Theoretical study of intermolecular interactions in crystalline arene–perhaloarene adducts in terms of the electron density

Abstract

The effect of halogen substitution on the intermolecular interactions and crystal packing of arene–perhaloarene adducts was studied by means of theoretical methods. Solid state density functional theory geometry optimizations with the LAPW methodology were carried out for the hexafluorobenzene–pyrene, hexafluorobenzene–triphenylene, hexachlorobenzene–pyrene and hexachlorobenzene–triphenylene complexes using as starting points the X-ray crystal geometries measured in our laboratory. The structures of hexachlorobenzene–pyrene and hexachlorobenzene–triphenylene are reported for the first time. Using the tools of the quantum theory of atoms in molecules, the following six types of intermolecular interactions were identified: π⋯π, π⋯X, π⋯H, X⋯H, H⋯H and X⋯X where X = F or Cl. The electron density and related properties at the bond critical points and the NCI index allow to classify them as closed-shell weak interactions. The alternating regions of positive and negative electrostatic potential of chlorine and a larger deformation density than the one observed in fluorine, allow to understand the stronger pairwise interactions involving Cl. The cohesion energies were also computed, being more negative those for the molecular crystals involving hexachlorobenzene. This observation was rationalized in terms of the properties of the electron density at the intermolecular contacts. It was also found that dispersion is the most stabilizing long-range contribution to the dimerization energies of several related model systems. These results suggest that the properties of the electron density and the energetic stabilizing contributions provide complementary viewpoints for the understanding of the intermolecular interactions in these crystals.

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1 Introduction

Understanding the nature of non-covalent interactions exhibited among aromatic compounds is relevant for the progress of chemical, biochemical and material sciences because of their key role in important processes such as biological recognition, protein folding, crystal packing and discotic liquid crystal formation.^1–3 Since it was discovered that hexafluorobenzene and benzene form a 1 : 1 adduct in which both molecules co-crystallize in an almost parallel arrangement,⁴ unlike the nearly perpendicular pattern of the pure compounds, interest has grown in studying the interactions that control the structural features of arene–perfluoroarene complexes and has extended to those involving other perhalogenated compounds.^5–9 These adducts are of theoretical importance because of the diversity of non-covalent interactions involved in their organization such as π–π, halogen–π and the so-called halogen–halogen bond.¹⁰ They have reached applications in diverse fields of study, for instance, in the catalytic control of macrocyclizations using ring-closed metathesis, synthesis of peptide-based materials, super electron donors formation, self-assembly of arene functionalized 2-aminopyrimidines and optoelectronic materials design.^11–15

In particular, much effort has been made to analyze the hexafluorobenzene–arene interactions either theoretically from molecular calculations^16–18 or experimentally (in thermodynamic, spectroscopic and X-ray diffraction studies).^19–22 These systems frequently crystallize in a parallel stacking motif, as in the case of the hexafluorobenzene–benzene adduct, and are considered as supramolecular synthons.^1,23 The C₆F₆–C₆H₆ dimer has been used as a benchmark for evaluating the energetic components of this type of interactions. From a variety of post Hartree–Fock and dispersion-corrected density functional theory calculations, it is now clear that electrostatics alone can not explain the parallel-displaced intermolecular arrangement of the hexafluorobenzene–benzene complex and other perhaloarene–arene interactions.^{9,16,18,24–26} For example, according to symmetry adapted perturbation theory investigations, the dispersion contribution is the largest stabilizing component in the hexafluorobenzene–pyrene complex and in the (C₆X₆)₂ and C₆X₆–C₆H₆ dimers, being X a halogen atom.^9,26,27

The molecular interactions in organic crystals of arenes, haloarenes or their mixtures have been analyzed with theoretical methods, mostly using empirical atom–atom potentials.^28–31 There are also quantum mechanical studies, for example, for the calculation of lattice parameters of the co-crystals of hexafluorobenzene with several polycyclic aromatic hydrocarbons (PAHs) and for the assessment of the stabilizing components in solid C₆Cl₆ and other halogenated compounds.^21,32,33 From the interactions taking place in this type of crystals, it has been suggested that halogen–halogen bonds may be as strong as hydrogen bonds.³⁴ However, the F⋯F contacts are often classified as closed-shell weak interactions.^35,36 In this context, the descriptors and partitioning scheme of three-dimensional space based on the properties of the electron density, ρ(r̄), within the quantum theory of atoms in molecules (QTAIM), has proved to be a suitable framework for the analysis of non-covalent interactions in molecular solids, solving some of the ambiguities of the interpretations based solely on geometric criteria.^37–39 As examples, the nature of the interactions in pure hexachlorobenzene, other chlorinated compounds and a fluorinated benzene derivative have been analyzed with theoretical or experimental electron densities.^32,33,40

