Molecular Interactions in Crystal Structures with Zâ² > 1 - Crystal

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Molecular Interactions in Crystal Structures with Z’ > 1 Robin Taylor, Jason Cole, and Colin R. Groom Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.6b00355 • Publication Date (Web): 15 Mar 2016 Downloaded from http://pubs.acs.org on March 31, 2016

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Crystal Growth & Design

Molecular Interactions in Crystal Structures with Z’ > 1 Robin Taylor,* Jason C. Cole and Colin R. Groom Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK

ABSTRACT: We describe an algorithm for enumerating and classifying molecular interaction motifs in crystal structures taken from the Cambridge Structural Database. It was used to determine which motifs have the highest propensity to occur in structures with Z’ > 1, and to show that these motifs also have a strong tendency to occur between symmetry-independent rather than symmetry-related molecules in such structures. In noncentrosymmetric structures, they are predominately stacking interactions and cyclic motifs containing strong or weak hydrogen bonds. In centrosymmetric structures, OH…O hydrogen bonds and C-H…π interactions predominate. The proportions of centrosymmetric and noncentrosymmetric structures with Z’ = 2 that might be explained by the presence of one or more of these motifs were estimated. Motifs with a strong tendency to form across inversion centers in structures with Z’ = 1 were also identified (“inversion-favoring motifs”). They are similar to those preferentially occurring between symmetry-independent molecules in noncentrosymmetric structures with Z’ = 2, but dissimilar to those in centrosymmetric structures. However, a significant number of centrosymmetric structures with Z’ = 2 have inversion-favoring motifs both across crystallographic inversion centers and between symmetry-independent molecules. Several other results were obtained: for example, crystallization with Z’ > 1 is more likely for molecules with 1 to 3 hydrogen-bonding atoms.

1. INTRODUCTION Crystal structures with Z’ > 1 have more than one formula unit in the asymmetric unit. When there is only type of chemical species present, this simply means that they have more than one molecule in the asymmetric unit. They have attracted considerable attention in recent years from, among others, those interested in crystal-structure prediction, polymorphism, crystal-structure 1 ACS Paragon Plus Environment


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modulation and crystal nucleation. The phenomenon was comprehensively reviewed in 2015 by Steed and Steed1 and, in comparison with their tour de force, anything we can say in this short Introduction runs the risk of appearing superficial. We therefore confine ourselves to a brief summary of the facts most relevant to our study, referring readers to the Steeds’ review for more details. Several factors may play a role in causing compounds to crystallize with Z’ > 1. Some molecules have awkward shapes, which might preclude them from effective close-packing unless two or more symmetry-independent molecules are included in the asymmetric unit.2,3 Exactly what constitutes an awkward shape remains very ill-defined. The possibility that many structures with Z’ > 1 are metastable has received attention.4,5 Evidence for this came from a delightful piece of lateral thinking by Nichol and Clegg.6 They speculated that crystal structure determinations using synchrotron radiation are more likely to be done on small, rapidly-grown crystals than those done with conventional X-ray sources; hence, the former structures should show a greater propensity to be metastable and have Z’ > 1. Conversely, neutron diffraction studies require large crystals that are grown slowly and should therefore have the opposite tendency. Searches of the Cambridge Structural Database (CSD)7 emphatically supported their argument. Intermolecular interactions that cause strong association of solute molecules prior to crystal nucleation (“pre-association”) are held to be important. For example, solute dimers bound together by strong hydrogen bonding might act as the effective packing unit during rapid crystallization and result in a metastable packing arrangement with Z’ > 1.8 The importance of intermolecular interactions was noted in a different context by Brock and Duncan.9 Noticing that monoalcohols and monoamines often crystallize with Z’ > 1, they looked into the steric problems associated with forming hydrogen-bond arrangements that exploit both the donor and acceptor capabilities of OH and NH2 groups. Sometimes, OH…OH or NH…NH hydrogen bonds can be arranged along screw axes or glide planes, but the presence of even moderately bulky

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substituents militates against this. Crystallization with Z’ > 1 is then one of the preferred ways in which these hydrogen bonds can be formed (use of high-symmetry space groups is the other). Another factor leading to structures with Z’ > 1 is termed “frustration” by Steed and his group,10 and also involves strong, directional interactions. Some interaction motifs are known to form preferentially across specific symmetry elements. For instance, amide and carboxylic acid hydrogenbonded rings:

usually occur across inversion centers.11 If a packing arrangement with the appropriate symmetry element is not practicable, crystallization with Z’ > 1 provides an alternative means of achieving the favored motif. An obvious example is when a compound is chiral and therefore constrained to crystallize in a space group without inversion centers (more specifically, a Sohncke space group). Figure 1 shows a typical example.

Figure 1. Packing of CSD entry AWUMUB. Molecules with different carbon-atom coloring (gray, blue) are crystallographically independent. Measurements indicate H…O distances in a hydrogen-bonded amide dimer motif.



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In this study, we focus on the relationship between intermolecular interactions and Z’. This has been the subject of several previous studies, summarized thoroughly in the Steeds’ review.1 The methodology usually employed is to search the CSD for structures containing a particular interaction motif, determine the percentage that have Z’ > 1, and establish whether it is significantly higher than the percentage in a suitable reference set. Additional factors are sometimes taken into account, such as the presence of chirality. A limitation of the approach is that it can only provide information about the interactions that are chosen for study. The methodology, while valuable, therefore has a degree of subjectivity. In the present work, we avoid this problem by enumerating interaction motifs algorithmically and surveying them all. In addition to measuring the percentage of structures containing a given interaction motif that have Z’ > 1, we also determine the propensity of motifs to form between symmetry-independent (SI) rather than symmetry-related (SR) molecules. We argue that an interaction with a strong tendency to occur between SI molecules is likely to play a significant role in stabilizing crystal-packing arrangements with Z’ > 1. The results of our survey are described below.

2. EXPERIMENTAL SECTION 2.1. Definition of metrics. We used an extended definition of Z’.12 Structures whose asymmetric units contain two molecules on special positions, one half of each molecule being crystallographically unique, were deemed to have Z’ = 2. In this situation, the value of the conventionally-defined Z’ parameter is 1, but we take the view that such structures effectively contain two symmetry-independent molecules. Similarly, we set Z’=3 for structures with three molecules on 3-fold axes, etc. Following earlier authors, Zr is the number of different types of chemical entity in a crystal structure.13 Given a sample of structures containing a particular interaction motif, we define the following terms. %Z’>1 is the percentage of the structures that have Z’ > 1. %SIO (standing for “percentage Symmetry-Independent Only”) is the percentage in which the given interaction motif occurs 4 ACS Paragon Plus Environment

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between SI molecules but not between SR molecules. %IO (“percentage Inversion Only”) is the percentage in which the interaction motif occurs between molecules related by crystallographic inversion symmetry but not between molecules related by any other type of symmetry, or between SI molecules. %SAO, %GPO and %CTO are defined analogously to %SIO and %IO but refer to the percentages of structures in which a motif occurs exclusively between molecules related by screw axes, glide planes and cell translations, respectively. 2.2. Data sets. The analysis of molecular interactions was performed using six data sets (Table 1), each containing crystal structures taken from version 5.36 of the CSD. All were restricted to structures with Zr = 1, thereby excluding salts, solvates and co-crystals. These latter are known to have a smaller propensity for Z’ > 1 than structures with Zr = 1.1 Excluding them therefore removes a factor extraneous to our focus of interest. All the data sets were required to satisfy the following constraints: R-factor ≤ 10% (structures excluded if no R-factor quoted); no reported disorder; no polymers; positional coordinates available for all atoms, including hydrogen. No restriction was placed on the Z’ value of the structures in the first pair of data sets (CENTRO-ALL, containing centrosymmetric structures, and NONCENTRO-ALL, containing noncentrosymmetric). The second pair, CENTRO-Z’2 and NONCENTRO-Z’2, were restricted to structures with Z’ = 2 and to centrosymmetric and noncentrosymmetric structures, respectively. The third pair contained structures with Z’ = 1; CENTRO-Z’1 contained only centrosymmetric structures and ALLSYMM-Z’1 contained both centrosymmetric and noncentrosymmetric. To reduce compute time, we restricted the larger data sets (CENTRO-ALL, NONCENTRO-ALL, CENTRO-Z’1 and ALLSYMM-Z’1) to structures containing molecules with ≤200 atoms (this excluded < 0.5% of structures that would otherwise be acceptable). We additionally restricted each of the sets CENTRO-Z’1 and ALLSYMM-Z’1 to 100,000 structures by random selection. When multiple structure determinations of the same chemical compound were available, only the one with the best R-factor was taken for sets CENTRO-Z’1, ALLSYMM-Z’1, CENTRO-Z’2 and NONCENTRO-Z’2. This restriction was not applied to CENTRO-ALL



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and NONCENTRO-ALL because we did not wish to exclude polymorphs with different Z’ values. CSD reference codes of all the data sets are deposited as Supporting Information. Hydrogen-atom positions were normalized by moving the hydrogen atom along its covalent bond vector, XH, so that the X-H distance was equal to the average neutron-diffraction value (C-H = 1.089, N-H = 1.015, O-H = 0.993Å).14,15 The relatively few hydrogen atoms that were bonded to elements other than carbon, nitrogen or oxygen were left in their published positions. Molecules were classified as chiral or achiral using an in-house program.16 Chirality due to restricted rotation was not taken into account, a molecule being assigned as chiral only if it had one or more carbon, phosphorus or sulfur stereocenters and was not a meso isomer. 2.3. Methodology. The method of analysis was to enumerate the interaction motifs in each data set, classify them by their chemistry and symmetry, and rank them into descending order of the metric of interest (%Z’>1, %SIO, etc.). Motifs that occurred in fewer than 50 structures were excluded. Motif geometries were investigated where necessary. Further details follow below.

