English

• 1. INTRODUCTION

  It is very important to determine the structure in investigating chemical compounds in the crystalline state in many fields, e.g., aggregation-induced emis-sion luminogens (AIEgens)^[1], metal-organic frame-works (MOFs)^{[2, 3]}, nanoclusters (NCs)^{[4, 5]}, transition-metal mixed-organic-ligand complexes (TMM-Cs)^[6], poly-oxometalates (POMs)^[7], and mechano-responsive luminescent (MRL) materials^[8], connec-ted with many applications in catalysis, optics, elec-tronics, magnetism, biochemistry, etc. Gaining insight into the relationship between structure and property, and then designing and constructing novel materials, are the necessary approach to prompt scientific development. Although X-ray, neutron, and electron diffraction studies can be applied to determine the structures on the atomic level using powder or crystal samples, single-crystal X-ray-diffraction analysis is still more facile, credible, and often preferentially selected as the structure deter-mination method for most researchers.

  After collection on a single-crystal X-ray diffrac-tometer, the diffraction data are stepwise reduced, corrected, and then solved and refined using software packages such as APEX3^[9], Shelxtl^[10], WinGX^[11], Olex2^[12] with Saint^[13], Sadabs^[14], Shelxt^[15], Shelxl^[16], Superflip^[17] or others. The final cif file containing the structure and refinement information is yielded, then checked to judge the result as accepted or not. With hardware and software improvements, this process is programma-tically performed in advanced machines, and the crystal structure can be directly given for simple or small molecules. This makes single-crystal structure determination look like very easy work, and some unreasonable structures have been published in the scientific journals. In fact, crystallographic knowle-dge, experimental facts, refinement skills, and experience are still vital for obtaining high-quality, publication-grade crystal structures, as pointed out by Müller^[18].

  The phenomenon of structural fragments randomly spread over two or more sites, or co-residing within the same site with partial site occupancy in the space, is known as disorder. Refining the disordered structure is difficult and challenging and has not been significantly addressed by tutorials so far^[18]. In contrast, the entire structure, possessing two or more sets of sites with different structural tropisms, may be assigned to a twinning crystal.

  Atomic thermal vibrations can be enhanced with increasing the temperature and the large displace-ment parameters, which could be overcome to a certain extent at low temperature by usually using liquid nitrogen, while the disordered atoms or assemblies, such as dissociative halogen, guest water molecules, nitrate, perchlorate, alkyl, and aromatic assemblies, are usually refined by adjusting the occupancies or splitting the sites, which are generally indelible at low temperature. If necessary, some orders, especially dfix, sadi, simu, rigu, isor and part^[16], are used to constrain/restrain the sites and displace parameters for better refinement parameters and structure shapes^[19]. Generally, these shapes are linear, V-shaped, trigonal, tetrahedral, and octahedral. Of course, the shapes must be reasonable and identical to the actual structures. It is not meaningful to excessively pursue good refinement parameters and shapes. In this work, the emblematic disordered fragments reported in the references are selected to be re-refined using the newest Shelxl-2018^[16] sub-program, and some approaches to refine this disorder are given, favoring a deeper insight into a key feature of disorder and serving as a meaningful complement to the highly recommended tutorials^{[18, 19]}.

  2. DISORDERED ASSEMBLIES

  2.1 Disordered nitryl (-NO₂) assembly

  Nitryl often adopts a V shape. As shown in Fig. 1(a), nitryl can rotate around the single C(1)–N(1) bond, and this results in a disordered structure. Two O atoms (O(2) and O(3)) belonging to nitryl were refined over two sites in the reported reference^[20]. After remo-ving the original disordered treatments and re-refining the structure over only one site, the refine-ment parameters drastically increased from the original (R = 0.043, wR = 0.123, S = 1.076) to (R = 0.113, wR = 0.419, S = 2.001). Moreover, two Q peaks near the nitryl indicate the appearance of another O-atom site (Fig. 1(b)), demonstrating that the splitting was necessary.

