREA (Stoe & Cie, 2002); CrysAlis PRO (Rigaku OD, 2017); APEX3 (Bruker, 2016)] return welldefined unit cells that only partially account for the observed diffraction maxima, is often the culprit. Post data collection, reciprocallattice slice images a.k.a. `pseudoprecession pictures' can readily expose by nonmerohedry, but are usually less useful for (pseudo)merohedric twins (Fig. 2).
3. Twofold pseudoorthorhombic a straightforward example
3.1. Crystal and diffraction data assessment
Crystals of chlorotetrakis(imidazole)copper(II) chloride, C_{12}H_{16}Cl_{2}CuN_{8}, (Otieno et al., 2001), XUBNIR in the CSD (Groom et al., 2016), are monoclinic, with of type P2_{1}/n and unitcell parameters a = 8.8434 (2) Å, b = 13.2093 (4) Å, c = 13.8658 (5) Å, β = 90.0072 (18)°. Since the β angle is so close to 90°, the is metrically orthorhombic, even though the underlying symmetry is monoclinic. This situation corresponds to criterion (a) in the list of classic symptoms of outlined by HerbstIrmer & Sheldrick (1998). The molecular structure of XUBNIR, which consists of a squarebased pyramidal Cu^{II} with four Nbound imidazoles, a bound chlorine ligand and one free chloride anion (Fig. 3), seems innocuous. Data collection and processing for XUBNIR were also unremarkable. To allow the reader to follow along, a dataset is available in the supporting information.
Under the initial assumption that this is a routine structure, XPREP (Sheldrick, 2008) was used to set up files for structure solution. Not surprisingly, the program suggests a primitive orthorhombic which has a seemingly respectable R_{sym} of 4.2% (Table 1a). Analysis of however, does not lead to an acceptable orthorhombic (Table 1b). Impossible are another classic symptom of corresponding to criterion (e) described by HerbstIrmer & Sheldrick (1998). XPREP does suggest 2_{1} screw axes associated with each of a, b, and c, as well as an nglide plane perpendicular to b. Thus, given the information at hand, even in the absence of any particular knowledge of the suspected the obvious way forward is to consider the next lowest symmetry monoclinic. In XPREP, this requires overriding the default suggestion (Table 1c), upon which the program suggests a of type P2_{1}/n, with R_{sym} = 2.3%. Other potential monoclinic settings can be sidelined as possibilities at this point because they are not consistent with the nglide and they each give a worse R_{sym}, similar to that of the rejected orthorhombic cell.

3.2. Twin operations by inspection of unitcell parameters for XUBNIR
The essence of _{1} screw, mirror, a or caxial, or ndiagonal glide planes associated only with the b axis (assuming the monoclinic bunique convention is respected). For orthorhombic crystals, these symmetry elements may each be associated with a, b, and c. In the translational parts of screw and glide operations manifest only as so in the context of we need only consider the pointsymmetry operations rotation, reflection, and inversion (vide supra, Section 1). For consideration as twin operations in XUBNIR, that limits the analysis to mirror and twofold rotation operations associated with the a and c axes. Such mirror operations change the sign of just one index, while twofold rotations flip the sign of two indices. The feasible twin operations, expressed as (3×3) transformation matrices, are thus:
in this structure lies in the difference between orthorhombic and monoclinic symmetry. For a primitive monoclinic crystal, there could be twofold rotation, 2Matrices m_{[100]} and m_{[001]} describe reflection across mirror planes perpendicular to a and c while 2_{[100]} and 2_{[001]} describe 180° (i.e. twofold) rotation about a and c, respectively. The effect of these operations on the unitcell axes are shown in Fig. 4a–e. Since the structure is centrosymmetric, m_{[100]} and 2_{[100]} are equivalent for this as are m_{[001]} and 2_{[001]}. Similarly, since monoclinic symmetry has either m, 2 or 2/m (by convention associated with the b axis), the sign of b (i.e. mirror plane perpendicular to b) or signs of a and c (i.e. twofold rotation about b) can be flipped. Thus, in for this structure, m_{[100]}, 2_{[100]} m_{[001]}, 2_{[001]}, all produce the same effect when used as twin matrices to transform reflection indices, and thereby constitute the twin law.