2.3.1. Motif enumeration. In each structure, we found the primary interaction17of each atom (Figure 2). For an atom A, this is defined as the shortest line-of-sight interaction that A forms to an atom in another molecule, where “shortest” is measured relative to van der Waals (vdw) radii; in other words, the interaction, A…B, with the smallest value of

∆ = dA…B – vA – vB

(1)

where dA…B is the A…B distance and vA, vB are the vdw radii. We used the vdw radii published by Alvarez,18 since they were determined from a very large set of crystal structures and he quotes a value for every element. An interaction is line-of-sight if both atoms can “see” each other because there is no third atom whose vdw sphere intersects the A…B line. On the reasonable assumption that an atom’s primary interaction will usually be energetically its strongest, or at the very least one of its strongest, our analysis is therefore biased heavily towards the interactions most likely to be significant in determining packing arrangements. Primary interactions with ∆ > 0.3Å were rejected on the basis that they are unlikely to be important; this eliminated about 20 percent of them. We 6 ACS Paragon Plus Environment

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acknowledge that our choice of ∆ threshold is to some extent arbitrary. However, test calculations with a higher threshold (0.6Å) suggested that it would lead us to similar conclusions to those presented below.

Figure 2. Top: Primary interactions (in purple) of the atoms of the two molecules (the “reference molecules”) in an asymmetric unit of CSD entry AANHOX01. Bottom: Expanded to show the molecules at the other ends of the interactions (the “contact molecules”, shown with thinner bonds). The picture now shows all the interactions from the top picture and all of the primary interactions of atoms in contact molecules that are to atoms in reference molecules. This constitutes the set of interactions used for this structure in our analysis.

The set of molecular interaction motifs generated for each structure comprised the primary interactions themselves, which we term acyclic motifs, and a set of cyclic motifs. Each cyclic motif consisted of a pair of primary interactions, A1…B1 and A2…B2, and the atoms on the shortest covalent



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bond paths between A1 and A2 and B1 and B2 (Figure 3). Obviously, A1 and A2 have to be in the same molecule, as do B1 and B2. All possible cyclic motifs with ring size ≤ 14 were included.

Figure 3. Conversion of primary interactions to motifs. The pair of C…H primary interactions shown gives rise to three motifs. Each individual interaction is considered an acyclic motif (left and center), and the pair give rise to a cyclic motif (right). The atoms shown as spheres define each motif’s chemical classification.

2.3.2. Motif classification. Each motif was classified by its chemistry and its symmetry. The chemistry of a motif was expressed by a canonicalized alphanumeric string. For an acyclic motif, this captured (a) the element types of the atoms making the interaction, (b) the element types of the atoms covalently bonded to the interacting atoms, and (c) the types of bonds formed by the interacting atoms to their covalently-bonded neighbors (single, double, triple or aromatic, the latter represented by the symbol ~). To reduce the number of possible motifs, metal atoms were assigned the generic symbol M rather than individual element symbols. The hybridization state of a carbon atom was explicitly specified (Csp3, Csp2 or Csp) unless obvious from the bond types specified in the string. So, for example, the acyclic motifs shown in Figure 3 are denoted (Csp2)H…C(O)(~C)(~C) and (Csp2)H…C(H)(~C)(~C). The strings assigned to cyclic motifs captured the element types of the atoms in the ring and the bond types between them. Thus, the cyclic motif shown in Figure 3 is denoted C~C~C…H-C~C-H and the amide dimer motif (I) is H-N-C=O…H-N-C=O, where the leading and trailing curly arrows indicate that the first and last atoms in the string form an interaction with each 8 ACS Paragon Plus Environment

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other. We apologize for introducing a new notation but feel it is more intuitive to those unfamiliar with line notations than existing systems, though far less extensible. Motif symmetry classification depended on the metric being investigated. For %SIO, each motif was classified as SR if the interacting atoms belonged to molecules related by any type of crystallographic symmetry, or SI if the molecules were symmetry independent. For %IO, each motif was either I (molecules related by inversion) or NI (not related by inversion); and analogously for %SAO, %GPO and %CTO. 2.3.3. Analysis of motif geometries. The geometry of motifs is sometimes important to the interpretation of our results. For example, a motif assigned the string C~C…H-C~C-H could arise from an edge-to-face interaction of aromatic rings or from two laterally displaced, parallel rings (Figure 4); other geometries are also possible. Examples of motifs of interest were therefore inspected visually using the crystal-structure viewing program Mercury.19 Many hundreds of structures were inspected during the course of the study, albeit each one briefly. Some motif geometries were further analyzed as follows. All examples of the motif of interest that belonged to the SI class (or a random subset of size 250 if they exceeded that number) were collated and subjected to sphere-exclusion cluster analysis.20,21 The dissimilarity matrix required by the cluster analysis was computed by taking each pair of observations, least-squares superimposing them (based on the atoms that contribute to the motif’s alphanumeric representation) and taking the resulting root mean -square deviation (RMSD) as the dissimilarity coefficient. For motifs with topological symmetry, all possible ways of superimposing the atoms were tried and the lowest RMSD thus obtained was accepted. To be assigned to the same cluster, any two observations were required to have RMSD < 0.4Å. Overlapping clusters were allowed. Representative examples from the clusters were made available for visual inspection in Mercury, along with their weights (the size of the cluster they represented). Thus, the geometric distribution of the motif could be rapidly and reliably discerned.



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Figure 4. Two of the possible geometries of the C~C…H-C~C-H motif.

3. RESULTS AND DISCUSSION 3.1. Introduction and statistical considerations. There are 23,097 different types of interaction motifs that occur in ≥50 structures in the CENTRO-ALL data set and 6,968 in NONCENTRO-ALL. We computed their %Z’>1 values. The resultant values for motifs common to both data sets show very little correlation. A correlation is somewhat more evident among motifs that occur in ≥500 structures in each data set, but is still rather weak; the correlation coefficient is 0.42 (here and elsewhere, the Pearson coefficient is used). This suggests two things: firstly, that there are significant errors in the %Z’>1 values due to random sampling effects, since the correlation improves noticeably when we focus on values based on large samples; but secondly, that the motifs associated with Z’ > 1 are appreciably different in centrosymmetric and noncentrosymmetric structures, since even with large samples the correlation is mediocre. A %Z’>1 value is effectively a binomial proportion, P, multiplied by 100. P is the ratio of the number of structures in the sample with Z’ > 1 to the total number of structures in the sample, N. The standard error of %Z’>1 can therefore be estimated as σ(%Z’>1) = 100√[P(1-P)/N]

(2)

Our %Z’>1 values have standard errors ranging from 0.1 to 7.1, the average being 3.4. In order to show at the 99% confidence level that a particular motif is associated with crystal structures that have Z’ > 1, we need to demonstrate that its %Z’>1 value is at least 3σ above the random


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expectation value, i.e. the value we would expect if the motif has no relationship with Z’. It is therefore necessary to estimate this expectation value. We found that 10.2% of the structures in the CENTRO-ALL data set have Z’ > 1. However, the average of the 23,097 motif-specific %Z’>1 values is much higher (14.6). A similar result is found for the noncentrosymmetric data set, the relevant figures being 15.7 and 24.3. A moment’s reflection suggests the likely reason: structures with Z’ > 1 tend to contain more atoms in their asymmetric units than do those with Z’ = 1 (the average values are 101 and 49, respectively, based on the combined CENTRO-ALL and NONCENTRO-ALL data sets). Therefore, they also contain more interaction motifs. Selecting structures that contain a particular type of motif, which is what we do when we calculate a motif’s %Z’>1 value, is therefore inherently biased towards structures with Z’ > 1. To quantify this effect, we enumerated the interaction motifs in all the structures of the CENTRO-ALL data set and altered the string representations of about 1 in 500 of them, chosen at random, to “X…X”. In two runs, the %Z’>1 value of the structures containing the “pseudo-motif” X…X was 14.7 and 15.2. The average of these values, 15.0, will serve as an estimate of the random expectation value of %Z’>1 for the CENTRO-ALL data set. The corresponding value for the NONCENTRO-ALL data set was found to be 26.3. Armed with this information, we then identified the motifs in each data set whose %Z’>1 values are greater than the random expectation value at the 99% confidence level (henceforth, “high %Z’>1 motifs”). There were 135 such motifs for the NONCENTRO-ALL data set and 404 for the CENTRO-ALL data set. They were arranged in descending order of %Z’>1. The top 25 motifs for each data set are given in Tables 2 and 3 (complete lists are available as Supporting Information). The column headed “type” in these tables, and also in Tables 3 and 4, are simple chemical descriptions of the motifs based on visual inspection of examples. While we can say with confidence that these motifs are associated with Z’ > 1, we wanted further evidence that they might cause crystallization with Z’ > 1. Two mechanisms by which they might do 11 ACS Paragon Plus Environment


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so were mentioned in the Introduction: solution pre-association and frustration. In both, the relevant motif occurs specifically between SI molecules. So our confidence that a particular type of motif plays a significant role in causing structures with Z’ > 1 will be increased if we can show that it has a strong tendency to form between SI rather than SR molecules, i.e. has a high %SIO value. We therefore determined the %SIO values of the motifs occurring in ≥50 structures in the NONCENTRO-Z’2 and CENTRO-Z’2 data sets (1,458 and 2,993 motifs, respectively). Tables 4 and 5 list the 25 motifs with the highest %SIO values in each data set; complete lists are available as Supporting Information. As with the %Z’>1 results, the motifs with high %SIO values are very different in centrosymmetric and noncentrosymmetric structures. The restriction of NONCENTRO-Z’2 and CENTRO-Z’2 to Z’ = 2 rather than Z’ > 1 was imposed to ensure constant stoichiometry. In every structure in the data sets, each molecule is surrounded by equal numbers of SI and SR molecules. If packing were random, the probability that an arbitrarily chosen interaction would be between SI molecules is therefore 0.5. However, the expectation value of the %SIO statistic is not 50. The statistic is defined as: %SIO = 100NSIO / N