  Figure 1

  

  Figure 1.  Disorder of nitryl by (a) splitting and (b) non-splitting of the site

  DownLoad: Full-Size Img PowerPoint

  2.2 Disordered nitrate (NO₃^–) assembly

  Different from the nitryl assembly, nitrate pos-sesses the typical planar trigonal structure. If only the peripheral O atoms are disordered (Fig. 2(a)), its geometry could be fixed by constraining/restraining the N–O and O⋅⋅⋅O distances, and the final refined structure is co-planar (Fig. 2(b)). If the entire struc-ture including the central N atom is disordered, all the atomic sites should be split (Fig. 2(c)). Another typical case is illustrated in Fig. 3(a), in which only the O(3) atom of the structure (denoted as compound 3 in Ref. [21]) is normal and over two sites with good final refinement parameters (R₁ = 0.065, wR₂ = 0.031, and S = 1.031). However, in this refined structure, all the atoms of nitrate are non-coplanar. More reasonably, the other atoms (N(4), O(4), and O(5)) should also be refined over two sites as shown in Fig. 3(b), giving the similar refinement para-meters (R = 0.065, wR = 0.031, and S = 1.020). BO₃, CO₃^2–, and other trigonally-shaped fragments could be refined following this suggestion.

  Figure 2

  

  Figure 2.  Shape of nitrate (a) before and (b) after constraint/restraint, and (c) after splitting the entire structure and constraint/restraint

  DownLoad: Full-Size Img PowerPoint

  Figure 3

  

  Figure 3.  Nitrate shape as (a) reported in the reference, and (b) for nitrate wholly refined over two sites

  DownLoad: Full-Size Img PowerPoint

  2.3 Disordered perchlorate, phosphate, tetrafluoroborate, and sulfate (ClO₄^-, PO₄^3-, BF₄^–, and SO₄^2–) assemblies

  ClO₄^–, PO₄^3–, BF₄^–, and SO₄^2– can be a kind of assembly due to their tetrahedral geometries, and are often disordered. Like nitryl, the terminal O/F atoms of these anions can be constrained/restrained or split to define the structural geometry if only these atoms are disordered. Herein, ClO₄^– was exemplified as shown in Fig. 4 (denoted as compound 5 in Ref. [22]). Different refinements of the ClO₄^– by non-constraint/non-restraint (Fig. 4(a)), constraining/res-training the geometry (Fig. 4(b)), splitting the peripheral O atoms (Fig. 4(c)), and splitting the whole atoms with the central Cl atom (Fig. 4(d)) yield similar refinement parameters, converged at R₁ = 0.036, wR₂ = 0.106, and S = 1.056 for the structure in Fig.4(a); at R = 0.038, wR = 0.112, and S = 1.104 in Fig. 4(b); at R = 0.038, wR = 0.113, and S = 1.108 in Fig. 4(c), and R₁ = 0.038, wR₂ = 0.109, and S = 1.079 in Fig. 4(d). Obviously, only confining the geometry of ClO₄^– without splitting is enough in successful refinement. However, if the entire struc-ture assumes a long and narrow ellipsoidal shape, all the atoms including the central Cl, P, B, and S atoms must be split.

  Figure 4

  

  Figure 4.  Disorder of ClO₄^–(a) with non-constraint/non-restraint, (b) constraining/restraining the geometry, (c) splitting the peripheral O atoms, and (d) splitting entire atoms with the central Cl atom

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  It is worth mentioning that the observed eight O atoms often exist around the P/B/Si/As atoms for PO_4­/BO_4­/SiO_4­/AsO_4­ in the special Keggin structure of POMs. In this case, the site-occupancy factors of the peripheral O atoms should be revised to a half of the full occupancy and then refined, and a correct shape of PO_4­/BO_4­/SiO_4­/AsO₄ with total four peri-pheral O atoms versus one central P/B/Si/As atom would be obtained. An example of the reported disordered PO₄^3– is depicted in Fig. 5 (denoted as compound 3 in the Ref. [6]). The original displace-ment parameters of each peripheral O atom with a site-occupancy factor of 1 are quite large and abnor-mal (Fig. 5(a)), while those of each atom with a site-occupancy factor of 0.5 are normal and desired (Fig. 5(b)). If each O atom has a site-occupancy factor of 1, PO₈ will be obtained, which is evidently false.

