Insights on the Similarity of Supramolecular Structures in Organic Crystals Using Quantitative Indexes.

The quest for concepts of isostructurality in organic crystals has been long and mostly based on geometric data, even with the development of modern software. This field of study is of great interest to the pharmaceutical industry and for the prediction of crystal structures. Despite this, there is still no methodology that provides broad quantitative and comparable similarity data between two complete crystalline structures. The present study demonstrated that the similarity between two crystalline structures could be estimated from the similarity between the two "supramolecular clusters". Quantitative indexes for similarity comparisons of crystal structures were shown using nine 5-aryl-1-(1,1-dimethylethyl)-1H-pyrazoles as a model. This proposal includes the quantitative data of a geometric parameter (I D), a contact area parameter (I C), and an energetic parameter (I G). The proposed indexes exhibited good perspective regarding the similarity data and distinct regions of similarity. The range of similarity was set at I X ≥ 0.80, 0.80 > I X > 0.60, and I X ≤ 0.60 (X = D, C, or G). Indexes with a value near 1.0 indicate systems with isostructural, isocontact, and isoenergetic behavior. The results indicated that supramolecular structures with high similarity must have high values for all three indexes (I D, I C, and I G).

Introduction

For many years, crystallographers have identified motifs in crystal structures and compared these to one known structure, which is important to understand to achieve certain applications, such as in the study of polymorphic forms,[1] cocrystals,[2] or molecules that may have pharmaceutical activities.[3] The field of structure prediction may also benefit with superior tools for supramolecular comparison and guide in designing new crystals. An improved understanding of isostructurality phenomenon can result in the design of similar structures, depending on the geometry and shape of the known structures.[4−7] Different substituents may be changed in the desired molecule without substantially changing the crystal lattice.

In the pursuit for isostructurality, Kitaigorodskii (1961)[8] stated that “isomorphism in organic crystals has some important features which force us to consider this vital topic from a viewpoint different from that for inorganic compounds”.[8] Three decades later, Kálmán et al. (1993)[9] referred to isostructurality as the similarity of spatial arrangements in crystalline structures. The phenomenon was interpreted three-dimensionally (isostructurality involving the whole crystal) and also observed one- and two-dimensionally. In this manner, the International Union of Crystallography states that “two crystals are said to be isostructural if they have the same structure but not necessarily the same cell dimensions nor the same chemical composition...”.[10] It is noteworthy that the two aforementioned concepts only consider the geometric parameter to compare complete structures.

The studies by Kálmán et al.[9,11,12] resulted in the first articulation with the phenomenon, and the authors recommended the use of two main descriptors: the unit-cell similarity index and the isostructurality index (Ii). Therefore, the Ii is obtained after full or partial least-squares fitting of the raw data, which was considered to be a direct measurement of the degree of approximate isostructurality between both molecules. Because of this, the authors recommended the classification of several types of supramolecular structures.[9]

Moreover, Fábián and Kálmán (1999)[13] indicated some limitations of the isostructurality index (Ii) and proposed a new volumetric isostructurality index (Iv) to overcome these restrictions. The authors suggest that the volumes occupied in the unit cell should be closely related in isostructural pairs. In this context, the application of extremely complicated formulae is necessary to obtain the data and a numerical algorithm to calculate Iv and compare structures of different space groups is necessary to transform both into a space group with lower symmetry.

Gelbrich and Hursthouse (2005)[14] described a method based on data from the XPac software that identifies similar supramolecular constructs (SCs), which are subcomponents of a complete crystal structure (0, 1, or 2 dimensionalities). This method can also identify 3D similarity in isostructural assemblies in different crystal structures. In the following year, the same authors used the SC concept to compare 25 related crystal structures and demonstrated that the XPac method is a suitable tool for investigating large sets of related crystal structures.[3] In 2012, the XPac was used in a large study of 14 structures of benzenesulfonamide-2-pyridines, which showed isostructurality of 1D (molecular chains), 2D (layers), and 3D, thus yielding the dissimilarity index (x).[15]

The parameters derived from the XPac program can be useful in establishing limits of isostructurality in polymorphs, which was showed by Coles (2014)[16] in the comparison of two forms of 3-chloromandelic acid. In the same year, the authors[17] also studied 27 monosubstituted mandelic acids and reported their degree of similarity and how all the structures are related.