The previous discussion shows the existing interest in characterizing the interactions involving halogenated aromatic compounds in the solid state. The study of hexahalobenzene–arene crystals provides a great opportunity for exploring a diversity of intermolecular interactions; the analysis of this type of systems has not been carried out before in terms of the electron density. It is also important to point out that crystal structures of hexachlorobenzene–arene complexes have not been published so far. Because of the similar properties of the hexachloro- and hexafluoro-benzene dimers, it is proposed here that the C₆Cl₆–arene crystals will have the same motif as the C₆F₆–arene ones, in which the Cl⋯H and Cl⋯Cl interactions should provide further stabilization that can be quantified with the properties of ρ(r̄). In this work, the adducts hexafluorobenzene–pyrene (1), hexafluorobenzene–triphenylene (2), hexachlorobenzene–pyrene (3) and hexachlorobenzene–triphenylene (4) are studied through solid-state DFT calculations. The corresponding crystal structures were optimized using as starting point the atomic positions obtained from X-ray diffraction experiments carried out in our laboratory. From these, the crystal structures of 3 and 4 are reported for the first time. The topological analysis of ρ(r̄) was carried out using the theoretical structure factors in order to evaluate the interactions stabilizing these molecular crystals. In addition, molecular calculations of selected clusters based on the crystal structures were performed to complement the solid state results. Additionally, an energetic analysis of the four adducts was performed. The cohesion energies of the crystals and interaction energies of selected model dimers were computed; the corresponding trends are discussed and compared with the properties of ρ(r̄).

2 Methodology

X-ray diffraction experiments were carried out in order to obtain the initial parameters used in the solid state computations. A crystallographic summary is provided in Table 1. Details about crystal growth and data collection of the crystals 1–4 can be found in the ESI.† The CCDC numbers for the new structures 3 and 4 are 1483082 and 1483083, respectively.†

Table 1 Crystal data and structure refinement

Complex	1	2	3	4
Formula	C6F6·C16H10	C6F6·C18H12	C6Cl6·C16H10	C6Cl6·C18H12
FW	388.30	414.34	487.03	513.07
T/K	100(2)	130(2)	240(2)	100(2)
Space group	P21/c	P21/c	P1̄	P21/c
a/Å	6.8764(1)	13.9028(10)	7.367(3)	17.2096(3)
b/Å	13.2307(2)	7.1867(4)	8.555(2)	7.2895(1)
c/Å	9.1980(1)	17.3403(14)	15.803(5)	18.0874(3)
α/°	90	90	94.02(2)	90
β/°	106.215(1)	92.478(6)	102.77(3)	116.651(2)
γ/°	90	90	89.86(3)	90
Z	2	4	2	4
λ/Å	0.71073	0.71073	0.71073	0.71073
θmax	30.507	25.248	25.253	30.438
Nref.	2454	3127	3494	6156
Rint	0.0250	0.0366	0.0295	0.0516
R1	0.0363	0.0967	0.0663	0.0255
wR	0.1205	0.3199	0.1847	0.0711
GooF	1.557	1.114	1.125	1.150

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Geometry optimizations of the crystal structures of the systems 1–4 were carried out with the Wien2K code⁴¹ using the PBEsol density functional employing the linearized augmented plane wave (LAPW) method. DFT-D3 gradients⁴² with three-body terms and the Becke–Johnson damping function were included in order to properly account for the long-range interactions in the optimization of the atomic positions.⁴³ The cell parameters were fixed to the experimental values given in Table 1 and the atomic coordinates obtained from the refinements were used as input data at the outset. The convergence of the integrations in reciprocal space was tested for the smallest unit cell using up to 10 k-points; it was found that 1 k-point was appropriate for the sizable cells and insulator features of all the solids. The following muffin-thin radii (RMT) were used for the atoms in all the structures: RMT(H) = 0.56, RMT(C) = 1.12, RMT(F) = 1.30 and RMT(Cl) = 1.60. The criterion used for the number of plane waves was RK_max = 3.0, because of the small size of the H atoms. Graphical representations of the crystalline structures were created with mercury.⁴⁴ Because of the difficulties inherent to an appropriate description of ρ(r̄) by the LAPW method, single point calculations at the optimized geometries (obtained from the Wien2K calculations) were carried out with the Crystal14 (ref. 45) code at the PBEsol/6-21G(d,p) level of theory. Static structure factors, corresponding to a 0.5 Å resolution, were generated with the XFAC keyword included in Crystal14. The theoretical ρ(r̄) was constructed by means of a multipole refinement fitted against the static structure factors with the XD2006 package,⁴⁶ which is based on the Hansen–Coppens formalism.⁴⁷ The deformation density maps, as well as the topological analysis of ρ(r̄) within the QTAIM framework, were carried out with the WinXPRO program.⁴⁸ The electrostatic potentials were visualized with Moliso.⁴⁹ A description of the results of the multipole refinement is provided in the ESI.†