  Figure 5

  

  Figure 5.  Disordered PO₄^3– unit in the center of the Keggin structure of POMs.

  Site occupancy factors of (a) 1 (a) and (b) 0.5 for peripheral O atoms of PO₄^3– unit

  DownLoad: Full-Size Img PowerPoint

  2.4 Disordered hexafluorophosphate (PF₆^-) assembly

  As shown in Fig. 6, PF₆^– with octahedral geo-metry can be refined like the aforementioned disordered fragments. In Ref. [23], four F atoms residing on the equator were refined over two sites, while the other two F atoms locating at the polar vertexes were over one site (Fig. 6(b)). If necessary, both the central P atom and two polar F atoms are also split over two or more sites. Other anions, such as SiF₆^2–, AsF₆^– and SbF₆^–, can be treated in a similar way.

  Figure 6

  

  Figure 6.  (a) Geometry of PF₆^- and (b) its disorder

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  2.5 Mixed-occupancy disorder

  It is a common phenomenon that two or more atoms/molecules reside on the same site. In particu-lar, different metal ions can statically occupy the same positions to coordinate the organic ligands. For example, Fe³⁺ and Fe²⁺ co-reside on the center of the octahedron (Fig. 7, denoted as MSF-3 in Ref. [24]). According to the result determined by Mössbauer spectroscopy, the molar percent of Fe³⁺/Fe²⁺ is 53.8%/46.2% with a total site-occupancy factor of 0.5. The refinement of Fe³⁺/Fe²⁺ atoms over two sites leads to site-occupancy factors of 0.269 and 0.231, respectively. During the final refinement, 0.731 H atom was added to the S atom of the asymmetric unit for the total charge balance.

  Figure 7

  

  Figure 7.  Fe³⁺ and Fe²⁺ statically residing at the same site

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  2.6 Disordered organic assembly

  The fragments of organic assemblies, including alkyl, alkoxyl and perfluoroalkyl chains and aroma-tic assemblies, can freely rotate around a C–C single bond, or be equivalently fixed at different directions by interacting with H bonds, π-π interactions, van der Waals forces, etc., so as to easily result in the disorder in its crystal^[25]. The ellipsoid of peripheral atoms is usually larger than that of the inside atoms. The disordered chain-/cycle-like organic assemblies can be refined over two or more sites with a twisted and interleaved shape. A typical example is the disorder of the n-butyl assembly (Fig. 8(a), denoted as HLSn1 in Ref. [26]). The t-butyl and phenyl assemblies could often be refined over two interla-ced sites with a total occupancy factor of "1" using "part" order, similar to the disordered alkyl (Fig. 8(b), denoted as compound 4a in Ref. [27]). Owing to the non-coplanarity of the phenyl ring, the split of the partial phenyl ring in Fig. 8(c) is inevitable, which should be treated just like its disordered pyridyl ring illustrated in Fig. 8(c) (denoted as polymorph I in Ref. [28]). In addition, the treatment of disorder of the entire molecule, such as 1-carboxymethyl-1-methyl-pyrrolidinium (Fig. 8(d))^[29] and 18-crown-6 ether (Fig. 8(e))^[30], is the same as these disordered organic assemblies.

  Figure 8

  

  Figure 8.  Disordered (a) n-butyl (a), (b) t-butyl and phenyl (b), (c) pyridyl and phenyl (c) assemblies, (d) 1-carboxymethyl-1-methyl-pyrrolidinium (d), and € 18-crown-6 ether (e)

  DownLoad: Full-Size Img PowerPoint

  2.7 Disordered assembly of MOFs

  For the fragments of MOFs, the linked ligands and coordinated solvent molecules are also probably disordered. Typically, dimethylformamide (DMF) donates its oxygen atoms to coordinate with the metal ions over two sites, and organic linkers con-nect with the metal ion to propagate the three-dimensional framework, also creating serious disor-der (Fig. 9, denoted as compound 2 in Ref. [31]). The disordered MOFs could be refined by referring to the simple disordered fragments.