More recently, the isostructurality of different crystalline forms in terms of nature and energy of the equivalent supramolecular building blocks associated with the presence of various intermolecular interactions in crystal packing has been studied by Dey et al.[1,18] The similarities have been investigated via XPac, two-dimensional Hirshfeld fingerprint,[19] and energy framework analysis.[20,21] These authors demonstrated the “energy framework analysis” with the purpose of indicating in the qualitative form 3D structural similarities in the interactions of studied crystalline forms. In addition, an average was presented between the net cluster energies to show the similarity between the systems.

From the emergence of the concept of isostructurality over 20 years ago up until recent studies, two main issues have been observed. First, most of the comparison studies are based on geometric data—pairs of molecules and/or fragments to compare systems—mainly with software such as XPac[15] and Mercury,[22] which generally provide quantitative isostructurality derived from absolute geometric data without any normalization. Although quantitative, these data do not enable an extension of the conclusions to different series of crystalline structures with important topological differences. On the basis of these observations, there is a lack of quantitative boundaries between high-similarity and low-similarity systems in organic crystals.

Recently, we proposed an approach for better understanding the crystallization phenomenon, which considers the topological and energetic properties of a crystal, explored as a lattice that grows from a supramolecular cluster.[23−28] Therefore, using this concept, we proposed a model that may lead to further understanding the similarity between crystal structures and enable classification in different ranges of similarity. The focus of previous analyses, such as XPac, is finding similar substructures, whereas the focus of this study is to compare complete structures. It is important to note that the goal is not to redefine the concept of isostructurality, which is defined as a geometric similarity, but to provide a tool able to identify similarities between complete structures using the geometric parameter along with other two parameters to observe similarity in a broad perception.

Similarity indexes using contact area, geometrical, and energetic parameters were considered for the first time. To achieve this purpose, a series of nine 5-aryl-1-tert-butylpyrazoles (1–9)[29] was used as the study model (Figure ).

Figure 1

Structures of compounds 1–9 used in this study. Pz = 1-tert-butylpyrazol-5-yl.

Results and Discussion

Supramolecular Cluster

By using the supramolecular cluster,[23] we considered that the similarity of two crystalline structures, which are called A and B, can be estimated from the similarity between the two supramolecular clusters. This comparison can be done for molecules from a series with different substituents or polymorphs. The spatial arrangement of molecules in a cluster can be described with internal coordinates (symmetry codes). Thus, the clusters of two different crystal structures containing molecules of a similar shape and size are immediately comparable. This can be applied regardless of the crystal system, space group, or number of crystallographically independent molecules (Z′). A representation of the supramolecular cluster in the infinite crystal lattice can be seen in Figure using compound 1 as an example. This representation can elucidate the main proposal of this study, which was how to obtain a final value of similarity between two crystalline structures. Current studies search for similarities in different organic crystals using systems that do not represent the whole crystal by comparing different portion sizes (unit cell, fragments or subclusters). In our approach, the comparisons are made with the whole supramolecular cluster (3D)—which has all the necessary information to obtain the similarity between systems—without neglecting molecules that are important for the differences between systems. We believe that it is only possible to assess similarities/differences by considering the 3D portion, because 1D or 2D fragments do not represent all the necessary information for a complete comparison of the crystal network.

Figure 2

Representation of the supramolecular cluster in the crystal network using compound 1 as an example.

From the supramolecular cluster, which was grown from the neighboring molecules around the reference molecule M1, the dimers that give us the contact area, geometric, and energetic information of the crystal structure were obtained. The neighboring molecules represent the molecular coordination number (MCN). These dimers are represented by the interaction between the monomers (molecules) M1 and MN (M1···M2, M1···M3, ..., M1···M15). All of the monomers that share surface contact area with the reference molecule M1 are considered part of the cluster. The supramolecular cluster and surface contact area were obtained using Voronoi–Dirichlet Polyhedra (VDP) by means of the ToposPro 4.0 software.[30]

Similarity Indexes

This proposal includes quantitative data for a geometric parameter (ID), a contact area parameter (IC), and an energetic parameter (IG). Data within the considered dimer of the supramolecular cluster were used for all indexes, which were the values between the monomers that form the dimer. The data were then correlated with the data of the equivalent dimer of the compared cluster. We considered that the two equal supramolecular clusters A and B should have a theoretical value of 1.0, and when comparing “real” systems, the deviation of similarity should be considered to discount from the ideal 1.0, and therefore, the root mean square error (RMSE) calculation was used. Our residual data are the difference between the actual values and the predicted ones, with the RMSE as a measure of the spread of the y values about the predicted y value.