The cohesive energy was calculated as E_coh = E_AB − E_A − E_B, where E_AB stands for the energy of the adduct in solid-state, and E_A and E_B for the molecular energies of the A and B monomers, respectively. Each of the latter values was obtained after optimizing the molecular geometry with one molecule per unit cell so that the closest contacts are at least 8 Å apart. The RK_max parameter for the perhalogenated species was set to 6.0 to achieve energetic consistency with the calculations of the adducts. In addition, dimerization energies of selected A–B pairs were computed with the PIXEL method using the PIXEL-CLP program package.^50–52

Single point calculations of molecular hexamers of 1–4 were carried out at the PBE/6-311++G(d,p) level of theory with Gaussian 09.⁵³ Each hexamer consists of 3 perhaloarene and 3 arene molecules fixed at the optimized crystalline geometry, selected in such a way that each type of non-covalent interaction appears at least once (see discussion below for the non-covalent interactions classification). The topological analysis of ρ(r̄) of the molecular clusters was carried out with the AIMALL package.⁵⁴ The NCI index⁵⁵ was obtained with the program NCIPLOT^55,56 and visualized with VMD.⁵⁷

3 Results and discussion

3.1 Crystal structure

Fig. 1 shows the intermolecular arrangements of the complexes in 1–4 in the solid state obtained from the theoretical calculations. The hexachlorobenzene molecules in 3 and 4 are arranged in a parallel-displaced configuration as their hexafluorobenzene–arene analogues. In each of the compounds 1, 2 and 4 there are one non-equivalent arene and one perhaloarene molecules per unit cell organized in stacked layers with interplanar angles of 2.30°, 1.78° and 1.14°, respectively. The corresponding mean interplanar distances are 3.356 Å, 3.294 Å and 3.404 Å. In 3 there are two non-equivalent pyrene and hexachlorobenzene molecules arranged with larger interplanar angles of 7.32° and 8.25° and mean interplanar distances of 3.603 Å and 3.554 Å. Complex 1 arranges in a laminar motif in which adjacent layers form a maximum angle of 2.45°. Complexes 2, 3 and 4 show rotated layers with maximum angles of 46.53°, 26.05° and 41.67°, respectively. A noticeable effect of the perhaloarene–arene interactions is a slight deviation of the arenes from planarity. The largest dihedral angles found for the arene moieties in 1–4 are 1.15°, 3.09°, 3.69° and 2.13°, respectively. The perhaloarenes also undergo a deviation from planarity. The largest dihedral angles found for these molecules in the compounds 1–4 are 1.08°, 1.17°, 2.50° and 3.08° respectively. In the case of 1, the optimized bond distances differ on average from experimental values by 0.006 and 0.002 Å for the C–F and C–C bonds respectively; in 4, the differences are 0.001 and 0.005 Å for the C–Cl and C–C bonds, respectively. For both complexes a difference of 0.147 Å was found for C–H bonds. Nevertheless, a more appropriate comparison should be done with the distances suggested from neutron diffraction experiments;⁵⁸ in this case, the discrepancy is of 0.014 Å. In spite of the apparent high value of this result, it has been reported that a difference of ∼0.01 Å does not have an important effect on the charge distribution of oligoacenes.⁵⁹ Positional disorder prevented from carrying out further comparisons of the geometries of 2 and 3 with available experimental values.

Fig. 1 Crystal packing of 1–4. The a, b and c cell axes are shown in red, green and blue, respectively.