3.3. Structure solution
The structure of XUBNIR does not solve with the correct SHELXT (Sheldrick, 2015a) or by chargeflipping (Oszlányi & Sütő, 2004) as implemented in PLATON (Spek, 2020). Recognizable, albeit rudimentary but ultimately usable solutions are, however, possible using the conventional programs SHELXS (Sheldrick, 2008) and SIR (Altomare et al., 1999), and possibly other programs not directly intended for twins. Nevertheless, the dualspace recycling algorithm used in SHELXD (Sheldrick, 2008) can include two twin components in a straightforward way, and results in a starting model that is quite easy to complete. The following instructions file for SHELXD was generated using XPREP, but has been hand edited to include twin matrix 2_{[001]}.
when using the iterative dualspace method inTITL XUBNIR in P2(1)/n
CELL 0.71073 8.8434 13.2093 13.8658 90.0000 90.0072 90.0000
ZERR 4.00 0.0002 0.0004 0.0005 0.0000 0.0018 0.0000
LATT 1
SYMM 0.5X, 0.5+Y, 0.5Z
SFAC C H N CL CU
UNIT 64 64 16 8 4
TWIN 1 0 0 0 1 0 0 0 1
FIND 15
PLOP 20 25 28
MIND 1.0 0.1
NTRY 1000
HKLF 4
END
In the above, 1000 trials (command NTRY) are overkill, but the structure is small so it runs quite quickly on modern computers. The default for twin component volume fractions is 0.5, but can be changed by a BASF parameter, as per SHELXL (Sheldrick, 2015b). The resulting preliminary model (Fig. 5a) is fairly complete. It is missing only three atoms and has the imidazole nitrogen atoms misassigned as carbon; all problems that are easily fixed. A few cycles of model building and (Fig. 5b–d) proves to be no more complicated than for a routine (nontwinned) singlecrystal structure. The final model has refined twin component volumes of about 54% and 46% and an R_{1} value of 2.5%
4. Threefold pseudohexagonal using a nonconventional setting, B2_{1}
Crystals of the chiral compound 1{(R)1[(3oxo2isoindolinoyl)methyl]2propenyl}5methyl2,3indolinedione, C_{21}H_{16}N_{2}O_{4}, Fig. 6, (Trost et al., 2020), WUGLES in the CSD (Groom et al., 2016), form as orange elongated hexagonal columnar needles. Initial indexing of the diffraction pattern gives a that appears to be primitive hexagonal. All attempts to solve the structure using hexagonal or trigonal symmetry, however, failed; reminiscent of criterion (d) described by HerbstIrmer & Sheldrick (1998). A chemically reasonable structure was eventually found that had twelve molecules in the of a of type P2_{1}. of this Z′ = 12 model as a threefold twin seemed promising, but became unstable when displacement parameters were made anisotropic. Subsequent analysis showed that for the pseudohexagonal setting, the individuals had to be Bcentred with an having Z′ = 6, and threefold twinned, requiring the unconventional (see, for example, Nespolo & Aroyo, 2016) setting B2_{1}, and similar twin component volumes. A thorough description of the in WUGLES was subsequently given by Nespolo et al. (2020). Detailed steps involved in structure solution, are given here. The full dataset is available in the supporting information.
4.1. Diffraction data analysis for WUGLES using XPREP
On reading the diffraction data into XPREP, the program suggests two plausible primitive unit cells, one hexagonal with R_{sym} = 7.8% and one monoclinic having R_{sym} = 2.0% (Table 2a). It also suggests five Ccentred orthorhombic and monoclinic unit cells, but these can be rejected immediately due to their unacceptable R_{sym} values of about 30% or so. All attempts to solve the structure using hexagonal or trigonal symmetry failed miserably. Indeed, since the R_{sym} for hexagonal is almost four times that of primitive monoclinic, XPREP suggests the latter as its default. The next task is to assign a tentative (Table 2b) indicate a 2_{1} screw axis parallel to b. Since the compound was known from the synthesis to be chiral and enantiopure, the suggestion of P2_{1}/m can be rejected, leaving only P2_{1}. Without additional information, this is the best we can do at this stage.