Normalized data were used instead of raw data because the proposal aims to compare systems with different N molecules in the cluster by using as raw data the atom–atom distances, surface areas, and stabilization energy that are distinct in different N molecules. Therefore, normalization is indispensable when comparing similarity in different systems. The use of raw data is one of the restrictions observed in other methods to provide not only quantitative but also comparable values. To understand the normalization process,[23,28] we can think in terms of an ideal symmetric system, where all the N molecules in the cluster should have the same contribution. For example, in a cluster with an MCN of 14, all dimers must have a contribution of 1/14 for all parameters considered (e.g., energy). However, this is far from the reality of most organic molecules, where distinct interactions occur leading to different dimeric contributions. The proposal needs to reduce all raw data to the same metric (scale) using the MCN as the reference for this normalization.

The geometric index of similarity uses the atom–atom distances within the considered dimer (formed by the interaction between the M1 and MN molecules of cluster A), i.e., atom–atom distance between the monomers of the dimer. Those distances are then correlated with the data of the equivalent dimer from cluster B. Geometric data normalization is in relation with the value obtained by i = m × N, where “m” is the number of atom–atom distances between the monomers of the considered dimer in the calculations and N is the number of molecules that form the cluster, i.e., around the reference molecule M1. From this point and for clarity purposes, the number of neighboring molecules from the MCN is named N. Eq is used to obtain the normalization of the raw distance of the dimer.

In eq , the ID value shows a geometric similarity between two supramolecular clusters A and B, where ND is the normalized atom–atom distance between the monomers of the considered dimer that is formed between molecule M1 and molecule MN of cluster A. NDpredict is the normalized atom–atom distance of the dimer formed between molecule M1 and MN, which is from cluster A, predicted from a linear equation of the normalized atom–atom distances of the dimeric equivalents of cluster B (see Supporting Information, section S1 for further information). A unique linear correlation must be performed in each cluster comparison.

For the contact area and stabilization energy parameters, raw data corresponding to the contact area (CM1···M) and energy (GM1···M) within the considered dimer, i.e., between the monomers of the dimer, are normalized in relation with the N molecules of the cluster, following eqs and 4, respectively.

Eqs and 6 show the contact area index (IC) and energetic index (IG) between clusters A and B, respectively.

In eqs and 6, both the NCM1···M—which is the normalized contact area—and the NGM1···M—which is the normalized intermolecular interaction energy of a dimer—are obtained between molecule M1 and a molecule MN of cluster A. NCpredict and NGpredict are the normalized contact area and the normalized stabilization energy, respectively, in a dimer of cluster A, predicted from the data of the dimeric equivalents of cluster B (see Supporting Information, section S1 for further information).

The steps of how to obtain the proposed indexes, including the normalization process, are demonstrated in the Supporting Information (section S1, Figures S1 and S2), where the contact area index (IC) in the supramolecular comparison between compound 3 (Cl) and 4 (Br) was used as an example. The indexes can be used for compounds with Z′ = 1 as demonstrated in this study and also for structures with Z′ > 1, where different supramolecular clusters can be isolated within the crystal structure, and in these cases, one cluster does not represent the whole structure. In those cases, a supramolecular cluster must be grown for each molecule present in the asymmetric unit, and comparisons between all constructed clusters must be carried following the same steps as structures with Z′ = 1.

The similarity values for any proposed index (eqs , 5, 6, and 7) will be between 1 (maximum similarity) and 0. Clearly, the theoretical minimum of Ix = 0 will never be obtained in a calculation because there is always some resemblance, even between less similar structures. The data from the proposed indexes are useful for comparing any supramolecular cluster because we consider normalized data from the equivalent dimers, that is, it is possible to evaluate several systems with different MCNs in the same similarity range. To the best of our knowledge, no quantitative data based on contact area and stabilization energy similarity of the supramolecular clusters of organic crystals have been reported in the literature.

Dissimilarity Index (x) and Geometric Index of Similarity (ID)

To demonstrate the use of the similarity indexes, the 5-aryl-1-tert-butylpyrazoles (1–9) were used as the study model. Initially, we performed a geometric approach. The XPac software was used as an initial screening for the study, only then those systems in which at least one dimension (1D) of similarity was found were considered for further investigation, as there is no reason to assess a degree of similarity for structures that do not share any resemblance in any direction. Therefore, systems identified as “0D supramolecular construct” or “no similarity” were discarded. All nonhydrogen atoms common to all structures were considered for the distance determination (Figure ).