3.2 Electron density analysis

The molecular graphs derived from the gradient of ρ(r̄) for hexamers in the crystals of 1 and 4 are shown in Fig. 2; for comparison, the corresponding clusters obtained from molecular calculations using the same geometries as in the solid state are also displayed. The diagrams for 2 and 3 are shown in Fig. S1 of the ESI.† In the case of the isolated clusters, geometry optimization leads to totally different intermolecular arrangements because of the lack of bulk effects; this is why the stationary electron densities from single point calculations are used. All the non equivalent intermolecular bond critical points (BCPs) were categorized in the 6 following groups of contacts: π⋯π, π⋯X, π⋯H, X⋯H, H⋯H and X⋯X, where X = F or Cl. Special emphasis is made on the X⋯X interactions because they are more sensitive to halogen substitution. The connectivity found in the molecular clusters is essentially the same as in the solid-state with some minor exceptions. For example: (1) some π⋯X BCPs only emerge at edge of the isolated molecular clusters and (2) a few π⋯H BCPs in these clusters are nearly structural catastrophes which by a small geometric variation could lead to the H⋯H BCPs observed in the solid. These differences are probably a consequence of bulk effects.

Fig. 2 Molecular graphs of (a) 1 and (b) 4 in the solid and in a molecular cluster using the same atomic coordinates. BCPs are depicted in red and bond paths in gray.

The electron density at the BCP, ρ_b, is used as a measure of non-covalent interaction strength and is the property emphasized in this study. In addition, other topological descriptors^60,61 evaluated at the intermolecular BCPs are also taken into consideration. Table S1 of the ESI† contains the average values the following scalar fields at the BCPs in the crystals: the Laplacian, ∇²ρ_b; the electronic energy density, H_b; the potential energy density, V_b; and its ratio with the positive definite kinetic energy density, |V_b/G_b|. All these properties are small and, except for V_b, positive for the four complexes allowing to classify the 6 groups of contacts as weak closed-shell interactions. The NCI index of W. Yang et al.⁵⁵ is also used as a complement for analysis of intermolecular interactions from a non-local perspective. The NCI isosurfaces shown in Fig. S2 of the ESI,† show similar trends and corroborate the weak nature of the interactions. These surfaces appear in the internuclear regions of all six types of contacts with those for π⋯π and π⋯X appearing jointly as a flat surface. Fig. S2† also shows the steric clashes in the middle of the rings, whose pattern is associated in this approach with ring stress.⁵⁵ In addition, the deformation density, Δρ(r̄), a useful tool for the visualization of the anisotropic distribution of ρ(r̄),⁶² is shown in Fig. 3 for a fluorine atom in 1 and a chlorine atom in 3, respectively; the electrostatic potential projected on ρ(r̄) = 0.1 e Å⁻³ envelopes is also displayed for dimers in these two systems. It is clear that the Cl atoms have the largest deformation of ρ(r̄). Moreover, in contrast to fluorine, they have a region of positive electrostatic potential values which certainly affects their interactions, as has been reported for other systems.⁶²

Fig. 3 (a) Electrostatic potential projected on ρ(r̄) = 0.1 e Å⁻³ envelopes (color code shown on the left) of dimers in the crystals of 1 and 3. (b) Deformation densities around fluorine and chlorine atoms in these complexes (the blue and red colors correspond to positive and negative values of |Δρ(r̄)| = 0.05 e Å⁻³ isosurfaces).

The ρ_b values for the crystalline structures are plotted in Fig. 4 versus those in the molecular clusters, . The slope and intercept are close to the ideal 1 and 0 values, respectively, although some dispersion of the data is also observed. This result is a consequence of the molecular nature of these solids. The largest deviation is found for the X⋯H interactions, for which is in general larger than ρ_b, although this might just be consequence of constraining all the κ parameters of the H atoms to the same value.

Fig. 4 Electron densities at the intermolecular critical points in the crystalline complexes, ρ_b, vs. the values for the molecular clusters, (e·Å⁻³). The solid line corresponds to a linear equation with slope = 0.9865 ± 0.03043, intercept = 4.4627 × 10⁻⁴ ± 0.00119 and R² = 0.9178.

Fig. 5 depicts the values of ρ_b as a function of the distance in each pair of atoms for the six groups of intermolecular contacts. In average, the strength of all the pairwise interactions decreases when increasing the internuclear distance, as expected. As can be seen in Fig. 5(a), the π⋯π contacts are stronger for the adducts formed by the fluorinated compounds because of their shorter intermolecular distances. The BCP associated with the largest distance in Fig. 5(a) corresponds to a non-parallel interaction. Conversely, according to Fig. 5(b), the chlorinated complexes have more and stronger π⋯X interactions than the fluorinated ones even for same distance values.