  Figure 9

  

  Figure 9.  Disordered linked ligands and coordinated solvent molecules of framework

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  2.8 Disordered assembly of the guest molecules

  For porous structures, such as MOFs and zeolites, many kinds of solvents or template molecules, acting as guests, exist in the channels/cages cons-tructed by the hosts. The guest should be found as much as possible, even though the reflections of the guest are usually very weak and disturb the refinement. If the research focuses on the host and the guests can be neglected, their reflection contri-butions could be removed by the squeeze function of the PLATON program^[32]. A new set of reflection files without guest contributions is used for further refinement. However, the information on the squeeze operation should be recorded in the cif file. The components of the guests can be deduced from the removed electron number combined with other analytical techniques, such as thermogravimetric analysis (TGA) and elemental analysis (EA), and renewing the formula in the cif file. However, this method is rough and often leads to a certain deviation. For example, the escape of guests from the host gives a large deviation compared with TGA/EA^[33] or the unmatched formula^[34].

  In some cases, it is very difficult to distinguish guests due to their disorders. For example, both disordered dimethylamine and protonized dime-thylamine with V shape apparently look like poly-hedra owing to their rotation around the polar axis. The four N atoms, each with a site-occupancy factor of 0.25, reside on the equatorial vertexes of an octa-hedron (Fig. 10(a)), or the three N atoms each with a site-occupancy factor of 1/3 reside on the equatorial vertexes of a trigonal bipyramid (Fig. 10(b)), in which two C atoms locate at the polar vertexes of a polyhedron or trigonal bipyramid. Indeed, the entire structure in total contains one N and two C atoms, namely a dimethylamine molecule or cation. In another instance, the structure depicted in Fig. 10(c), produced by inversion, is very confused, and the molecule in Fig. 10(d) looks more like the o-xylene; however, both could be considered toluene molecules. The structure in Fig. 10(c) is refined by "part -1" order, whereas the latter is refined with a total site-occupancy factor of 1 for the two methyl assemblies by "part 1" and "part 2" orders.

  Figure 10

  

  Figure 10.  Disorders of the dimethylamine or protonized (a, b) dimethylamine and (c, d) toluene as linked ligands and coordinated solvent molecules

  DownLoad: Full-Size Img PowerPoint

  2.9 Disordered assembly of clusters

  Some metal cores of the cluster also take on multi-fold disorders, leading to a great increase in the metal atomic number of the cores. For example, it seems that NUPF-2Y possesses an 18-metal-nucleus core (Y18) (Fig. 11a); however, this is actually caused by partial crystallographic occupan-cy using a twofold disordered Y9 cluster on the basis of the Y⋅⋅⋅Y non-bond distance^[35] (Fig. 11a). Similarly, threefold overlapping Zr6 was reported to form a fake Zr18 cluster by partial crystallographic occupancy^[36]. During the refinement procedure, "part" order can often be applied to well treat those disorders so as to demonstrate disordered assemblies separately. Sometimes, those disorders with different whole structure tropisms could arise by reticular merohedry or non-merohedry twinning crystals, and reverse to normal by extracting the reflections with the same tropism and merging them in the reciprocal space. This can be tentatively referred to as pseudo-disorder.

  Figure 11

  

  Figure 11.  Disordered assembly of clusters: (a) apparent Y18 and (b) actual Y9

  DownLoad: Full-Size Img PowerPoint

  3. ADDING H ATOMS TO DISORDERED FRAGMENTS

  Geometric calculation or found from the differen-tial Fourier map, while they are obtained with difficulty in disordered fragments. In some cases, the H atoms were not added to the disordered fragments, e.g., the disordered triethylamine (Fig. 12(a), deno-ted as compound 2 in Ref. [6]) and guest water molecules (Fig. 12(b), denoted as compound 2 in Ref. [37]). However, the complete formula must be provided in the final cif file, even though the checkCIF report gives an alert on the inconsistency between the calculated and reported formulas. It is well known that the sp³ orbitals of the O atom adopt tetrahedral geometry. In this case, the O atom locates at the center of the tetrahedron, and the H atoms bonded to the O atom and the metal atoms coordinated with the O atom occupy the vertexes of the tetrahedron. In a structure (Fig. 12(c)) reported by Wang and co-workers^[38], the O atom conjugated to the H atom in a correct geometry was rejected due to the B alert of "without acceptor, " while the final incorrect geometric conformation without any A/B alerts, in which all of the H, O and B atoms almost reside on the same plane, was accepted, possibly because of a better checkcif report.