Figure 3

Common set of atoms (points) in the studied molecule (indicated by blue circles) used for the geometric index (ID) and the XPac analysis.

For all the indexes, the equivalent dimers were established after an overlay of the clusters (Supporting Information, Figures S26–S40). The overlay of the clusters was done by considering the M1 molecule as the reference point. An overlay of the M1 molecule was carried out by choosing common atoms between the structures using the Mercury software. After the overlay of the reference molecules, all the other equivalent molecules of the cluster can be found. The geometric similarity data, which were obtained from the XPac program and through the ID index, are shown in Table .

Table 1

Geometric Similarity Data Obtained from the Comparison between the Supramolecular Clusters (“A” vs “B”) of Compounds 1–9, Using the XPac Program and the ID Index

parameter	Cl vs Br	F vs Cl	F vs Br	H vs F	H vs Cl	H vs Br	Me vs OMe	Me vs NO2
xa	0.6	1.7	2.1	2.1	3.2	3.5	5.8	4.9
Nb	14	14	14	14	14	14	10	6
IDc	0.993	0.979	0.974	0.971	0.950	0.945	0.951	0.809
N	14	14	14	14	14	14	16	16

Dissimilarity index (x) obtained from the XPac software.[15]

Considered number of neighbors (N) around the reference molecule (M1 or Kernel).

Geometric similarity parameter (ID) defined by eq .

0D similarity observed by the XPac program, consequently removed from this study.

For the purpose of clarity, all compared systems were referred to as substituent “X” (Figure ). In relation to compounds 1–4, the dissimilarity index (x) provided by the XPac software was between 0.6 and 3.5, while our geometric similarity parameter (ID) was between 0.993 and 0.945. For these compounds, 14 neighboring molecules (N = 14), which correspond to the supramolecular cluster, were considered to be equivalent dimers.

An excellent correlation was observed between the dissimilarity index (x) and the parameter ID, for compounds 1–4, with a linear equation of x = −57.45ID + 57.853, with r = 0.988 (Figure a). This result is expected, given that we are considering very similar supramolecular structures with 3D isostructurality.

Figure 4

(a) Correlation between the dissimilarity index (x) and the index ID (1–4), and (b) correlation between the effective distance (Δdeff) and the index ID (1–4).

For the set of compounds (5–9) with N = 16, the ID index was between 0.717 and 0.945, while the dissimilarity index (x), which was provided by the XPac program, was between 1.3 and 10.0. However, the “x” data, although quantitative, do not follow an appropriate trend. In other words, the “x” values do not provide clear quantitative information regarding the similarity of the compared structures. It is possible to have systems of high similarity (3D) with values close to 10.0 and lower similarity values closer to 1.0, because the method considers a different number of molecules in each example and thus do not allow a range of comparison. The dissimilarity index does not follow an appropriate trend of maximum and minimum. In our molecules when XPac was used, lower dissimilarity values (x) were obtained in systems with large differences between them (e.g., “I vs NO2”, with x = 2.0). The same has been observed in recent studies,[18,31] in which high dissimilarity values (up to 16.0) have been observed for 3D systems. In the compounds 5–9, the number of neighboring molecules considered by the software differs between 12 and 1, in which the 16 molecules of the supramolecular cluster should be considered. This expected difference in the N and x values occurs because it is considered only subunits for a systematic comparison of the clusters.

The relationship between the ID data and those of the dissimilarity index could not be provided for this series of compounds (5–9) because we are dealing with supramolecular comparisons for which XPac considers only the portion of similarity in 1D and 2D of the cluster, and our method considers the whole 3D portion. Our contribution is that we compare the complete structure to provide a quantitative value for the compared system while the XPac method seeks similar substructures.

A correlation between the absolute differences of the effective distance (Δdeff) associated with each substituent, and the parameter ID for compounds 1–4 are shown in Figure b. Additional information regarding the effective distance calculation can be seen in Supporting Information (Tables S8–S9). The Δdeff was already mentioned by Gelbrich et al. (2012),[15] in which good correlation with the dissimilarity index was reported. The effective distance can explain the geometric similarity between these systems with a high correlation (r = 0.971), which is in accordance with Gelbrich et al. (2012)[15] (Figure b).