(3)

Here, N is the total number of structures that contain the motif of interest, and NSIO is the number in which the motif occurs only between SI molecules. If packing were random, we would expect the latter quantity to be equal to its analog, NSRO, the number of structures in which the motif occurs only between SR molecules. However, there is a third category, structures in which the motif occurs both between SI and between SR molecules. These structures are counted when N is calculated; hence, at random, %SIO is expected to be less than 50. To quantify this, we repeated the analysis of the NONCENTRO-Z‘2 data set, but assigning the SI or SR label to each interaction arbitrarily, each having equal probability of being chosen. The average %SIO value thus obtained was 39.0, the 99th percentile was 55.8, and the highest value was 64.3. A corresponding experiment on the CENTRO-Z’2 data set gave similar results: average = 38.3, 99th


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percentile = 54.5, maximum = 66.1. Taking the more conservative of the 99th percentiles, we therefore conclude that any true %SIO value exceeding 55.8 is unlikely to have occurred by chance. 3.2. Relationship between %Z’>1 and %SIO. The %Z’>1 and %SIO values show a moderate degree of correlation with each other, the correlation coefficients being 0.50 and 0.57 for motifs in centrosymmetric and noncentrosymmetric structures, respectively. Of most interest, however, is the %SIO distribution among high %Z’>1 motifs. Since the CENTRO-Z’2 data set is relatively small, %SIO values are available for only 220 of the 404 high %Z’>1 motifs identified for centrosymmetric structures. Of these, most (155) have %SIO above the 99th percentile of the randomly-generated %SIO distribution (see previous section) and almost all do (20 of 21) in the subset of these motifs that have %Z’>1 exceeding 25. For the noncentrosymmetric structures, %SIO values are available for 63 of the high %Z’>1 motifs, of which all but 1 are higher than the 99th percentile and most are much higher (45 of the 63 have %SIO ≥75). We conclude that high %Z’>1 motifs in noncentrosymmetric structures have an extremely strong tendency to occur exclusively between SI molecules, consistent with them causing crystallization with Z’ > 1. The tendency is not as strong for the high %Z’>1 motifs in centrosymmetric structures, but it is still clearly discernable. 3.3. Motifs associated with Z’ > 1 in noncentrosymmetric structures. In a nice vindication of earlier studies,10,22 the motifs at the top of the noncentrosymmetric %Z’>1 ranking (Table 2, ranks 1 to 5) are hydrogen-bonded rings involving nitrogen or oxygen donors and nitrogen, oxygen or sulfur acceptors. They include the motifs I and II, which also have very high %SIO values (≥ 95, Table 4). The other three of these motifs are missing from Table 4 because they occurred in 1 have topological symmetry, i.e. forms such as ABCD…ABCD, suggestive of local pseudo-inversion symmetry in the three-dimensional structure. Cluster analyses of motif geometries confirmed this. The result is



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consistent with earlier studies that have emphasized the prevalence of pseudo-inversion between SI molecules in structures with Z’ > 1.23-25 Some cyclic motifs involving CH…O hydrogen bonds appear in Table 2 (ranks 7, 11 and 16) and they also appear at slightly lower ranks, e.g. H-C~C-N=O...H-C~C-N=O and H-Csp3-N-C=O...HCsp3-N-C=O at ranks 34 and 49, respectively. Their %Z’>1 values range as high as 60. They have the topological symmetry noted above. %SIO values are available for three of these motifs and are also high (≥87). Examples are shown in Figure 5 (the motifs shown in this and subsequent figures are all between SI molecules). These interactions are weak,26 but our results suggest they may have a significant influence in stabilizing structures with Z’ > 1. A similar conclusion was reached by Babu and Nangia.27

Figure 5. Examples of cyclic motifs containing C-H…O hydrogen bonds that have high %Z’>1 values in noncentrosymmetric structures. They are (clockwise from top left) the motif types ranked 16, 11, 49 and 34. The examples are taken from CSD entries TIRGEI, TARGAU, AGAXOX and ISUKAJ.

About half (11) of the top 25 motifs, and many further down the list, are various types of stacking interactions, most commonly involving aromatic rings28,29 and heterocycles (which may be metalbound) and also exocyclic groups such as nitro and carbonyl. Unsaturated bonds arranged with


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antiparallel dipoles such as N=O…N=O (rank 52, %Z’>1 = 45) are included in this category. The motif ranked 22 mainly involves stacked esters and anisoles. The highest-ranked stacking interaction has %Z’>1 = 59. Many stacking motifs also have high %SIO values, e.g. Csp2-C~C~C~C...Csp2C~C~C~C has %Z’>1= 57, %SIO = 89. The aromatic stacked interactions sometimes show pseudoinversion, but by no means always. The interactions can vary from the sandwich arrangement (one ring directly over the other) to geometries showing substantial lateral displacement (Figure 6), but the ring planes are usually parallel or nearly so.

Figure 6. Typical geometries of aromatic stacking interactions between symmetry-independent molecules in noncentrosymmetric structures. Examples taken from CSD entries MAWGIC, YUQTEK, EDOFEI, AFOWEZ.

So almost all of the top-ranked motifs are either hydrogen-bonded rings, which may involve weak or strong hydrogen bonds, or stacking interactions. However, a single stacked arrangement might give rise to more than one motif. For example, if a pair of stacked aromatic rings gives rise to the motif C~C~C…C~C~C, it may also give rise to the smaller motif C~C…C~C, though this is not



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guaranteed. The large number of stacking motifs among the top ranks may therefore be somewhat misleading. To better understand the relative importance of stacking and hydrogen-bonded rings, we identified all high %Z’>1 motifs in the noncentrosymmetric list (see Supporting Information) that we felt confident were strong hydrogen-bonded rings and determined the number of structures in the NONCENTRO-Z’2 data set that contained at least one of these between SI molecules. There were 507. Repeating the procedure for weak hydrogen-bonded rings (CH donors) and stacking interactions found 494 and 1,772 structures, respectively. Thus, although the strong hydrogenbonded ring motifs tend to have higher %Z’>1 and %SIO values, stacking may be a more important cause of Z’>1 because aromatic rings and other π-systems are so ubiquitous. Our definition of stacking motifs was broad, including metal systems and stacked antiparallel bond dipoles. When we restricted the definition to motifs that we were confident corresponded to aromatic systems only, the structure count was 587, which is much reduced but still higher than the hydrogen-bonded rings. There were 2,418 structures containing at least one of the stacking or hydrogen-bonded ring motifs between SI molecules. Given that the NONCENTRO-Z’2 data set contains 8,623 structures in all, this suggests that about a quarter of the structures in the data set may owe their Z’ > 1 value to the presence of one or more of these high Z’>1 motifs. As noted earlier, it has been hypothesized that some molecules crystallize with Z’ > 1 because they are chiral, and therefore constrained to space groups which lack the inversion centers across which many types of motifs preferentially form. Instead, it is argued, these motifs form between SI molecules. We divided the NONCENTRO-ALL data set into two subsets, depending on whether structures contained chiral or achiral molecules. %Z’>1 values were determined for the 1,698 motifs that occur in ≥50 structures in both subsets. As would be expected from the above hypothesis, the mean %Z’>1 value for the chiral subset was significantly higher than that for the achiral subset (26.6 versus 22.9, p < 0.001, paired t-test) and 71.6% of the motifs (1,216) had higher %Z’>1 values in the chiral subset. However, it is noteworthy that there are many motifs with large %Z’>1 values even in 16 ACS Paragon Plus Environment

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the achiral subset; for example, the motifs H-O-C=O...H-O-C=O, H-N-C=O...H-N-C=O, HC~C-O...H-C~C-O and Csp2-C~C~C~C...Csp2-C~C~C~C have %Z’>1 = 76, 70, 48 and 48, respectively. 3.4. Motifs associated with Z ’ > 1 in centrosymmetric structures. The top-ranked motifs for the centrosymmetric subset (Table 3) tend to have lower %Z’>1 values than those in noncentrosymmetric structures, but all in the top 25 are ≥ 38. In emphatic support of the conclusions of Brock and Duncan (see Introduction), visual inspection of examples confirms that several of the highest-ranked motifs are various types of OH…OH hydrogen bond (ranks 1, 2, 3, 4, 6, 7, etc.; also note rank 17 in the noncentrosymmetric structures, Table 2).9 Most of the remaining motifs in Table 3, and many with slightly lower %Z’>1 values, are various types of CH…π interactions.28,30 They sometimes involve alkyl CH groups interacting with aromatic π-systems, but most commonly arise from two aromatic ring systems, very frequently polycyclic, in edge-to-face dispositions. Geometrically, the ring planes may be mutually orthogonal or tilted (Figure 7); there is a fuzzy dividing line between interactions that are unequivocally aromatic edge-to-face and CH…π contacts that might better be described as incidental artefacts of close packing, e.g. the phenyl…phenyl interactions of inter-meshed triphenylphosphine groups.



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Figure 7. Typical examples of an aromatic-aromatic CH…π motif in centrosymmetric structures, showing that the interacting ring systems can be approximately orthogonal (left, CSD entry DUMVUC02) or tilted (right, RUFMAG).