  Figure 12

  

  Figure 12.  Disorders of (a) triethylamine and (b) guest water without H atoms, and (c) incorrect geometry of the O atoms

  DownLoad: Full-Size Img PowerPoint

  4. CHECKING STRUCTURE

  Some workers have their own criteria by which to judge the refinement according to the checkCIF report, but researchers should remember that there is no "error" parlance or absolute right/wrong regarding the checkCIF consequence since alerts with different levels are mechanically given by the checkCIF report. Higher-level alerts mean greater probability of serious problems existing in the structure. Researchers should carefully check and further refine the structure if necessary. The "alerts" do not mean a false structure and thus have no validity. On the contrary, "no A/B alerts, " even "no C/G-level alerts, " do not represent correct structure. Our emphases should focus on the aim of the work, structural rationality, and mutual relation between the structure and property. The above-mentioned position of the H atom is a typical example. For another instance, the as-published structures were re-submitted for checking, and the checkCIF report gave "B alerts" about Flack parameters, while the chirality was confirmed by the CD spectrum^[39]. The heavy Br atoms were isotropically refined in another published paper^[40]. It is a good way to add notes about A/B alerts to the cif file, since it is very helpful for readers to know the reasons for the alerts.

  5. CONCLUSION

  In summary, some typical disordered fragments are presented and discussed in this paper. These examples and suggestions are given to better understand disorder refinement. The disorders are common and unavoidable in crystal-structure deter-mination, and even sometimes the unprecedented properties of crystal materials, such as phase transitions^[41], magnets^[42] and spectroscopy^[43], result exactly from the disordered structures. If the final refined structure is rational and identical to the experimental facts, the result is acceptable. We hope practitioners can treat similar fragments following our suggestions.

  
  ACKNOWLEDGEMENT: Thanks for the data support from each researcher and the programs from all the compilers. Thanks for the help from Dr. Z. Y. Zhang and Dr. Y. T. Wang of Bruker AXS Incorporation in Shanghai.
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  Figure 1  Disorder of nitryl by (a) splitting and (b) non-splitting of the site

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  Figure 2  Shape of nitrate (a) before and (b) after constraint/restraint, and (c) after splitting the entire structure and constraint/restraint

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  Figure 3  Nitrate shape as (a) reported in the reference, and (b) for nitrate wholly refined over two sites

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  Figure 4  Disorder of ClO₄^–(a) with non-constraint/non-restraint, (b) constraining/restraining the geometry, (c) splitting the peripheral O atoms, and (d) splitting entire atoms with the central Cl atom

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  Figure 5  Disordered PO₄^3– unit in the center of the Keggin structure of POMs.

  Site occupancy factors of (a) 1 (a) and (b) 0.5 for peripheral O atoms of PO₄^3– unit

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  Figure 6  (a) Geometry of PF₆^- and (b) its disorder

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  Figure 7  Fe³⁺ and Fe²⁺ statically residing at the same site

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  Figure 8  Disordered (a) n-butyl (a), (b) t-butyl and phenyl (b), (c) pyridyl and phenyl (c) assemblies, (d) 1-carboxymethyl-1-methyl-pyrrolidinium (d), and € 18-crown-6 ether (e)

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  Figure 9  Disordered linked ligands and coordinated solvent molecules of framework

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  Figure 10  Disorders of the dimethylamine or protonized (a, b) dimethylamine and (c, d) toluene as linked ligands and coordinated solvent molecules

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  Figure 11  Disordered assembly of clusters: (a) apparent Y18 and (b) actual Y9

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  Figure 12  Disorders of (a) triethylamine and (b) guest water without H atoms, and (c) incorrect geometry of the O atoms

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