Contact Area (IC) and Energetic (IG) Indexes

The contact area and energetic approach was chosen to achieve the two other indexes. First, VDP analysis was used to determine the contact area between the M1 and MN molecules (CM1···M) (see Supporting Information, Tables S10–S18). In the second step, the crystal energy stability was determined based on the stabilization of the different dimers formed between the M1 and the MN molecules (see Supporting Information, Tables S10–S18). Finally, eqs and 6 were applied to our contact area and energetic data, while taking into consideration the equivalent dimers. Quantitative data regarding the comparison of several different supramolecular structures were obtained, and the values for all indexes can be seen in Table . A better visualization of the data of the three similarity indexes (ID, IC, and IG) is presented in Figure .

Table 2

Data of Geometric (ID), Contact Area (IC), and Energetic (IG) Parameters of the Supramolecular Comparison for Compounds 1–9

parameter	Cl vs Br	F vs Cl	F vs Br	H vs Br	H vs Cl	H vs F	Me vs OMe
N	14	14	14	14	14	14	16
ID	0.993	0.979	0.974	0.945	0.950	0.971	0.951
IC	0.981	0.992	0.978	0.921	0.928	0.931	0.853
IG	0.941	0.939	0.891	0.895	0.841	0.795	0.764

Figure 5

Overview of geometric (ID), contact area (IC), and energetic (IG) indexes for compounds 1–9.

The comparisons between compounds 1–4 and the Me versus OMe system showed the parameters with the highest values of similarity, with values ranging from around 0.80 to 0.99 for the three main parameters (ID, IC, and IG). These data confirm that these systems can be considered similar, thus demonstrating isostructural, isocontact, and isoenergetic behavior (Figure ). According to the XPac software, the aforementioned comparisons have a 3D degree of similarity. The H versus Br, H versus Cl, and H versus F systems had the following order of similarity for the geometric and contact area parameters: F > Cl > Br. However, the energetic index (IG) had an inverse order. In addition to the subtle difference between these systems (2.4–4.7%), this kind of information shows that the energetic and contact area parameters are crucial in defining the similarity between two crystalline structures.

The systems with N = 16 had a lower degree of similarity and showed two other regions of similarity—one between 0.8 and 0.6 and the other below 0.6—when considered the contact area and stabilization energy parameters. The Me versus NO2 and OMe versus NO2 systems had values in the intermediate region between 0.80 and 0.60 for the contact area and stabilization energy parameters—they have a 2D geometric similarity. However, in this case, XPac indicated a 3D profile. Me versus I and OMe versus I also present values within this intermediate region for the IC and IG.

The geometric data show a relatively high value (around 0.7), even in the less similar systems, which can mislead the interpretation of similarity when only this parameter is considered. Therefore, for a system to be considered similar, it must have high similarity values (>0.80) for all three indexes (ID, IC, and IG), because we should see the similarity as a balance between the three parameters (Figure ). In the cases where only the geometric parameter has higher values than the other two parameters (IC and IG), one should consider this comparison with a lower degree of similarity.

The comparisons with lower similarity involve the Pz structure, which can be observed in the low values of IC and IG. The comparisons Pz versus Me and Pz versus OMe showed a unique behavior where the IC exhibited almost twice the values of the IG, which demonstrates that in these systems there is no direct relationship between area and energy. Therefore, these systems have fewer similarities in contact area but an even less similarity regarding the intermolecular interactions, which results in more distinct stabilization energies (Figure ).

Because of being in agreement with what was suggested by Kálmán et al. (1991),[11] we can establish that there is a similarity range, in which IX ≥ 0.80, 0.80 > IX > 0.60, and IX ≤ 0.60 (X = D, C, or G) are considered to be high, mean, and low values, respectively. The contact area and energetic parameters clearly lead us to a final value that corroborates with what is visually observed (qualitative analysis) in the cluster overlay. Thus, the interpretation using the different indexes shows good perspective in relation to the similarity data, with distinct regions of similarity, especially when using the contact area and stabilization energy parameters. The three main indexes present physical meaning for the understanding of the crystal packing. The contact area and energetic indexes present information regarding the complementarity of surfaces and stabilization energies between molecules, respectively. The use of the three distinct parameters allows a complete comprehension of system similarity.

Additionally, a multiparameter similarity index (Imp) can be composed between more than one parameter between two supramolecular clusters (A and B). The Imp drags the inheritance of the similarity in different parameters to provide a holistic understanding of the comparison. The multiparameter index can be obtained following eq , in which n is the number of parameters considered. When the three parameters (X = D, C, and G) shown in eqs , 5 and 6 are considered, the multiparameter index will be named IDCG.