The motifs ranked 8 and 11 involve a CH…π contact and an N-H…N hydrogen bond. The latter can involve various types of nitrogen donor and acceptor but frequently is a weak hydrogen bond between two aromatic -NH2 groups. To assess the overall importance of OH…OH hydrogen bonds, we identified all high %Z’>1 motifs in the centrosymmetric list (see Supporting Information) that correspond to this type of interaction, and determined the number of structures in the CENTRO-Z’2 data set that contained at least one of them between SI molecules. There were 573, representing about 3.3% of the structures in the data set. Hence, although OH…OH hydrogen bonds are undoubtedly associated with Z’ > 1 structures, they can explain only a small proportion of them. Repeating the analysis for motifs that we believe are of the CH…π type revealed that 3,177 structures in the data set contain at least one such motif between SI molecules. However, we did not visually inspect examples of all these motifs, there being too many of them, so this result should be interpreted with caution. Confining our search to the subset of motifs that were specifically of the aromatic edge-to-face type resulted in 2695 structures (15.4% of the data set). Overall, it seems likely that the formation of CH…π contacts and, in particular, aromatic edge-to-face interactions may be a causative factor behind the formation of at least some centrosymmetric structures with Z’ > 1. However, we must emphasize the approximations of this and the similar analysis performed for noncentrosymmetric structures (see preceding section). We have confined ourselves only to motifs whose %Z’>1 values are statistically significant at a very high level, so the numbers we have quoted may be artificially low. On the other hand, we cannot assume that every high %Z’>1 motif that occurs between SI molecules is necessarily important in causing the structure to crystallize with Z’ > 1.


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3.5. Some previously studied motifs. Some motifs which are missing from Tables 2 and 3 have been suggested by other workers to be influential in causing structures with Z’ > 1.1 Of course, they may have high %Z’>1 values but be outside the top ranks or not statistically significant at the 99% confidence level. Motifs of the form (Csp2)Hal…Hal(Csp2) where Hal = I, Br, Cl have %Z’>1 values in centrosymmetric structures of 21, 19 and 17, respectively, the first two of these being significant at the 95% (I) and 99% (Br) confidence levels. The corresponding values in the noncentrosymmetric data set are 26, 34 and 28, only the bromine value being statistically significant. The “urea tape motif”10,31 does not have a statistically significant %Z’>1 value in either data set (11 and 23 in centrosymmetric and noncentrosymmetric structures, respectively). By assigning the same symbol to all metal atoms, we cannot detect metal-specific interactions such as aurophilic contacts.10, 32 3.6. Crystallographic inversion symmetry and Z’. It is known that pseudo-inversion is often found in structures with Z’ > 1,23-25 and we have seen that the motifs associated with Z’ > 1 are markedly different in centrosymmetric and noncentrosymmetric structures. We therefore wished to understand better the relationship between the interaction motifs that form between SI molecules and those that tend to occur across crystallographic inversion centers. 3.6.1. Identification of inversion-favoring motifs. We identified inversion-favoring motifs by determining the %IO values of the 11,240 motifs that occur in ≥50 structures in the CENTRO-Z’1 data set. The decision to use only structures with Z’ = 1 was taken in the light of exploratory searches of the CSD using the ConQuest program.33 They showed that the cyclic amide dimer (I) is far more likely be on a crystallographic center of symmetry when Z’ = 1 than when Z’ = 2. We conclude from this that the intrinsic preference of motifs for inversion centers is masked when Z’ >1, presumably because they can instead occur between SI molecules. Therefore, it is better to perform the %IO analysis on structures with Z’ = 1. 1,056 motifs were found with %IO ≥ 80. They will be referred to from now on as “inversionfavoring motifs”. Table 6 lists the most common inversion-favoring motifs, where “common” means that they occur in at least 1% of the structures in the CENTRO-Z’1 data set. A complete list is 19 ACS Paragon Plus Environment


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available as Supporting Information. These motifs are of considerable interest but will only be discussed briefly here. Predictably enough,34 they include many aromatic stacking interactions, other stacked interactions of unsaturated systems, cyclic hydrogen-bonded motifs involving strong or weak hydrogen bonds, and antiparallel bond dipoles. Relevant examples may easily be picked out in Table 6; others not in the table include H-N-P=O...H-N-P=O (occurring in 118 structures, %IO = 92), H-N-C=S...H-N-C=S (712, 86), C≡N…C≡N (242 , 87), N=O…N=O (546, 86) and O=S…O=S (153, 94). But there are also motifs such as Csp3-Csp3-Csp3-Csp3-H...Csp3-Csp3Csp3-Csp3-H (825, 84) and H-Csp3-Csp3-O-Csp3-Csp3-H...H-Csp3-Csp3-O-Csp3-Csp3-H (117, 81). These probably have high %IO values because inversion is often an effective way of achieving close packing.35 Although something of a digression, we could not resist looking at other symmetry operations. Using the ALLSYMM-Z’1 data set, we determined %SAO, %GPO and %CTO values for our algorithmically-generated motifs, to probe motif preferences for screw axes, glide planes and cell translations, respectively. Surprisingly, there was only one motif with %SAO > 80, one with %CTO > 80 and none with %GPO > 80 (the highest %GPO value was only 55). This is in huge contrast to the results obtained for inversion and emphasizes the unique importance of inversion centers in smallmolecule crystal packing. More than one explanation can be suggested, but the most compelling is probably that molecules related by inversion symmetry tend to have many bond dipoles arranged in an energetically favorable, antiparallel orientation.36 Of course, we are not detecting chains of interactions, which can only occur between molecules related by the other types of symmetry, but we might have expected the component acyclic motifs of such chains to be picked out by our analysis and this was not the case. 3.6.2. Relationship between %SIO and %IO. Figure 8 shows scatterplots of %SIO against %IO for centrosymmetric and noncentrosymmetric structures. The (non)centrosymmetric plot is based on motifs occurring in ≥50 structures in both CENTRO-Z’1 and (NON)CENTRO-Z’2. For centrosymmetric structures, the correlation coefficient is -0.67 and for noncentrosymmetric structures it is 0.65. 20 ACS Paragon Plus Environment

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Therefore, and perhaps unsurprisingly, the motifs that tend to occur between SI molecules complement those that are available through crystallographic symmetry. In noncentrosymmetric structures with Z’ > 1, SI molecules tend to interact via inversion-favored motifs and in centrosymmetric structures they tend to avoid these motifs. The curious little “tail” at the bottom left of the centrosymmetric plot contains many acyclic motifs involving (Csp3)H or (Csp2)H atoms.

Figure 8. Scatterplot of %IO against (left) %SIO in centrosymmetric and (right) %SIO in noncentrosymmetric structures.

3.6.3. Relationship between %Z’>1 and %IO; frustration. Figure 9 shows a histogram of the %IO values of high %Z’>1 motifs in centrosymmetric and noncentrosymmetric structures. The bars are normalized to show percentages rather than absolute numbers of motifs so that the two distributions can more easily be compared. It is clear that motifs associated with Z’ > 1 in noncentrosymmetric structures tend to be inversion-favoring whereas those in centrosymmetric structures tend to be the opposite.



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Figure 9. Histogram showing distribution of %IO values of high %Z’>1 motifs in centrosymmetric (blue bars) and noncentrosymmetric structures (red bars).

Motifs that preferentially form across inversion centers cannot do so if a molecule is forced to crystallize in a Sohncke group because it is optically pure (and also, perhaps, when effective close packing mandates a noncentrosymmetric space group?). It is suggested that, in such circumstances, crystal structures with Z’ > 1 are likely because it enables the motifs to form between SI molecules with local pseudo-inversion.1 We sought to obtain further evidence to test this “frustration” hypothesis. We established that, of the 5,026 structures in the NONCENTRO-Z’2 data set that contain chiral molecules in Sohncke space groups, 2,064 (41%) contain motifs between SI molecules that are inversion favoring. They include several types of hydrogen-bonded ring motifs and aromatic stacking interactions, together with the motifs Csp3-Csp3-Csp3-Csp3-H...Csp3-Csp3-Csp3-Csp3-H and H-Csp3-Csp3-O...H-Csp3-Csp3-O. The latter two emphasize that one function of SI molecules may be to achieve close packing of hydrophobic, aliphatic groups. Overall, our results strongly support the frustration hypothesis as it applies to noncentrosymmetric structures with Z’ = 2. Frustration is also a possibility in centrosymmetric space groups. A structure may contain more inversion-favoring motifs than can be accommodated around the crystallographic inversion centers, in which case one or more of the motifs might instead form between SI molecules.1,10 We investigated this using the same methodology as above. Of the 17,494 structures in the CENTRO-Z’2


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data set, there are 3445 (19.7%) in which inversion-favoring motifs occur both across crystallographic inversion centers and between SI molecules. The motifs most often involved include the cyclic amide dimer, ring stacking interactions such as C~C~C-Csp3-H...C~C~C-Csp3-H and Csp2-C~C~C...Csp2-C~C~C, and the antiparallel stacking of metal-bound carbonyl groups. Of course, frustration is only one possible explanation of inversion-favoring motifs between SI molecules in centrosymmetric structures. This outcome might also be due to solution preassociation. 3.7. Further observations on %SIO. 3.7.1. Other motifs with high %SIO values. Some motifs with high %SIO values, and therefore a strong tendency to occur between SI rather than SR molecules, are of types not yet mentioned. The cyclic motifs in Table 5 with ranks 2, 3, 13 and 15 (also one not listed at rank 26) involve a CH group and either an NH or OH group donating to the same oxygen or nitrogen acceptor. This type of motif can also have high %SIO values in the noncentrosymmetric data set, though none fall in the top 25 listed in Table 4. Examples are shown in Figure 10. We noticed that these motifs are sometimes perpetuated in chains throughout the structure. Acyclic motifs involving N-H…N and N-H…O hydrogen bonds can also have high %SIO values in both centrosymmetric and noncentrosymmetric structures, e.g. the motif at rank 22 in Table 5. The %Z’>1 values of these cyclic and acyclic motifs are usually moderately high, e.g. %Z’>1 = 33 for (C=)(N)N...H(N) in noncentrosymmetric structures. The motifs are not inversion-favoring. Other motifs with high %SIO but low %IO that involve NH groups include (N)H...O(=N), H-Csp3-Csp2-NH...O and (M)Cl...H(N), the latter being an NH hydrogen bond to a metal-bound chlorine.37 Visual inspection of some examples suggested that these types of motifs frequently connect molecules arranged along pseudo-screw axes (e.g. CSD entries HUDYUA, BEZRUV, RIZYEE, AQIRIC, BALYAP) but may also be components of pseudo-glide planes (e.g. ABESEG01) or interactions reminiscent of basepairing (e.g. BIVVIL). Examples are shown in Figure 11.