4.2. Suspected threefold and a plausible for WUGLES
The unitcell parameters for the asindexed monoclinic P setting have a ≃ c and β ≃120°, so threefold pseudomerohedric about the b axis is a reasonable supposition and is consistent with criterion (d) of HerbstIrmer & Sheldrick (1998). A threefold rotation requires successive rotational increments of 120°, with the third step reproducing the starting position. For positive rotation (anticlockwise) about b, this corresponds to the following matrices:
The transformational effects of these matrices on the unitcell axes are illustrated in Fig. 7.
4.3. Initial structure solution for WUGLES using P2_{1}
A single molecule of WUGLES has 27 nonhydrogen atoms. Given the relatively large volume of the primitive monoclinic cell [V = 10357.0 (9) Å^{3}], for P2_{1} to be correct, the for the pseudohexagonal cell could accommodate twelve molecules, i.e. 234 (C, N, O) atoms. Perhaps not surprisingly in view of the expected and large number of similar sized atoms, neither conventional nor iterative dualspace algorithms (SHELXT or in PLATON) are able to readily solve this structure. The dualspace recycling algorithm in SHELXD, however, is able to provide a starting model with recognizable chemical fragments. Although the current version of SHELXD (v2013/2) is restricted to just two twin components (rather than the suspected three in WUGLES), its success rate is higher if two twin components of equal volume are included (rather than just a single component), i.e. using the transpose of either the 3^{+}_{[010]} or matrices defined above. A further increase in the success rate is possible via random omission of some fraction of the atoms during the dualspace recycling using the command `WEED' in SHELXD, a form of randomized `omit' map (Bhat & Cohen, 1984), which has the effect of reducing phase bias for noncentrosymmetric structures. Thus, in the following input file for SHELXD written by XPREP, the added TWIN and WEED commands enable a dramatic reduction in the number of trials (command NTRY) from the default 1000 to about 20.
TITL WUGLES in P2(1)
CELL 1.54178 15.5993 49.062 15.6099 90.000 119.896 90.000
ZERR 24.00 0.0008 0.002 0.0008 0.000 0.002 0.000
LATT 1
SYMM X, 0.5+Y, Z
SFAC C H N O
UNIT 504 384 48 96
FIND 194
PLOP 259 324 362
MIND 1.0 0.1
TWIN 0 0 1 0 1 0 1 0 1
NTRY 20
WEED 0.3
HKLF 4
END
The resulting structure solution prior to any model building and a. This model is far from complete, but aside from a few disconnected parts, there are many recognizable molecular fragments, including the expected six and fivemembered rings. A few rounds of model building and isotropic with threefold included rapidly generates the whole for the P2_{1} cell (Fig. 8b), resulting in an R_{1} value of about 9.5%. Under normal circumstances, the next step would be to complete the structure by including anisotropic displacement parameters (ADPs) and adding hydrogen atoms. However, all attempts to refine ADPs for this model were wildly unstable. A search for missed symmetry using ADDSYM in PLATON, however, proved to be fruitless. Even with inclusion of an extensive battery of restraints, there were still hefty correlations between pairs of similargeometry molecules in the leastsquares At this stage, therefore, it seemed likely that the actual of each individual was only half as large as required by the current P2_{1} model (i.e. Z′ ought to be 6, not 12). Possible causes therefore included each component having a with either a or c (but not both) only half as long, or alternatively, Bcentring. Given the threefold about b, each scenario would result in a diffraction pattern for the twin that is indexable as apparently primitive and pseudohexagonal. To investigate further, the `LIST 8' command in SHELXL was used to generate `detwinned' data. The resulting SHELXL format fcf file contains the following information for each reflection: h, k, l, , , , , d, . The first five fields may be easily converted to an `HKLF 4' format data file using the unix (Linux, MacOS, etc.) utility awk [also available for Windows via the Cygwin project (Cygwin, 2020)], as follows:
is shown in Fig. 8awk '{printf "%4d%4d%4d%8.2f%8.2f\n", $1,$2,$3,$4,$5}' in.fcf > out.hkl