The multiparameter index can be adapted according to the desired study. In a study regarding the relation only between the contact area and stabilization energy parameter, it is possible to obtain the ICG. The furnished value in the multiparameter index (Table S7) corroborates with what is visually observed (qualitative analysis) in the cluster overlay. The overview of all IDCG data, with the three distinct regions of similarity, is presented in Figure S3.

The multiparameter index can yield the contribution of each parameter (each distinct physical meaning) in a series of compounds. Applying a multiple linear regression fit to the contribution of each parameter to the IDCG similarity index allows us to observe which parameter is more relevant to the similarity of the studied systems. This multiple linear correlation provided the following: IDCG = 0.434IG + 0.339IC + 0.170ID + 0.052. From this correlation, it is possible to acquire important information regarding the high contribution of the energetic parameter (47%) to the final similarity value, followed by the contact area index (35%). The geometric index contributes with around 18% of the final value of IDCG. This multiple correlation provides some final information that corroborates with the previous data presented, where the geometric parameter alone does not provide reliable data regarding similarity because even in less similar systems a high value of geometric similarity can be observed (ID). The correlation shows for the studied compounds that the energetic and contact area parameters are more sensitive to differences between the systems.

Dimers Contribution—ΔNparameter

The indexes (Iindex) of each comparison were correlated with the absolute average of the normalized data between the equivalents dimers considered in the clusters (ΔNparameter). Eq obtained the absolute average of the normalized data from each dimer considered in the clusters (ΔNparameter).

In eq , AN and BN are the normalized data from the dimer of cluster A and the normalized data from the equivalent dimer of cluster B, respectively. Here, “m” is the number of data considered in each dimer. For the geometric parameter, where the atom–atom distances of each dimer are used, “m” is equal to the number of distances considered. For the contact area and energetic parameters, “m” is equal to 1 because each dimer will always have only one data regarding topology and energy.

The ΔNparameter can be ΔNd, ΔNC, or ΔNG, using the distance, contact area, and energy data, respectively. This approach assists in observing the different degrees of similarity between the supramolecular structures studied (Figure ). In addition, there was an excellent view of dimer influence on the similarity index of the cluster.

Figure 6

Correlation between the similarity parameters—ID (a), IC (b), and IG (c)—and the differences in intermolecular distances (ΔNd), surface contact area (ΔNC), and energy (ΔNG), respectively, of the equivalent dimers of the supramolecular clusters A and B of compounds 1–9.

The correlation in Figure a shows the distinct isostructurality region (3D) close to the origin for compounds 1–4 and the “Me versus OMe” system. Moving away from the origin, there is less similarity for the geometric parameter (ID)—which has 1D and 2D systems—and, consequently, there is an increase in the absolute value of ΔNd. The same tendency is observed for the geometric parameter (Figure a), with isocontact and isoenergetic crystal structures close to the origin demonstrated in Figure b,c. Nevertheless, their data are more distributed than in Figure a with only subtle differences in the position between them. This correlation allows us to observe the importance of data traceability from each dimer that were used to achieve the final indexes of similarity.

Correlations were made between the proposed indexes and several parameters in the search for variables that have an influence on the similarity phenomenon. Some of the parameters used were related to the substituent (e.g., the Hammet substituent effect and electronegativity), while others were related to the molecule (e.g., melting enthalpy and sublimation enthalpy). Parameters related to the unit cell (e.g., crystal packing efficiency—CPE[27]) and to the cluster (e.g., the total cluster surface contact area, energy, energy per area, volume, surface, globularity, and asphericity (Ω)) were also correlated. Correlations were done with all the indexes (ID, IC, IG, and IDCG), and the absolute difference was determined between the values of the parameters considered from each compared cluster. No significant correlation was found (i.e., r > 0.700) in most comparisons; however, a significant trend was identified when the CPE and the Ω were considered.

Asphericity is a measure of the anisotropy of an object, in which values around 0.0, 0.25, and 1.0 correspond to isotropic objects (spherical), prolate objects (cigar-shaped), and oblate objects (disk-shaped), respectively. The cluster asphericity (Ωcluster) was obtained directly from the data provided by the software CrystalExplorer[32] through the Hirshfeld surface. In the cell, void volume determination was used with an isovalue of 0.0020 a.u. using the software CrystalExplorer.[32] The asphericity of the cluster was considered instead of the molecule because our indexes are based on the cluster and not on the individual molecule. For the CPE, it was necessary to use data from the unit cell and to compare with our indexes because there is no tool that allows us to obtain the total volume and void volume of the supramolecular cluster. The correlations between the CPE and asphericity, and the similarity indexes are shown in Figures S21 and S22, respectively. The ΔCPE (Supporting Information, Figure S21) and the ΔΩcluster (Supporting Information, Figure S22) correlations had r values ranging from 0.718 to 0.792 and 0.740 to 0.814, respectively. These correlations showed that the CPE, and especially the asphericity of the supramolecular cluster, could be related to the similarity of the organic crystal systems.