Fig. 5 ρ_b (e Å⁻³) as a function of the distance (Å) between the atoms involved for the six groups of interactions: (a) π⋯π, (b) π⋯X, (c) π⋯H, (d) X⋯H, (e) H⋯H and (f) X⋯X with X = F or Cl. A different scale is used in each case.

π⋯H contacts are only observed in 2 and 4, perhaps because their crystal structures involve rotated adjacent layers (Fig. 2). According to the values of ρ_b displayed in Fig. 5(c), the topological properties at the intermolecular BCPs shown in Table S1† and because of the large deviation of the C–H⋯C fragments from linearity, it is not possible to classify these interactions as weak hydrogen bonds. It is also worth mentioning that most of these interactions are close to conflict structural catastrophes that could yield non-parallel π⋯π interactions upon small geometric perturbations.

X⋯H is the most copious interaction in the four complexes for which Fig. 5(d) shows the broad distribution of ρ_b values with respect to X–H distances. The larger values for the perchlorinated complexes suggest that substitution of fluorine by chlorine results in stronger X⋯H interactions. This observation is in agreement with findings in a molecular beams study of CH₂FCl⋯H₂CO clusters where these two interactions are present and compete with each other; accordingly, the Cl⋯H interaction is favored over the one involving fluorine.⁶³ The presence of C–H⋯Cl contacts is common in crystals⁶⁴ and they have been classified as a non-conventional hydrogen bonds.⁶⁵ However, the NCI surfaces of the complexes in Fig. S1† are characteristic of very weak hydrogen bonds⁶⁶ and the properties ρ(r̄) do not allow for their unambiguous identification as hydrogen bonds. In addition, as Table S1† shows, both C–H⋯F and C–H⋯Cl interactions are close to the van der Waals limit.

Fig. 5(e) shows the ρ_b values for the H⋯H interactions, the so-called hydrogen–hydrogen bonds.⁶⁷ Whereas 1–3 have intermolecular contacts of this type, these appear in the intramolecular regions of 2 and 4, as can be seen in Fig. 2 for both the crystals and the molecular clusters. It has been argued that the hydrogen–hydrogen bond might be an artifact of the closeness between hydrogen atoms without any stabilizing effect.^68,69 Nevertheless, there is increasing evidence derived from theoretical and experimental works, in both molecular and solid-state complexes, concluding that this type of interactions has an actual stabilizing effect.^61,70 For instance, Hathwar et al.⁷¹ estimated a stabilization of up to 6.6 kJ mol⁻¹ by H⋯H bonds in the crystal structure of orthorhombic rubrene, which shares some features with the arenes studied in this work. The topological properties at BCPs for these weak interactions are about the same order of magnitude as for the other contacts. The largest ρ_b values observed in Fig. 5(e) involve distances shorter than 2.40 Å, the sum of van der Waals radii. In addition, some disc-shaped NCI isosurfaces are similar to those found for the X⋯H interactions. All these results support the stabilizing effects of the intermolecular H⋯H bonds in the complexes.

The most contrasting difference between perfluorinated and perchlorinated complexes is found in the X⋯X interactions, Fig. 5(f). In comparison to the other groups, the ρ_b values at the Cl⋯Cl BCPs follows a monotonous attenuation. Interestingly, this behavior has also been found for Cl⋯Cl interactions in other chlorinated crystals.³³ Whereas the F⋯F contacts appear only at separations smaller than 3.6 Å, the Cl⋯Cl interactions occur at distances as big as 4.4 Å, which are greater than for any other group. Compared to fluorine, the larger size of the intermolecular NCI surfaces, the alternating regions of positive and negative values of electrostatic potential and the larger deformation density of chlorine (both shown in Fig. 3) contribute to the ability of this atom to form stronger halogen–halogen bonds at larger distances.

In summary, Fig. 5 shows that in 3 and 4 the stabilizing effects are achieved at larger distances than in 1 and 2. At a fixed distance, ρ_b can be up to 5 times larger in chlorinated species in comparison with the fluorinated ones, which involves stronger interactions. Although it is incorrect to attribute the crystal packing of these compounds to the halogen–halogen interactions only, it is undeniable that they directly affect the structure of the chlorinated versus the fluorinated solids.