where in.fcf and out.hkl are the input `LIST 8' format fcf file (after removal of its format header) and output `HKLF 4' format hkl files, respectively. Comparison of intensities for the full dataset with this `detwinned' data (Table 3a,b) clearly show that for this unitcell setting, the individual is Bcentred. Thus, the pseudohexagonal originally indexed as primitive using reflections from the whole threecomponent twin, actually corresponds to threefold rotational of Bcentred cells of the individuals, each requiring an unconventional B2_{1}. Thus, in WUGLES is by reticular pseudomerohedry with zero obliquity (the twin axis is coincident with the unitcell b axis), but nonzero twin misfit (since β is not exactly 120°, twinrelated lattice points do not exactly superimpose). A rigorous analysis is given by Nespolo et al. (2020). This Bcentred cell may, of course, be transformed to a conventional with half the volume. However, such a smaller P2_{1} cell is pseudoorthorhombic, and thus the threefold nature of the becomes far less intuitive than for the larger pseudohexagonal cell. Moreover, using SHELXL would then require conversion of the `HKLF 4' format twinned datafile to a much larger `HKLF 5' format dataset. That is possible (e.g. by adaptation of the scheme in Appendix A), but for WUGLES, use of the unconventional Bcentred setting is far more elegant, at least for using SHELXL. Technical details of the equivalence of the B2_{1} versus P2_{1} description are described at length in Nespolo et al. (2020).

4.4. Structure solution and for WUGLES using B2_{1}
The initial (subsequently shown to be incorrect and unrefinable) P2_{1}, Z′ = 12 model in the larger primitive pseudohexagonal could quite easily be pared down by removing one member of each symmetryequivalent pair of molecules. Nonetheless, given the speed of modern computers, it is perhaps easier to simply resolve the structure using the B2_{1} setting. For SHELXD this requires one trivial edit to the instructions file, namely, changing the LATT command from `LATT 1' (primitive noncentrosymmetric) to `LATT 6' (Bcentred noncentrosymmetric). Using the previously obtained detwinned data and the symmetry of B2_{1}, SHELXD easily finds all the nonhydrogen atoms (Fig. 9a). In spite of the circuitous route taken to solve the structure, subsequent refinement carried out against the full dataset with threefold included proceeds smoothly for the B2_{1} model. It requires no constraints or restraints, even for a fully anisotropic model, with all hydrogen atoms having been found in differenceFourier maps and included in the (Fig. 9b). The could also be determined from the diffraction data using established methods (Flack, 1983; Hooft et al., 2008; Parsons et al., 2013).
5. Fourfold pseudotetragonal via an Icentred supercell
Not all cases of SHELXL. The of pinacol monohydrate has primitive monoclinic symmetry of type P2/n (space group 13), but is fourfold twinned by virtue of a pseudotetragonal Icentred The monohydrate phase of crystalline pinacol was identified by Pushin & Glagoleva (1922), but its structure (Fig. 10) remained unsolved until 2003 (Hao et al., 2005; SAXDUR in the CSD).
by pseudomerohedry can ultimately be accounted for using the TWIN command in5.1. Diffraction pattern indexing and data analysis for SAXDUR
The crystal used for SAXDUR initially indexed to give cell parameters a = 12.9001 (7) Å, b = 12.8941 (7) Å, c = 12.8917 (9) Å, α = 107.517 (3)°, β = 110.359 (3)°, γ = 110.581 (3)°. This triclinic setting was used for data collection to ensure that no experimental information was inadvertently lost or skipped, but without making any assumptions about crystal symmetry. The resulting dataset is available in the supporting information. The similarity of the above three axis lengths and of the β and γ angles, however, immediately portend transformation to a higher symmetry cell. A search for higher symmetry using XPREP returned eight possible centred cells; one tetragonalI, one orthorhombicI, three monoclinicI, and their three monoclinicC equivalents (which were dismissed as they have β ≃ 134°, thereby obscuring the pseudotetragonal symmetry). The Icentred cases are reproduced in Table 4. The tetragonalI cell (option A) was dismissed due to its much higher R_{sym} and because were inconsistent with any tetragonal [criterion (e) of HerbstIrmer & Sheldrick, 1998]. Moreover, the crystal itself did not exhibit optical extinction characteristic of tetragonal symmetry (Hao et al., 2005). For orthorhombicI, XPREP suggests a of type Ibca (Table 5a), but this also proved to be a dead end. All attempts to find a chemically reasonable structure for orthorhombicI failed, with or without consideration of This leaves the three monoclinicI settings (options D, F, H in Table 4), each having similar R_{sym} and cell angles all ∼90°. As a worstcase scenario, all three settings would need to be considered, with only the right one expected to yield a viable structure model. It makes sense to first consider the setting that gives the lowest R_{sym} (option F); with hindsight it also happens to be the correct choice. This cell has the same transformation matrix (from the initial primitive cell) as the orthorhombicI option and has consistent with space groups of type I2/a and Ia (Table 5b).