Conclusions

There is a lack of additional parameters in the investigation of packing comparisons in molecular organic crystals in the crystal engineering. The demarcation issue related to the size of the system studied is the starting point of the problem in packing similarity studies. This study showed that the supramolecular cluster could be an alternative for comparisons between complete structures. For the first time, geometric, contact area, and energetic parameters were considered for proposing indexes to assess the understanding of the packing similarity of complete crystal structures. Here, a comparison between two crystal structures to be considered similar must have an isostructural (geometric parameter), isocontact (contact area parameter), and isoenergetic (stabilization energy parameter) behavior. We believe that any similarity comparison tool must consider these three parameters to achieve a consistent and coherent result. A similarity range with distinct regions of similarity was established, in which IX ≥ 0.80, 0.80 > IX > 0.60, and IX ≤ 0.60 (X = D, C or G) are considered to be high, mean, and low values, respectively. The multiparameter index was shown to be a quantitative index to assess the understanding of the contribution of each considered parameter. We expect these indexes to contribute as a new tool to assess future studies regarding the comparison of crystalline systems.

Experimental Section

XPac Analysis

All measurements were carried out using the XPac 2.0 software using default values and filter settings of 10, 14, and 1.50, for the angular deviation (a), interplanar angular deviation (p), and corresponding molecular centroid distance deviation (d), respectively. All the results are presented in the Supporting Information (Figures S4–S19).

Contact Area Analysis

VDP was used in the ToposPro[30,33] software for the construction of the supramolecular cluster. The MCN found for each cluster in compounds 1–4 and 5–9 was 14 and 16, respectively. The face of the molecular VDP was considered a set of atomic VDP faces corresponding to the adjacent contacts between the atoms of two molecules. From this, we established that the area of the face of a VDP corresponds to M1···MN interactions, and the considered parameter granted by the software was the area (in Å2) in contact between the monomers of the considered dimer.[34]

Stabilization Energy Calculation

The intermolecular interaction energies of the dimers from the supramolecular clusters of compounds 1–9 were determined by single-point calculations performed with geometries obtained from X-ray diffraction. The optimization of the molecular geometry was not performed. Density functional theory at the ωB97X-D/cc-pVDZ level of theory, with basis set superposition error (BSSE) correction, was used for the calculation. The calculation for compound 5 was done with ωB97X-D/cc-pVDZ-PP, because it is necessary to correct relativistic effects of the iodine atom. All calculations were performed with the aid of the Gaussian 09 software package.[35] The counterpoise method of Boys and Bernardi (1970)[36] was employed to minimize the BSSE. The stabilization energy for M1···MN was obtained by the energy values of the considered dimer minus the sum of the monomer energies of M1 and MN.

Single Crystal X-ray Diffraction

Diffraction measurements of compounds 1–4 and 6 were performed using graphite monochromatized Mo Kα radiation with λ = 0.71073 Å, on a Bruker SMART APEX II diffractometer with a charge-coupled device detector. The diffraction measurements of compound 7 were carried out using Cu Kα radiation with λ = 1.54080 Å on a Bruker D8 QUEST diffractometer, with a KAPPA four-circle goniometer, equipped with a PHOTON II CPAD area detector. Anisotropic displacement parameters for nonhydrogen atoms were applied. Integration, scaling correction, and data reduction were performed using Bruker APEX-II. Absorption corrections were performed using Gaussian and multiscan methods. The structures were solved and refined using the package WinGX.[37] The structures were solved using the SHELXS program[38] and refined based on the full-matrix least-squares method using the SHELXL program.[39] ORTEP projections of the molecular structures were generated using the ORTEP-3 program,[37] which are presented in the Supporting Information (Figure S24). Additional information regarding the data collection and structure refinement can be observed in Tables and 4. Compounds 5, 8, and 9 were previously reported.[23,29] Crystallographic data for the structural analysis of the compounds have been deposited at the Cambridge Crystallographic Data Center with the deposition numbers of 780275 (1), 1541209 (2), 1541207 (3), 1541208 (4), 993665 (5), 1541211 (6), 1541212 (7), 721171 (8), and 993672 (9). Copies of the data can be obtained, free of charge, on application to CCDC, 12 Union Road, Cambridge CB21EZ, United Kingdom; Fax: +44 1223 336033 or deposit@ccdc.cam.ac.uk.