From the structural point of view, halogen–halogen interactions are commonly classified as type I (θ₁ ≈ θ₂) or type II (θ₁ ≈ 90°, θ₂ ≈ 180°), where θ₁ and θ₂ are the C–X₁⋯X₂ and X₁⋯X₂–C angles (X = halogen), respectively.⁷² The former are believed to be a consequence of close packing and the latter to the action of electrostatic forces. Table 2 contains the average values of ρ_b, interatomic distances and angles θ₁ and θ₂ for the X⋯X contacts in all four complexes. From this table, it seems troublesome to classify the geometrical arrangements of the complexes as type I or II. It has been suggested that these characteristics of the Cl⋯Cl interactions entail an interplay between a decrease in the repulsion and an increase of the attractive electrostatic atom–atom interactions, although dispersion and other contributions among entire molecules should be taken into account.^27,62,73,74

Table 2 Average values of ρ_b (e Å⁻³), intermolecular distance (Å), θ₁ and θ₂ for X⋯X in all four complexes

Complex	ρ(r̄)	Distance	θ1	θ2
1	0.009	3.436	132.28	140.97
2	0.009	3.471	108.93	121.24
3	0.046	3.650	132.37	149.62
4	0.021	4.038	115.46	133.462

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3.3 Energetic analysis

It has been proposed⁶⁸ that the study of molecular cohesion should go beyond the consideration of only pairwise interactions as significant parts of the molecules determine the crystal packing of a substance. In addition, the pertinence of evaluating the stability of a crystal in terms of atom–atom interactions is a current debate in the literature.^75–77 In what follows, we explore the non-local character of the interactions in the solids on energetic grounds and compare with the conclusions obtained from the properties of ρ(r̄) at the BCPs used to measure the strength of intermolecular interactions.

The cohesion energies calculated at the present level of theory for the crystal adducts are presented in Table 3. They show that the interactions involving C₆Cl₆ are stronger than those with C₆F₆, in agreement with the general trends of ρ_b in Fig. 5. Further information on how much of the cohesive energies is recovered by the atomic contact pairs can be obtained from potential energy density at the intermolecular BCPs using the Espinosa–Molins–Lecomte (EML) equation,⁷⁸ in this case applied as E = 0.5∑_biV_bi, where the sum runs over all the intermolecular atom–atom contacts of one molecule of C₆X₆ and one of PAH in the unit cell. Further details are provided in the ESI.† As a result of using the V_b values from Table S1,† this procedure recovers on average 50% of the DFT cohesive energy. The empirical nature of the EML equation has been addressed in the literature emphasizing that coefficients other than 0.5, which was obtained for hydrogen bonded crystals, have been used elsewhere.^79,80 It should also be emphasized that most of the times this equation underestimates E_coh. Interestingly, the use of E′_coh = ∑_biV_bi instead, essentially recovers the DFT cohesive energy, as shown in Table 3. In other words, the trends obtained with this approximation to the cohesive energies in terms of the atom–atom contacts are correct because the larger values of E′_coh correspond to the C₆Cl₆–PAH crystals, advocating for a relevant role of the pairwise interactions as a whole in the stabilization of the complexes 1–4.

Table 3 Cohesive energies from solid state DFT calculations (E_coh) and estimated from potential energy densities (E′_coh). The dimerization energies of selected pairs obtained with the PIXEL method are also displayed. Values in kcal mol^−1a

Complex	Ecoh	E′coh	(C6X6)2	(PAH)2	C6X6–PAH^a	C6X6–PAH^b
a In ^astacked and ^bedge–edge intermolecular arrangements.
1	−45.7	−46.8	−0.3	−3.7	−7.8	−1.6
2	−42.0	−40.6	−0.7	−5.4	−7.4	−1.8
3	−60.2	−61.6	−1.9	−3.5	−12.4	−2.9
4	−58.9	−61.8	−2.0	−5.2	−13.9	−4.0