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Figure 10. Examples of cyclic motifs featuring NH and CH groups donating to a shared acceptor atom. These can have high %SIO values in both noncentrosymmetric and centrosymmetric structures. Examples (clockwise from top left) are taken from CSD entries DEYFOD, INEJUG, DPYRAM02, DOBHEJ and AZUSAQ. The first two exemplify motifs ranked 38 and 71 in the NONCENTRO-Z’2 data set, the last three represent ranks 13, 3 and 2 in the CENTRO-Z’2 data set.

Figure 11. Examples of pseudo-screw axes (top left, CSD entry HUDYUA; right, BEZRUV) and glide plane (bottom left, ABESEG01) in structures with Z’ = 2. Symmetry-independent molecules are distinguished by different carbon-atom coloring (gray, orange). For clarity, hydrogen atoms are 24 ACS Paragon Plus Environment

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omitted except those involved in the N-H…O or N-H…N hydrogen bonds connecting the chains of molecules.

Motifs involving a terminal oxygen atom (e.g. carbonyl or nitro) interacting with a trigonal atom, such as carbonyl carbon or a conjugated nitrogen in an electron withdrawing environment, appear at ranks 17 and 19 in centrosymmetric structures (Table 5). They also occur at ranks 39 and 45 (not shown in table). Their %SIO values range from 72 to 77. Typically, the direction of interaction is approximately orthogonal to the plane of the trigonal atom. In some motifs there is a subsidiary CH…O contact. Examples are shown in Figure 12. The importance of this type of contact in both small-molecule crystal structures and protein-ligand binding has been noted in the literature.38 The corresponding %Z’>1 values are not particularly high, ranging from 16 to 19, Two motifs in Table 5 (ranks 18 and 21) primarily involve alkyl…alkyl contacts, which are unlikely to be energetically strong or have pronounced directional requirements.39 They may be incidental to stronger motifs elsewhere, but it seems more likely to us that they are the by-products of the close-packing of SI molecules.

Figure 12. Examples of high-scoring %SIO interactions in centrosymmetric structures that involve the approach of a terminal oxygen atom to an electron-deficient trigonal carbon or nitrogen atom. They are taken from CSD structures ETOBAQ and GARBIM.



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3.7.2. Comparison of %SIO values in different space groups. We have seen that the propensity for an interaction motif to occur between SI molecules varies dramatically depending on whether structures are centrosymmetric or noncentrosymmetric. Do they also vary from space group to space group within these subsets? The five most common space groups in the CENTRO-Z’2 and NONCENTRO-Z’2 data sets are P1, P1ത, P21, P21/c and P212121 (varying between 945 and 8,671 structures). We determined %SIO values for interaction motifs in each of them. Taking each pair of space groups which are either both centrosymmetric or both noncentrosymmetric, we then computed the correlation coefficient, r, of the motif %SIO values. Results were as follows: P1ത versus P21/c, r = 0.72; P1 versus P21, r = 0.85; P1 versus P212121, r =0.81; P21 versus P212121, r = 0.83. Thus, there is a fairly good correlation for all four pairs, especially when compared with the r value for the centrosymmetric versus noncentrosymmetric data sets (r = 0.11). However, some interesting differences were noticed between the P1ത and P21/c results. Most importantly, there are several motifs that fall towards the bottom of the %SIO rankings in P1ത but have high ranks in P21/c, and therefore show a much greater propensity to occur between SI molecules in the latter space group. Examples are H-N-C=O...H-N-C=O (%SIO = 37 in P1ത, 72 in P21/c), Csp2-N-C~C...C~C (37, 68) and H-O-C=O...H-O-C=O (32, 68). Many of these discrepant motifs appear to be the types of interactions – stacking or topologically-symmetric hydrogen-bond ring motifs – that tend to have high %SIO values in noncentrosymmetric structures. Closer investigation of the H-N-C=O...H-N-C=O motif revealed that, in P1ത, it occurs exclusively between SR (specifically, inversion-related) molecules in 91 structures and exclusively between SI molecules in only 57. The corresponding figures for P21/c are very different, being 38 and 109, respectively. It is an interesting result that invites further research. 3.8. Symmetry-independent molecules and close-packing. Previous work has suggested that, in structures with Z’ > 1, tighter contacts and a greater surface area of contact tend to occur between SI molecules than between SR molecules.25,27,40 Our results support this. We calculated the percentage of primary interactions in each structure in the combined CENTRO-Z’2 and NONCENTRO26 ACS Paragon Plus Environment

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Z’2 data sets that involve atoms in SI molecules. A histogram of the results is shown in Figure 13. It is skewed towards values higher than the random expectation value of 50%, the average being 57.7%. Distributions calculated for structures in each of the five most common space groups were all similar, the average varying only from 56.7% (P21) to 63.6% (P1). While the primary focus of this paper has been on specific interaction motifs, we must not lose sight of the strong possibility that close packing considerations often suffice to explain crystallization with Z’ > 1.

Figure 13. Histogram of the percentage of primary interactions in a structure that occur between symmetry-independent molecules.

Despite the tighter packing that tends to occur between SI molecules, there has been speculation that structures with Z’ > 1 may tend to have lower densities than those with Z’ = 1.1 This might be expected if many of the former are metastable. However, Gavezzotti compared two samples of structures, one with Z’ = 1 and one with Z’ = 2, and found their average densities to be similar.25 We concur. We searched the CSD for pairs of polymorphs, one with Z’ = 1 and one with Z’ > 1, both determined at ambient temperature and pressure. When more than one pair was available for a given compound, the pair with the best R-factors was used. Application of a paired t-test on the resulting sample of 792 pairs showed that there was no significant difference between the densities of the Z’ = 1 and the Z’ > 1 polymorphs.



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3.9. Other results. Two further results were obtained by analyzing CSD structures containing only r

one type of chemical entity (Z = 1). Firstly, we investigated the relationship between Z’ and the number of atoms in a molecule that are capable of donating or accepting strong hydrogen bonds (NHB). Any NH or OH group was considered a donor. Any oxygen or nitrogen atom was deemed to be an acceptor, other than conjugated oxygen (e.g. in esters or furans) and nitrogen atoms that are quaternary or planar trigonal. An atom that can both donate and accept, such as hydroxyl oxygen, was counted only once. The mean values of NHB were 3.1 for the 296,964 structures with Z’ = 1 and 2.8 for the 39,766 structures with Z’ > 1. The difference is statistically significantly (p < 0.005, unpaired t-test). Inspection of the distributions (Figure 14) reveals that it is entirely attributable to structures with small but non-zero NHB values (NHB = 1 to 3), where there are relatively more structures with Z’ > 1. Molecules with a small number of hydrogen-bonding atoms may have difficulty achieving close packing while simultaneously ensuring that the necessary hydrogen bonds are formed (remembering Etter’s first rule41). It seems feasible that the problem becomes less severe as the number of hydrogen-bonding atoms increases, as there will then be a range of possible hydrogen-bond patterns to choose from.

Figure 14. Distribution of number of atoms capable of donating or accepting hydrogen bonds (NHB) in CSD structures with Zr = 1 and Z’ = 1 (blue) or Z’ > 1 (red). Bar heights indicate percentage of sample rather than actual numbers of structures so that the distributions can be more easily compared. Final pair of bars shows percentage for NHB > 10.


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Secondly, we investigated the relationship between Z’ and the solvent from which crystals were formed. The nature of the solvent is known to influence the extent and type of solute preassociation,42-44 which in turn may affect the probability of solute clusters (e.g. dimers) acting as the effective packing unit and leading to metastable crystal structures with Z’ > 1. The analysis was a little awkward because solvent is stored as a free-text field in the CSD and the same solvent may be referred to in several different ways (e.g. “ethanol”, “absolute alcohol”, “EtOH”). Therefore, only the most common solvents were considered. Their logP values were retrieved from the literature.45 All mixture solvents were excluded. We found that the average logP of the solvent is somewhat higher for structures with Z’ > 1 than for those with Z’ = 1 (1.25 versus 1.15) and the difference is statistically significant (p < 0.005, unpaired t-test). Closer inspection of this result (Figure 15) revealed that it is almost entirely due to %Z’>1 being high for crystals grown from very nonpolar solvents (17.5, 14.7, 13.9 and 13.7 for heptane, hexane, cyclohexane and pentane, respectively) and low for the polar solvents dimethylformamide (7.3) and dimethyl sulfoxide (7.1). Petroleum ether is not included in Figure 15 because it is a mixture, but it also has high %Z’>1 (15.2). It is tempting to speculate that solute-solute interactions tend to be enthalpically preferred over solute-solvent in the paraffin solvents, though they will obviously be entropically disfavored. Visual inspection of the Z’ > 1 structures crystallized from these nonpolar solvents suggested that they are structurally diverse, and substructure searching with ConQuest showed that they do not contain large clusters of lipophilic molecules such as long-chain alkane derivatives or steroids. Over half of them (54%) are metal complexes, a ratio similar to that in the CSD as a whole.