Table 3

Data Collection and Structure Refinement for Structures 1–3

compound	1	2	3
chemical formula	C14H15F3N2	C14H14F4N2	C14H14ClF3N2
Mr	268.28	286.27	302.72
crystal system, space group	orthorhombic, Cmc21	orthorhombic, Cmc21	orthorhombic, Cmc21
temperature (K)	293	293	293
a, b, c (Å)	9.4818 (2), 13.8288 (3), 10.4259 (2)	9.3091 (6), 13.4521 (6), 11.0257 (5)	9.3114 (4), 13.3006 (4), 11.5459 (4)
α, β, γ (°)	90, 90, 90	90, 90, 90	90, 90, 90
V (Å3)	1367.06 (5)	1380.71 (13)	1429.93 (9)
Z	4	4	4
F(000)	560	592	624
Dx (mg m–3)	1.303	1.377	1.406
radiation type	Mo Kα	Mo Kα	Mo Kα
Μ (mm–1)	0.11	0.12	0.29
crystal size (mm)	0.55 × 0.27 × 0.27	0.40 × 0.28 × 0.17	0.66 × 0.19 × 0.14
data collection
diffractometer	X8 APEX II	X8 APEX II	X8 APEX II
absorption correction[40]	Gaussian XPREP	Gaussian XPREP	Gaussian XPREP
Tmin, Tmax	0.664, 0.745	0.893, 1	0.831, 0.960
no. of measured, independent and observed [I > 2σ(I)] reflections	5934, 1458, 1288	11043, 2395, 1388	6670, 1656, 1480
Rint	0.022	0.035	0.028
θmax (°)	26.40	33.56	27.14
(sin θ/λ)max (Å–1)	0.626	0.778	0.642
refinement
R[F2 > 2σ(F2)], wR(F2), S	0.074, 0.228, 1.10	0.099, 0.349, 1.22	0.043, 0.115, 1.08
no. of reflections	1458	2395	1656
no. of parameters	114	110	128
no. of restraints	1	1	1
Δρmax, Δρmin (e Å–3)	0.61, −0.44	1.22, −0.82	0.42, −0.27

Table 4

Data Collection and Structure Refinement for Structures 4, 6, and 7

compound	4	6	7
chemical formula	C14H14BrF3N2	C15H17F3N2	C15H17F3N2O
Mr	347.18	282.31	298.31
crystal system, space group	orthorhombic, Cmc21	monoclinic, C2	triclinic, P1̅
temperature (K)	293	293	293
a, b, c (Å)	9.3635 (5), 13.3247 (5), 11.7516 (5)	16.5186 (17), 9.6555 (8), 12.094 (2)	9.5481 (11), 9.6346 (15), 10.364 (2)
α, β, γ (°)	90, 90, 90	90, 128.318 (5), 90	106.890 (11), 92.607 (17), 119.317 (12)
V (Å3)	1466.20 (11)	1513.4 (3)	775.0 (2)
Z	4	4	2
F(000)	696	592	312
Dx (mg m–3)	1.573	1.239	1.278
radiation type	Mo Kα	Mo Kα	Cu Kα
Μ (mm–1)	2.83	0.1	0.90
crystal size (mm)	0.56 × 0.23 × 0.16	0.36 × 0.30 × 0.13	0.31 × 0.13 × 0.13
data collection
diffractometer	X8 APEX II	X8 APEX II	D8 QUEST
absorption correction[40]	Gaussian XPREP	Gaussian XPREP	Gaussian XPREP
Tmin, Tmax	0.579, 0.802	0.705, 0.745	0.651, 0.751
no. of measured, independent and observed [I > 2σ(I)] reflections	6399, 1563, 1452	9561, 2782, 1377	8452, 1931, 1632
Rint	0.023	0.038	0.022
θmax (°)	27.17	26.81	55.1
(sin θ/λ)max (Å–1)	0.642	0.635	0.532
refinement
R[F2 > 2σ(F2)], wR(F2), S	0.027, 0.071, 1.04	0.068, 0.240, 1.01	0.071, 0.206, 1.38
no. of reflections	1563	2782	1931
no. of parameters	128	182	192
no. of restraints	1	1
Δρmax, Δρmin (e Å–3)	0.43, −0.22	0.28, −0.19	0.37, −0.26