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Finally, the PIXEL method was used to estimate the long-range contributions to the dimerization energies (DE) of selected intermolecular pairs taken from the crystal structures. The resulting DE values are displayed in Table 3 and the corresponding long-range contributions are included in Table S2 of the ESI.† Four types of dimers were considered and the strongest one of each was analyzed. In all cases, the dispersion energy is the largest stabilizing component. The stacked C₆X₆–PAH dimers have the largest DE values and, from these, the chlorinated complexes are the most stable ones. The other dimers have smaller interaction energies in part because of their lateral arrangements dictated by the crystal packing. However, the remaining dimers of each type (for example, there are six C₆Cl₆ molecules around each C₆Cl₆ in 3) should also be taken into account to fully explain the cohesive energy. Concerning the atom–atom contacts defined in terms of the molecular graphs, not all six types are present simultaneously in these complexes: whereas the stacked C₆X₆–PAH dimers present π⋯π and π⋯X contacts, the edge–edge complexes have X⋯H interactions; in (PAH)₂ only H⋯H and C⋯H contacts appear; and (C₆X₆)₂ only have halogen–halogen bonds. If exclusively the ρ_b values were considered, the (C₆Cl₆)₂ dimers should have the largest DE values. Despite that the energetics of complex formation cannot be fully accounted for by the Cl⋯Cl bonds, it is clear from these results that the charge distribution of the Cl atom (discussed before in terms of the electrostatic potential and the Δρ envelopes shown in Fig. 3) affects the crystal packing of 3 and 4 compared to the fluorinated solids.

4 Concluding remarks

The rich variety of intermolecular interactions stabilizing the hexafluorobenzene–pyrene, hexafluorobenzene–triphenylene, hexachlorobenzene–pyrene and hexachlorobenzene–triphenylene crystals was analyzed in terms of the properties of the electron density. Although this approach has been applied in the case of solid hexachlorobenzene and other chlorinated compounds, it is used here to assess the nature of perhaloharene–arene interactions. The experimental crystal structures of the complexes, used as starting point for this theoretical study, were obtained from X-ray diffraction experiments carried out in our laboratory. From these, the crystallographic data for the hexachlorobenzene–arene adducts are reported for the first time.

The electron densities obtained from a multipole refinement of the theoretical structure factors were used to evaluate the nature of the intermolecular pairwise interactions taking place in the adducts by means of the quantum theory of atoms in molecules; the NCI index was also used to gain non-local complementary information. In addition, model molecular clusters were studied in order to make comparisons with the results for the solid state. The molecular graphs defined in terms of the gradient of the electron density and the topological descriptors evaluated at the bond critical points allowed to identify six types of closed-shell weak interactions in the crystals: π⋯π, π⋯X, π⋯H, X⋯H, H⋯H and X⋯X being X = F or Cl. Apart from minor discrepancies, they are also present in the corresponding molecular clusters, in agreement with the molecular nature of these solids. Another interesting finding is that, as in a wide variety of other types of crystals, the hydrogen–hydrogen bonds detected are of an stabilizing nature. In spite of weakening the π⋯π interactions, halogen substitution of fluorine by chlorine enhances the stabilizing contribution of the π⋯X, X⋯H and the halogen–halogen interactions. The cohesion energies of the crystals were also calculated. As an attempt to understand the observed trends of this property in terms of atom–atom contacts, a variant of the Espinosa–Molins–Lecomte equation was used. In agreement with the cohesive energies calculated using the DFT method, this procedure assigned the largest values to the crystals involving C₆Cl₆. In order to explore the relevance of non-local interactions between the charge distributions, the dimerization energies of several model systems were also computed. The dispersion contribution was found as the most influential stabilizing factor in all cases, being the stacked-displaced the most stable arrangement for all the dimers. Although the non-local features of the interaction between the charge distributions affect the stabilization of the crystals, the atom–atom contacts, accounted for by the properties of the electron density at the intermolecular bond critical points, play a key role in the crystal packing of the complexes.

Acknowledgements

The authors gratefully thank DGTIC-UNAM for supercomputer resources (project SC16-1-IR-71), UNAM-DGAPA-PAPIIT (project IN115215) and PAIP Facultad de Química UNAM (numbers 5000-9043 and 5000-9004) for financial support. Dr Marcos Flores Álamo from USAI Facultad de Química UNAM is acknowledged for carrying out the X-ray diffraction experiments of the molecular complexes of this study. Professors Jancik Vojtech and Joel Ireta Moreno are gratefully acknowledged for their useful suggestions on this investigation. B. L.-R. acknowledges financial support from CONACyT (Mexico) through fellowship 245584.

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Footnote

† Electronic supplementary information (ESI) available. CCDC 1483082 and 1483083. For ESI and crystallographic data in CIF or other electronic format see DOI: 10.1039/c6ra14957j

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