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Figure 15. Plot of %Z’>1 against logP of solvent from which structures were crystallized.

4. CONCLUSIONS By analyzing several thousand different types of molecular interaction motifs in the CSD, we have identified those that show the strongest tendency to (a) occur in crystal structures with Z’ > 1, and (b) form between symmetry-independent molecules in those structures. Predictably enough, the two tendencies largely go hand-in-hand. Results vary dramatically depending on whether structures are centrosymmetric or not. Motifs that have a strong association with Z’ > 1 in centrosymmetric structures are most commonly OH…OH hydrogen bonds, edge-to-face interactions of aromatic rings and other interactions of the C-H…π type. In noncentrosymmetric groups, they include hydrogenbonded ring motifs – which may involve strong or weak (C-H…O) hydrogen bonds – and a variety of stacking interactions, most importantly between aromatic ring systems. We have shown that a significant proportion of structures with Z’ = 2 may be due to the presence of one or more of the aforementioned motifs. We have also shown that symmetry-independent molecules in noncentrosymmetric structures with Z’ = 2 often interact via motifs that are “inversion favoring” (i.e. also have a strong propensity to occur between molecules related by crystallographic inversion in structures with Z’ = 1). In centrosymmetric structures, they tend to avoid those motifs. The results therefore suggest that the role of symmetry-independent molecules may often be to form energetically favorable interactions


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that are complementary to those that can readily be formed between the symmetry-related molecules. The latter are determined in part by the space group symmetry, which may be largely dictated by close-packing or other considerations. The most obvious space-group constraint occurs for optically pure chiral compounds, which necessarily crystallize in Sohncke groups. Our study has shown that about 40% of single-enantiomer structures with Z’ = 2 contain a motif between symmetry-independent molecules that is inversion-favoring. But we have also found that almost 20% of centrosymmetric structures with Z’= 2 contain inversion-favoring motifs across crystallographic inversion centers and between symmetry-independent molecules. It seems likely that these molecules have difficulty finding a packing arrangement with Z’ = 1 that achieves close packing and allows these motifs all to form across crystallographic inversion centers. A cynic might say that all we have done is rediscover the well-known fact that symmetryindependent molecules are often related by pseudo-inversion. But, to borrow a phrase from Cicero, this would be to put the cart before the horse. It is molecular interactions that cause symmetry (or pseudo-symmetry), not the other way round. We have determined the interactions most likely to be responsible for symmetry-independent molecules adopting a pseudo-inversion relationship, and with an objective methodology. Of course, non-specific dispersion and repulsion interactions – the causative agents of close packing – are also important. Indeed, while our main focus has been on specific interaction motifs, we must acknowledge that compounds often – perhaps usually – crystallize with Z’ > 1 simply to achieve close-packing. In that context, we obtained evidence to support the conclusion drawn by others that symmetry-independent molecules in structures with Z’ > 1 pack more tightly than do symmetry-related molecules. We also found that molecules with small numbers of hydrogen-bonding atoms have an enhanced tendency to crystallize with Z’ > 1, as do crystals grown from nonpolar solvents. These observations can be rationalized (Section 3.9). Of all motifs, the ones with the largest propensity to occur between symmetry-independent molecules are strong hydrogen-bonded ring motifs in noncentrosymmetric space groups. But aromatic interactions are much more common and also favor symmetry-independent molecule pairs 31 ACS Paragon Plus Environment


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(specifically, stacking interactions in noncentrosymmetric groups and edge-to-face interactions in centrosymmetric). Because they are so common, they may well be the motifs that most often play a role in causing compounds to crystallize with Z’ > 1, and we have obtained evidence to support this (Sections 3.3, 3.4). Some aromatic systems prefer to interact with each other by stacking (e.g. nitrobenzene46) and some by forming edge-to-face contacts (e.g. benzene47). A chiral molecule containing an aromatic ring that favors stacking is likely to be particularly predisposed towards crystallizing with Z’ > 1. Recent work suggests that the optimum orientation of aromatic rings depends on electrostatic through-space interactions involving or induced by substituents, and is amenable to calculation.48,49 This takes us to the question of predictability. It is still not possible to predict whether a molecule will crystallize with Z’ > 1. What appears more achievable is to make this prediction if the space group is known, constrained (e.g. must be a Sohncke group) or hypothesized. This is relevant to crystal structure prediction, where a common methodology is to search for good packing arrangements in each of the most common space groups in turn. Gavezzotti once remarked that every structure with Z’ = 2 may be a story in itself.25 Nevertheless, some clear trends have emerged from this work and that of our predecessors in the field.

ASSOCIATED CONTENT Supporting Information CSD reference codes of data sets; lists of high %Z’>1 motifs, motifs with high %SIO values and inversion-favoring motifs.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Tel: +44 1923 775972.


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REFERENCES (1) Steed, K. M.; Steed. J. W. Chem. Rev. 2015, 115, 2895-2933. (2) Anderson, K. M.; Probert, M. R.; Goeta, A. E.; Steed, J. W. CrystEngComm 2011, 13, 83-87. (3) Bishop, R.; Scudder, M. L. Cryst. Growth Des. 2009, 9, 2890-2894. (4) Desiraju, G. R. CrystEngComm 2007, 9, 91-92. (5) Anderson, K. M.; Steed, J. W. CrystEngComm 2007, 9, 328-330. (6) Nichol, G. S.; Clegg, W. CrystEngComm 2007, 9, 959-960. (7) Allen, F. H. Acta Crystallogr., Sect. B: Struct. Sci. 2002, 58, 380-388. (8) Kuleshova, L. N.; Antipin, M. Yu.; Komkov, I. V. J. Mol. Struct. 2003, 647, 41-51. (9) Brock, C. P.; Duncan, L. L. Chem. Mater. 1994, 6, 1307-1312. (10) Anderson, K. M.; Goeta, A. E.; Steed, J. W. Cryst. Growth Des. 2008, 8, 2517-2524. (11) Eppel, S.; Bernstein, J. Acta Crystallogr., Sect. B: Struct. Sci. 2008, 64, 50-56. (12) Bond, A. D. CrystEngComm 2008, 10, 411-415. (13) Anderson, K. M.; Probert, M. R.; Whiteley, C. N.; Rowland, A. M.; Goeta, A. E.; Steed, J. W. Cryst. Growth Des. 2009, 9, 1082-1087. (14) Taylor, R.; Kennard, O. Acta Crystallogr., Sect. B: Struct. Sci. 1983, 39, 133-138. (15) Allen, F. H.; Bruno, I. J. Acta Crystallogr., Sect. B: Struct. Sci 2010, 66, 380-386. (16) van de Streek, J.; Motherwell, S. CrystEngComm 2007, 9, 55-64. (17) Taylor, R. CrystEng Comm 2014, 16, 6852-6865. (18) Alvarez, S. Dalton Trans., 2013 42, 8617-8636. (19) Macrae, C. F.; Bruno, I. J.; Chisholm, J. A.; Edgington, P. R.; McCabe, P.; Pidcock, E.; Rodriguez-Monge, L.; Taylor, R.; van de Streek, J. ; Wood, P. A. J. Appl. Cryst. 2008, 41, 466470. (20) Taylor, R. J. Chem. Inf. Comput. Sci. 1995, 35, 59-67. (21) Butina, D. J. Chem. Inf. Comput. Sci. 1999, 39, 747-750.



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(22) Anderson, K. M.; Afarinkia, K.; Yu, H.; Goeta, A. E.; Steed, J. W. Cryst. Growth Des. 2006, 6, 2109-2113. (23) Marsh, R. E.; Schomaker, V.; Herbstein, F. H. Acta Crystallogr., Sect B: Struct. Sci. 1998, 54, 921-924. (24) Marsh, R. E. Acta Crystallogr., Sect B: Struct. Sci. 1999, 55, 931-936. (25) Gavezzotti, A. CrystEngComm 2008, 10, 389-398. (26) Desiraju, G.R.; Steiner, T. The Weak Hydrogen Bond in Structural Chemistry and Biology, Oxford University Press, Oxford, 2001. (27) Babu, N. J.; Nangia, A. CrystEngComm 2007, 9, 980-983. (28) Salonen, L. M.; Ellermann, M.; Diederich, F. Angew. Chem. Int. Ed. 2011, 50, 4808-4842. (29) Martinez, C. R.; Iverson, B. L. Chem. Sci. 2012, 3, 2191-2201. (30) Tsuzuki, S. Annu. Rep. Prog. Chem. C: Phys. Chem. 2012, 108, 69-95. (31) Todd, A. M.; Anderson, K. M.; Byrne, P.; Goeta, A. E.; Steed, J. W. Cryst. Growth Des. 2006, 6, 1750-1752. (32) Schmidbaur, H. Nature 2001, 413, 31-33. (33) Bruno, I. J.; Cole, J. C.; Edgington, P. R.; Kessler, M.; Macrae, C. F.; McCabe, P.; Pearson, J.; Taylor, R. Acta Crystallogr. Sect. B: Struct. Sci. 2002, 58, 389-397. (34) Taylor, R.; Allen, F. H.; Cole, J. C. CrystEngComm 2015, 17, 2651-2666. (35) Brock, C. P.; Dunitz, J. D. Chem. Mater. 1994, 6, 1118-1127. (36) Lee, S.; Mallik, A. B.; Fredrickson, D. C. Cryst. Growth Des. 2004, 4, 279-290. (37) Aullón, G.; Dellamy, D.; Orpen, A. G.; Brammer, L.; Bruton, E. A. Chem. Commun. 1998, 653654. (38) Paulini, R.; Müller, K.; Diederich, F. Angew. Chem. Int. Ed. 2005, 44, 1788–1805. (39) Li, A. H.-T; Chao, S. D. J. Chem. Phys. 2006, 125, 094312. (40) Pidcock, E. Acta Crystallogr. Sect. B: Struct. Sci. 2006, 62, 268-279. (41) Etter, M. C. Acc. Chem. Res. 1990, 23, 120-126.