5.2. Suspected for SAXDUR
The unitcell metrics of the chosen Icentred monoclinic cell are consistent with pseudotetragonal fourcomponent about its c axis. For positive rotation, four successive 90° steps about c are required, yielding the following four matrices:
Two 90° steps generate a twofold rotation (as for a pseudoorthorhombic twin), while the fourth step regenerates the starting position. The presence of pseudotetragonal
would result in significant populations for four individuals, as opposed to two if it were a pseudoorthorhombic twin.5.3. Initial structure `solution' for SAXDUR using I2/a
The following instructions file for SHELXD created by XPREP has been edited to include matrix 4^{2+}_{[001]}, but 4^{+}_{[001]} or 4^{3+}_{[001]} could also be tried.
TITL monI in I2/a
CELL 0.71073 14.6876 14.7273 15.2443 90.0000 90.027 90.0000
ZERR 16.00 0.0010 0.0011 0.0011 0.0000 0.005 0.0000
LATT 2
SYMM 0.5X, Y, Z
SFAC C H O
UNIT 96 256 48
TWIN 1 0 0 0 1 0 0 0 1
FIND 15
PLOP 21 26 29
MIND 1.0 0.1
NTRY 1000
HKLF 4
END
Since the structure is quite small, even a thousand trials (`NTRY 1000') runs quickly. On completion of the SHELXD run, the resulting model is not complete (Fig. 11a), but shows enough of the structure to easily build two pinacol molecules and assign two water oxygens (Fig. 11b). Fourfold can then be tested by fourfold application of either twin matrix 4^{+}_{[001]} or 4^{3+}_{[001]}, for example, by including (for the former) the following commands in an ins file for SHELXL.
TWIN 0 1 0 1 0 0 0 0 1 4
BASF 0.25 0.25 0.25
The component fractions in the above SHELXL `BASF' command are just initial guesses and will refine. After a few cycles of leastsquares the model is dramatically improved. Even anisotropic (with restraints) is possible, as is addition of riding methyl hydrogens (Fig. 11c). All of the refined BASF parameters are significant, indicating that fourfold is appropriate. Consequently, the Rvalue drops well into single digits.
5.4. Search for missed symmetry
In spite of the progress, the current I2/a model has demonstrable problems. A careful inspection reveals hefty correlation between atoms related through the central bond of each pinacol molecule, suggestive of missed inversion symmetry. Thus, a careful check for missing symmetry, visually and for example, using ADDSYM in PLATON (Spek, 2020) is warranted. For the latter, PLATON requires a CIF and a `LIST 4' format fcf file, which are written by SHELXL if both `ACTA' and `LIST 4' commands are specified in the SHELXL ins file. ADDSYM predicts a primitive monoclinic (P2/n) cell with a volume only a quarter as large as the current Icentred cell (Fig. 12), and supplies a transformation matrix from the I2/a setting to P2/n, namely:
It also gives the option to save a copy of the transformed model (option ADDSYMSHX).