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(42) Davey, R. J.; Allen, K.; Blagden, N.; Cross, W. I.; Lieberman, H. F.; Quayle, M. J.; Righini, S.; Seton, L.; Tiddy, G. J. T. CrystEngComm 2002, 4, 257-264. (43) Davey, R. J.; Blagden, N.; Righini, S.; Alison, H.; Quayle, M. J.; Fuller, S. Cryst. Growth Des. 2001, 1, 59-65. (44) Prentice, G. M.; Pascu, S. I.; Filip, S. V.; West, K. R.; Pantos, G. D. Chem. Commun. 2015, 51, 8265-8268. (45) Sangster, J. J. Phys. Chem. Ref. Data 1989, 18, 1111-1229. (46) Trotter, J. Acta Crystallogr. 1959, 12, 884-888. (47) Bacon, G. E.; Curry, N. A.; Wilson, S. A. Proc. R. Soc. London, Ser. A 1964, 279, 98-110. (48) Wheeler, S.E. Acc. Chem. Res. 2013, 46, 1029-1038. (49) Wheeler, S.E.; Houk, K.N. Mol. Phys. 2009, 107, 749–760.



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Table 1. Data sets used in analysis of interaction motifs. name

contents

number of structures

CENTRO-ALL

centrosymmetric structures with Zr = 1

193,342

NONCENTRO-ALL

noncentrosymmetric structures with Z = 1

r

64,403

CENTRO-Z’1

centrosymmetric structures with Zr = 1 and Z’ = 1

100,000

ALLSYMM-Z’1

structures with Zr = 1 and Z’ = 1, no symmetry restriction

100,000

CENTRO-Z’2

centrosymmetric structures with Zr = 1 and Z’= 2

NONCENTRO-Z’2

noncentrosymmetric structures with Zr = 1 and Z’ = 2


17,494 8,623

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Table 2. The motifs most strongly associated with Z’ > 1 in noncentrosymmetric structures. typea

rank motif

Nb %Z'>1

1 H-O-C=O...H-O-C=O

shr

211

75

2 H-N-C=N...H-N-C=N

shr

60

72

3 H-N-C=O...H-N-C=O

shr

381

71

4 H-N-C=S...H-N-C=S

shr

73

70

5 H-N-S=O...H-N-S=O

shr

52

63

6 H-Csp3-Csp3-Csp3-N-C=O...H-Csp3-Csp3-Csp3-N-C=O

o

52

60

7 H-C~C-C=O...H-C~C-C=O

whr

94

60

8 H-Csp3-N-M...H-Csp3-N-M

st

68

59

9 C=N-Csp3...Csp2

st

55

58

10 Csp2-C~C~C~C...Csp2-C~C~C~C

st

129

57

11 H-C~C-S=O...H-C~C-S=O

whr

106

57

12 Csp2-C~C~C...C~C~C~C-Csp2

st

63

56

13 Csp2-C~C~C...Csp2-C~C~C~C

st

115

55

14 C~C-N...C~C-N

st

116

54

15 Csp3-Csp3-C~C~C...H-Csp3-Csp3-C~C~C~C

CHπ

50

54

16 H-C~C-O...H-C~C-O

whr

144

53

17 H-Csp3-Csp3-Csp3-Csp3-O-H...O-Csp3-Csp3-Csp3-Csp3-H

OHO

79

53

18 Csp2-C~C~C~C...C~C~C

st

98

53

19 C~C-N=O...C~C~C~C

st

66

53

20 C~C~C~C...C~N

st

53

53

21 H-N-Csp3-C=O...H-N-Csp3-C=O

shr

154

52

22 Csp2-O-Csp3-H...Csp2-O-Csp3-H

st

148

51

23 H-Csp3-Csp2-N-Csp3-C=O...H-N-Csp3-C=O

o

53

51



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24 C~C~C...H-Csp3-N-M

st

63

51

25 Csp2-N-Csp2...Csp2-N-Csp2

st

87

51

a

Type of motif: shr = strong hydrogen-bonded ring; whr = weak hydrogen-bonded ring; st = stacking

interaction; CHπ = CH…π interaction; OHO = OH…OH hydrogen bond; o = other. bNumber of structures containing motif.


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Table 3. The motifs most strongly associated with Z’ > 1 in centrosymmetric structures. motif

typea

Nb

%Z'>1

1

C~C-Csp3-O-H...O-Csp3-C~C-H

OHO

61

54

2

H-Csp3-Csp3-O-H...O-Csp3-Csp3-H

OHO

165

52

3

(H)(Si)O...H(O)

OHO

73

51

4

H-Csp3-Csp3-Csp3-O...H-O-Csp3-Csp3-H

OHO

107

48

5

C~C~C~C~C~C~C...H-C~C~C~C~C-H

CHπ

63

48

6

H-Csp3-Csp3-Csp3-O-H...O-Csp3-Csp3-H

OHO

97

45

7

H-Csp3-Csp3-Csp3-O-H...O-Csp3-Csp3-Csp3-H

OHO

109

44

8

Csp2-N...H-N-C~C-H

CHπ, NHN

98

44

9

C~N-M-N~N~C...H-C~C-H

CHπ

70

43

10

Csp2-C~C~C...H-C~C-C~C

CHπ

100

42

11

C~C-N...H-N-C~C-H

CHπ, NHN

131

42

12

C~C~C-H...C~C~N-M-N~C-H

CHπ

67

42

13

Csp2-C~C~C...H-C~C-Csp2-H

CHπ

53

42

14

C~C-Csp3-O...H-O-Csp3-H

OHO

53

42

15

C~C-C~C~C-C~C-H...C~C-C~C~C

CHπ

53

42

16

C~C-O...H-O-C~C-H

OHO

51

41

17

Csp2-H...C~N~N-M-N~C-H

CHπ

61

41

18

C~C~N-B-N...H-C~C-H

CHπ

77

40

19

Csp-M...M

o

50

40

20

C~C~C-C~C~C~C...H-C~C-C~C~C-H

CHπ

53

40

21

Csp2-C~C~C~C...H-C~C-C~C-H

CHπ

90

39

22

Csp2-N-Si-Csp3-H...H-Csp3-H

CHπ

54

39

rank



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Page 40 of 64

23

C~N-B-N~N~C...H-C~C-H

CHπ

88

39

24

H-N-Csp2-C=N...H-N-Csp2-C=N

shr

83

39

25

C~C-H...C~C~C~C-Csp3-Csp3-C~C-H

CHπ

73

38

a

Type of motif: OHO = OH…OH hydrogen bond; CHπ = CH…π interaction; NHN = N=H…N hydrogen b

bond; shr = strong hydrogen-bonded ring; o = other. Number of structures containing motif.


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Table 4. Motifs with highest propensity to occur between symmetry-independent molecules in noncentrosymmetric structures with Z’ = 2. typea

rank motif

Nb

%SIO

1 H-O-C=O...H-O-C=O

shr

125

98

2 H-N-C=O...H-N-C=O

shr

224

95

3 H-N-Csp3-C=O...H-N-Csp3-C=O

shr

61

93

4 H-C~C-O...H-C~C-O

whr

62

90

5 C~C~C~C-Csp3-H...C~C~C~C-Csp3-H

st

51

90


st

65

89

7 H-C~C-S=O...H-C~C-S=O

whr

50

88

8 Csp2-C~C~C...Csp2-C~C~C

st

83

88


st

64

88

10 H-Csp3-N-C=O...H-Csp3-N-C=O

whr

70

87

11 Csp2-C~C~C...C~C~C~C

st

154

87

12 (Csp2)(C~)(C~)C...C(Csp2)(~C)(~C)

st

116

85

13 H-Csp3-Csp3-Csp3-O...H-Csp3-Csp3-Csp3-O

whr

53

85

14 C~C-N...C~C-N

st

52

85

15 C~C~C-O-Csp3-H...C~C~C-O-Csp3-H

st

54

83

16 Csp2-C~C...Csp2-C~C

st

119

83

17 Csp2-Csp3-Csp3-H...Csp2-Csp3-Csp3-H

alk

89

83

18 H-Csp3-Csp3-C=O...H-Csp3-Csp3-C=O

whr

93

83

19 H-Csp3-C=O...H-Csp3-C=O

whr

103

83

20 Csp2-Csp3-Csp3-H...C~C~C-Csp3-Csp3-H

alk

50

82

21 Csp2-H...Csp2-O

o

55

82

22 C~C~C~C...C~C~C~C~C

st

71

82



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Page 42 of 64

23 Csp2...M-N

st

53

81

24 C~C~C-Csp3-H...C~C~C~C-Csp3-H

st

79

81

25 Csp2-Csp3-H...Csp2-Csp3-H

alk

163

81

a

Type of motif: shr = strong hydrogen-bonded ring; whr = weak hydrogen-bonded ring; st = stacking

interaction; alk = alkyl…alkyl interaction; o = other. bNumber of structures containing motif.