Since the volume of the P2/n cell is only a quarter the size of the I2/a cell, its is half as big, which means the for SAXDUR is 2. Thus, the is reticular; each individual only contributes to half of the measured diffraction maxima of the twin. Transformation of the dataset for P2/n by successive 90° rotations about the fourfold twin axis (Fig. 13) would therefore generate noninteger reflection indices for half the data. Normally, that is not a problem as they'd be simply ignored or deleted; noninteger indices do not represent actual Bragg peaks. Nevertheless, it causes such `impossible' indices to coincide with actual Bragg maxima from other individuals. Similar to WUGLES, we should not simply discard them, but for SAXDUR, there is no setting of 13, conventional or otherwise, that would allow use of the SHELXL `TWIN' command. The way forward is to make a data file in SHELXL `HKLF 5' format that preserves all the information. One approach to creation of such an `HKLF 5' format datafile for SAXDUR is given in a series of straightforward steps in Appendix A. The resulting datafile is available in the supporting information.
5.5. Complete of SAXDUR as a fourfold twin using P2/n
After minor editing, the P2/n model supplied by ADDSYM in PLATON refines smoothly as a fourcomponent twin against the `HKLF 5' format dataset without the need for restraints (Fig. 10). When viewed down the [01] direction of the P2/n cell, which corresponds to the pseudotetragonal c axis of the I2/a the approximate fourfold symmetry is readily apparent (Fig. 14).
6. Conclusions
Any ultimately correct
of a twinned crystal requires use of the proper spacegroup symmetry and complete treatment of the Once these criteria are met, final of the structure is usually no more problematic than a nontwinned structure of similar complexity. Nevertheless, the route taken to solve the structure and assign the true spacegroup symmetry and might in practice be rather indirect. As such, whatever tools and tricks are used as means to delivering a valid crystallographic end result are fair game.APPENDIX A
Manual construction of a SHELXL `HKLF 5' format datafile for SAXDUR
The SHELX `HKLF 5' format for twinned diffraction data is well described in the SHELXL manual and elsewhere (see, e.g. the supporting information for Sevvana et al., 2019). Briefly, each individual is assigned a separate batch number and twinrelated reflections are grouped together sequentially. Batch numbers are set negative except for the last member of each group, for which the batch number is positive. There is no best way to generate such an `HKLF 5' format datafile, but equivalents should be already merged because the format mandates suppression of merging within SHELXL. One could write a short program or script (e.g. Bolte, 2004), but the task itself is straightforward and does not require programming skills beyond text file reformatting. The following describes a logical stepbystep approach to generate an `HKLF 5' format datafile from an `HKLF 4' format datafile. It uses only common unix commandline tools awk and paste [these are available on Windows via the Cygwin project (Cygwin, 2020)] and is adaptable to other cases. For the pseudotetragonal of SAXDUR in Section 5 above, the four individuals of the twin are related by successive rotation of 90° about the caxis of a pseudotetragonal Icentred A positive rotation of 90° (anticlockwise about c) is achieved by the following matrix:
Repeated application generates four matrices. Starting with the identity matrix, these are:
A1. Split data into separate components
For each of the four individuals, write separate files with indices transformed by matrices A2–5 using awk. The order does not matter, but a logical approach is to go stepwise about the (pseudo) fourfold. From a commandline terminal, issue the following commands:
awk '{print $1,$2,$3,$4,$5,1,NR}' monI.hkl > step11.txt
awk '{print $2,$1,$3,$4,$5,2,NR}' monI.hkl > step12.txt
awk '{print $1,$2,$3,$4,$5,3,NR}' monI.hkl > step13.txt
awk '{print $2,$1,$3,$4,$5,4,NR}' monI.hkl > step14.txt
In the above, the matrix transformation is accomplished by manipulation of the fields ($1, $2, $3) within the print commands. Also note that a batch number (1–4) and the line number (NR) of each reflection are appended to each line in the output text files. As stated above, in an `HKLF 5' file, the last member of a group of twinrelated reflections needs a positive batch number, with the rest negative. That will be fixed in a later step using these additional fields.
A2. Combine components
The individual files must be combined, with lines interlaced in sequence. An easy way to achieve this is with the unix utility paste, which does a `parallel merge'. To trick paste into putting each component on consecutive lines, the delimiter is changed from the default (tab) to a newline (\n) character:
paste d"\n" step11.txt step12.txt step13.txt step14.txt > step2.txt
The above writes a file in which contributions to each measured intensity are grouped, with sequential batch numbers.
A3. Transform from monoclinicI to monoclinicP
Transformation from the Icentred setting to primitive monoclinic is via the following matrix:
This is the same matrix suggested by ADDSYM in Section 5.4 (Note: for other cases, this matrix would be different). To apply it, again use awk:
awk '{print ($1+$3)/2,$2/2,($1$3)/2,$4,$5,$6,$7}' step2.txt > step3.txt
The index transformation inevitably writes some lines that have noninteger indices for each individual, which must be discarded.
A4. Eliminate noninteger indices
The following line of awk only accepts lines that have two dot `.' characters.
awk F. 'NF==3' step3.txt > step4.txt
In the above, the field separator is first changed to a `.' character. Lines in the input file are only transferred to the output file if the number of fields is 3. This works here because an `HKLF 5' file should only have two decimal points [in the F^{2} and σ (F^{2}) fields] per reflection. [Note, however, some hkl files written by PLATON store F^{2} and σ(F^{2}) as integers, and thus should have no decimal points at all, and would require a modified condition (i.e. 'NF==1') in the above awk command.]
A5. Flag last member of each twinrelated group
The batch number of the last member of each group of twinrelated reflections must be positive and the rest negative. The following awk oneliner gets part way there:
awk 'NR==1{printf "%s", $0; next}; {print " " $NF; printf "%s", $0}' step4.txt > step5.txt
The above writes a new file with each line appended with an extra field containing the (original) line number of the next reflection in the file.
A6. Generate 'HKLF 5' format datafile
The final step sets batch numbers for the last member of each group positive and the rest negative:
awk '{if ($(NF1)==$(NF)) {m=1} else {m=1}; printf "%4d%4d%4d%8.2f%8.2f%4d\n", $1,$2,$3,$4,$5,$6*m}' step5.txt > step6.hkl
In the above, (original) line numbers of consecutive reflections are compared and their batchnumber signs modified, as per the `HKLF 5' format rules. Lastly, it writes an hkl file formatted as `3I4, 2F8.2, I4'. The fastidious might want to manually edit the file termination line(s), but that is not necessary for proper SHELXL operation. For convenience, the above commands are available in the supporting information as a plain text file, suitable for copy/paste. They may also be combined into a single script along with additional commands to delete intermediate files.
Supporting information
Diffraction data for XUBNIR in SHELX HKLF 4 format. DOI: https://doi.org/10.1107/S205698902100342X/hb7973sup1.hkl
Input file for structure solution of XUBNIR by SHELXD. DOI: https://doi.org/10.1107/S205698902100342X/hb7973sup2.ins.txt
Diffraction data for WUGLES in SHELX HKLF 4 format. DOI: https://doi.org/10.1107/S205698902100342X/hb7973sup3.hkl
Input file for P21 https://doi.org/10.1107/S205698902100342X/hb7973sup4.ins.txt
structure solution of WUGLES by SHELXD. DOI:Diffraction data for SAXDUR indexed as triclinicP, HKLF 4 format. DOI: https://doi.org/10.1107/S205698902100342X/hb7973sup5.hkl
Diffraction data for SAXDUR indexed as monoclinicI, HKLF 4 format. DOI: https://doi.org/10.1107/S205698902100342X/hb7973sup6.hkl
Input file for I2/a https://doi.org/10.1107/S205698902100342X/hb7973sup7.ins.txt
solution of WUGLES by SHELXD. DOI:Diffraction data for SAXDUR indexed as monoclinicP, HKLF 5 format. DOI: https://doi.org/10.1107/S205698902100342X/hb7973sup8.hkl
Steps for generation of an HKLF 5 format datafile from an HKLF 4 format datafile for structure SAXDUR (pinacol monohydrate), in plain ASCII text suitable for copy/paste. DOI: https://doi.org/10.1107/S205698902100342X/hb7973sup9.txt
Acknowledgements
I am grateful to Dr Rebecca Smaha (Stanford University) for the opportunity to work on WUGLES, and along with Dr Paul Boyle (Western University) for careful critique of my website tutorials on https://xray.uky.edu/Tutorials/tutorials.html) that formed the basis of this paper.
(References
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