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Table 5. Motifs with highest propensity to occur between symmetry-independent molecules in centrosymmetric structures with Z’ = 2. typea

Nb

%SIO

1 C~C~C~C~C...H-C~C~C-H

CHπ

52

88

2 H-C=N-N-H...O

hbca

59

83

3 H-C~C-Csp2-N-H...O

hbca

152

83

4 C~C...H-C~C-N-H

CHπ, NHπ

75

83

5 Csp2-C~C...H-C~C-C~C-H

CHπ

90

81

6 Csp2-C~C~C...H-C~C-C~C-H

CHπ

77

81

7 C~C-C~C~C...H-C~C~C-C~C-H

CHπ

59

80

8 C~C~C~C...H-C~C~C~C~C-H

CHπ

53

79

9 H-Csp3-Csp3-O-H...O-Csp3-Csp3-H

OHO

62

79

10 C=C-C~C~C...H-C~C-H

CHπ

76

79

11 C~C~C~C-Csp3-Csp3-H...H-Csp3-Csp3-H

CHπ

57

79

12 C~C~C~C...H-C~C~C-H

CHπ

151

79

13 H-C~C-N-H...N

hbca

61

79

14 C~C~C...H-C~C-N-H

CHπ, NHπ

65

78

15 Csp3-Csp3-N-H...O

hbca

51

78

16 Csp3-C~C~C...H-C~C-Csp3-H

CHπ

95

77

17 Csp2-N-Csp3-H...O

O…Cδ+

56

77

18 H-Csp3-Csp3-Csp3-O...H-Csp3-Csp3-H

alk

79

76

19 Csp2-N-Csp3...O

O…Cδ+

54

76

20 C~C~C~C...H-Csp2-C~C-H

CHπ

69

75

73

75

213

75

rank motif

21 Csp2-N-Csp3-Csp3-H...H-Csp3-Csp3-Csp3-H alk 22 (C=)(Csp2)N...H(N)

NHN



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 64

23 Csp2-Csp3-C=O...H

o

52

75

24 Csp2-H...N-C~C

o

112

75

25 Csp2...H-C~C-C~C-H

CHπ

68

75

a

Type of motif: CHπ = CH…π interaction; NHπ = NH…π interaction; hbca = one strong and one weak

hydrogen bond to a common acceptor; OHO = OH…OH hydrogen bond; NHN = N=H…N hydrogen bond; alk = alkyl…alkyl interaction; O…Cδ+ = oxygen interacting with electron deficient carbon; o = other. bNumber of structures containing motif.


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Table 6. Common motifs with the highest propensity to occur between inversion-related molecules in structures with Z’ = 1. Na

%IO


1006

92


1686

91

3 H-Csp3-Csp3-C=O...H-Csp3-Csp3-C=O

1289

91

4 C~C~C~C-Csp3-H...C~C~C~C-Csp3-H

2067

90

5 C=O...C=O

1221

90

6 H-Csp3-C=O...H-Csp3-C=O

1518

90

7 C~C~C-N...C~C~C-N

1123

90

8 H-N-C=O...H-N-C=O

1650

89


1451

88

10 C=C-Csp3-H...C=C-Csp3-H

1226

88

11 Csp-M-M-C≡O...Csp-M-M-C≡Ob

1111

88


2692

87

13 H-O-C=O...H-O-C=O

1136

87


1723

87

15 H-Csp3-O-C=O...H-Csp3-O-C=O

1326

86

16 C~C~C-O-Csp3-H...C~C~C-O-Csp3-H

1492

86

17 C~C-N...C~C-N

1187

86


1500

86

19 C=C-Csp2...C=C-Csp2

1756

86

20 C~C-P-C~C-H...C~C-P-C~C-H

1320

85

21 H-C~C-O...H-C~C-O

1309

84

22 H-Csp3-O...H-Csp3-O

3008

84

rank motif



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 64

23 C~C-O-Csp3-H...C~C-O-Csp3-H

1435

84

24 M-C≡O...M-C≡Ob

1394

84

25 H-Csp3-N...H-Csp3-N

1056

84

26 C~C~C-Csp3-H...C~C~C-Csp3-H

4673

83

27 H-Csp3-Csp3-O...H-Csp3-Csp3-O

1839

83

28 Csp2-N-Csp3-H...Csp2-N-Csp3-H

1252

83

29 H-C~C-C=O...H-C~C-C=O

1071

82

30 C=C...C=C

1236

82

31 Csp2-N...Csp2-N

1540

81

32 C~C~C-Csp3-Csp3-H...C~C~C-Csp3-Csp3-H

1644

81

33 Csp2-Csp3-H...Csp2-Csp3-H

3537

81


3436

81

35 C~C~C-P-C~C-H...C~C~C-P-C~C-H

1560

81

36 C~C~C~C...C~C~C~C

4903

81


2438

80

38 C~C~C-Csp3-H...C~C~C~C-Csp3-H

1400

80

a

Number of structures containing motif. bM = metal.


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Table of Contents Graphic:

SYNOPSIS: We have exhaustively enumerated molecular interactions motifs in crystal structures taken from the Cambridge Structural Database. This has enabled us to identify motifs that show a statistically significant tendency to occur in crystal structures with Z’ > 1, along with motifs that preferentially form between symmetry-independent molecules in such structures. These motifs may be instrumental in stabilizing structures with Z’ > 1.



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Figure 1. Packing of CSD entry AWUMUB. Molecules with different carbon-atom coloring (gray, blue) are crystallographically independent. Measurements indicate H…O distances in a hydrogen-bonded amide dimer motif. 508x326mm (96 x 96 DPI)

ACS Paragon Plus Environment

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Figure 2. Top: Primary interactions (in purple) of the atoms of the two molecules (the “reference molecules”) in an asymmetric unit of CSD entry AANHOX01. Bottom: Expanded to show the molecules at the other ends of the interactions (the “contact molecules”, shown with thinner bonds). The picture now shows all the interactions from the top picture and all of the primary interactions of atoms in contact molecules that are to atoms in reference molecules. This constitutes the set of interactions used for this structure in our analysis. 508x326mm (96 x 96 DPI)



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Figure 3. Conversion of primary interactions to motifs. The pair of C…H primary interactions shown gives rise to three motifs. Each individual interaction is considered an acyclic motif (left and center), and the pair give rise to a cyclic motif (right). The atoms shown as spheres define each motif’s chemical classification. 508x241mm (96 x 96 DPI)


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Figure 4. Two of the possible geometries of the ΝC~C…H C~C-HΟ motif. 517x324mm (96 x 96 DPI)



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Figure 5. Examples of cyclic motifs containing C-H…O hydrogen bonds that have high %Z’>1 values in noncentrosymmetric structures. They are (clockwise from top left) the motif types ranked 16, 11, 49 and 34. The examples are taken from CSD entries TIRGEI, TARGAU, AGAXOX and ISUKAJ. 517x324mm (96 x 96 DPI)


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Figure 6. Typical geometries of aromatic stacking interactions between symmetry-independent molecules in noncentrosymmetric structures. Examples taken from CSD entries MAWGIC, YUQTEK, EDOFEI, AFOWEZ. 508x326mm (96 x 96 DPI)



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Figure 7. Typical examples of an aromatic-aromatic CH…π motif in centrosymmetric structures, showing that the interacting ring systems can be approximately orthogonal (left, CSD entry DUMVUC02) or tilted (right, RUFMAG). 508x326mm (96 x 96 DPI)


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Figure 8. Scatterplot of %IO against (left) %SIO in centrosymmetric and (right) %SIO in noncentrosymmetric structures. 254x190mm (96 x 96 DPI)



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Figure 9. Histogram showing distribution of %IO values of high %Z’>1 motifs in centrosymmetric (blue bars) and noncentrosymmetric structures (red bars). 254x190mm (96 x 96 DPI)


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Figure 10. Examples of cyclic motifs featuring NH and CH groups donating to a shared acceptor atom. These can have high %SIO values in both noncentrosymmetric and centrosymmetric structures. Examples (clockwise from top left) are taken from CSD entries DEYFOD, INEJUG, DPYRAM02, DOBHEJ and AZUSAQ. The first two exemplify motifs ranked 38 and 71 in the NONCENTRO Z’2 data set, the last three represent ranks 13, 3 and 2 in the CENTRO Z’2 data set. 508x326mm (96 x 96 DPI)



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Figure 11. Examples of pseudo-screw axes (top left, CSD entry HUDYUA; right, BEZRUV) and glide plane (bottom left, ABESEG01) in structures with Z’ = 2. Symmetry-independent molecules are distinguished by different carbon-atom coloring (gray, orange). For clarity, hydrogen atoms are omitted except those involved in the N-H…O or N-H…N hydrogen bonds connecting the chains of molecules. 517x324mm (96 x 96 DPI)


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Figure 12. Examples of high-scoring %SIO interactions in centrosymmetric structures that involve the approach of a terminal oxygen atom to an electron-deficient trigonal carbon or nitrogen atom. They are taken from CSD structures ETOBAQ and GARBIM. 517x324mm (96 x 96 DPI)



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Figure 13. Histogram of the percentage of primary interactions in a structure that occur between symmetryindependent molecules. 254x190mm (96 x 96 DPI)


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Figure 14. Distribution of number of atoms capable of donating or accepting hydrogen bonds (NHB) in CSD structures with Zr = 1 and Z’ = 1 (blue) or Z’ > 1 (red). Bar heights indicate percentage of sample rather than actual numbers of structures so that the distributions can be more easily compared. Final pair of bars shows percentage for NHB > 10. 254x190mm (96 x 96 DPI)



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Figure 15. Plot of %Z’>1 against logP of solvent from which structures were crystallized. 254x190mm (96 x 96 DPI)


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254x190mm (96 x 96 DPI)



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TOC graphic 508x326mm (96 x 96 DPI)


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Molecular Interactions in Crystal Structures with Zâ² > 1 - Crystal

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Molecular Interactions in Crystal Structures with Zâ² > 1 - Crystal