Introduction

Understanding of supramolecular architecture of the molecular crystals remains a difficult problem despite the great success of modern crystallography. X-ray diffraction study of crystals provides very accurate data about molecular structure and arrangement of molecules in the crystal phase. However, it is not able to give information about the reasons of packing of molecules in such or another way as well as preferable packing motif of the crystal.

Significant step towards understanding of supramolecular architecture of molecular crystals was provided by supramolecular synthons concept [1–4]. According to this approach molecules in the crystal phase are bonded by limited set of intermolecular interactions formed by specific functional groups or molecular fragments. Interactions between these groups or fragments may be considered as supramolecular synthons. Arrangement of molecules in the crystal is determined by the character and mutual orientation of synthons. Recognition of supramolecular synthons allows finding some fragments of the crystal containing strongly bonded molecules. Thus, the structure of the molecular crystal may be presented as packing of such fragments.

Further development of supramolecular synthons concept requires more reliable approach for the recognition of synthons. Usually, supramolecular synthons are based on specific intermolecular interactions in the crystals—for example, hydrogen or halogen bonds, and stacking interactions. Existence of such interactions is determined from an analysis of interatomic distances which should be shorter than van der Waals radii sum. However, if such interactions are absent the recognition of supramolecular synthons becomes quite ambiguous task. For example, any specific intermolecular interactions are absent in the crystals of unsubstituted hydrocarbons. Therefore, it is very difficult and almost impossible to recognize preferred packing motif in these crystals especially in the case of more or less isometric molecules.

It is quite clear that the packing of the molecules in the crystal is governed by energies, and directionality of intermolecular interactions and interatomic distances represent only some reflection of this energetics (see, for example, Ref. [5]). However, the analysis of relationship between the crystal structure and energy of intermolecular interactions requires some tools. Recently [6–8] a simple approach allowing easy recognition of preferred packing pattern of molecular crystals based on quantum chemical calculations of pairwise interaction energies between molecules was suggested. It includes the determination of the most strongly bonded dimers formed by the basic molecule located in asymmetric part of unit cell and the recognition of infinite fragment of crystal containing these dimers. Such fragment may be considered as a basic structural motif (BSM) of the crystal because total energies of interaction of the basic molecule with the molecules belonging to the same BSM and to neighbouring BSM are significantly different. Thus, any molecular crystal may be considered as packing of basic structural motifs where molecules are considerably strongly bonded to each other within BSM compared to interaction between molecules belonging to different BSMs.

Thus, the calculation of intermolecular interaction energies is a key step in the analysis of supramolecular architecture of the molecular crystals. However, it is well known (see, for example, references. [9, 10]) that values of these energies strongly depend on level of theory applied. Therefore, it is important to consider an applicability of different theoretical methods for the analysis of the crystal structure. It is well known that the most reliable data about energy of intermolecular interactions are provided by coupled clusters methods with inclusion of triplets (CCSD(T) and higher levels). However, these methods of calculations are very resource-consuming and may be applied only to relatively small molecules. From this viewpoint the benzene–acetylene co-crystal represents very suitable structure for the investigation because it is formed by rather small molecules [11]. This makes possible an application of high level methods for the calculations of the energy of interactions between the molecules which may be used as reference data for other methods.

In present paper we consider the topology of intermolecular interactions in the benzene–acetylene co-crystal based on results of quantum chemical calculations by different methods and analysis of electron density distribution using Bader’s ‘Atoms in Molecules’ theory [12]. According to all methods the hydrogen-bonded benzene–acetylene chains represent the basic structural motif of the crystal. It is also demonstrated that the energetic approach to the analysis of the crystal structure almost does not depend on theoretical method applied because it is based on ratio between interaction energies instead of differences. This allows using computationally cheap DFT-D methods for the investigation of supramolecular architecture of molecular crystals.

Methods of calculations

The crystal structure of the benzene–acetylene co-crystal was taken from Cambridge Crystal Structure Database [13] (Release 2010). Before calculations of intermolecular interactions energies the C–H bond lengths were fixed to 1.082 Å for benzene and 1.060 Å for acetylene molecule without changing the corresponding bond and torsion angles.

In order to analyse characteristics of intermolecular interactions all neighbouring molecules belonging to the first coordination sphere of each of the basic molecule located in asymmetric part of unit cell have been determined. This was performed on the basis of calculations of the Dirichlet domains and coordination polyhedron of the basic molecule in the crystal. The Dirichlet polyhedron of molecule was constructed as sum of the Dirichlet polyhedrons of all atoms of this molecule. Two molecules in a crystal are considered as neighbouring if their polyhedrons have at least one common facet. Set of the common facets of the Dirichlet polyhedrons of two neighbouring molecules forms the boundary surface between molecules. According to Fischer and Koch [14] the coordination number of a molecule in the crystal is determined as a number of the boundary surfaces with each surface area exceeding 2 % of a total area of the Dirichlet polyhedron of a molecule. Calculations of the Dirichlet polyhedron and their boundary surfaces were performed by Panov’s method [15]. The Dirichlet polyhedrons were constructed for benzene and acetylene molecules separately.

Energy of the interaction between molecules in each dimer was obtained by single-point calculations using Density Functional Theory [16], second-order Moller–Plesset perturbation theory [17], Local Pair Natural Orbital/Coupled Electron Pair Approximation (LPNO-CEPA/1) [18, 19] and CCSD(T) [20] methods with different basis sets as difference between the energy of a dimer and the energy of monomers and it has been corrected for basis set superposition error using Boys–Bernardi counterpoise procedure [21]. Data for complete basis set were obtained by an extrapolation of results of calculations for double and triple zeta basis sets according to Truhlar procedure [22]. The D3 empirical correction for dispersion interactions [23] was used with DFT methods (DFT-D methods). Total energy of interactions of the basic molecule with its environment in the crystal was determined as sum of pairwise interactions with each molecule of the first coordination sphere as it was suggested recently [24]. All calculations were performed using ORCA 2.8.0 programme [25].

The analysis of the topology of intermolecular interactions in the crystal is based on vector properties of intermolecular interactions as it was described earlier [8]. According to this approach intermolecular interaction between two molecules in the crystal may be described by vector originated in geometrical centre of one molecule and directed towards geometrical centre of second molecule. Length of this vector is calculated using the following equation:

$$ [{L_{\text{i}}} = \left( {{R_{\text{i}}}}{{E_{\text{i}}}} \right) / 2{E_{\text{str}}}] $$

where R i is a distance between geometrical centres of interacting molecules, E i is the energy of interaction between these two molecules and E str is the energy of the strongest pairwise interaction in the crystal.

Application of this approach makes it possible to construct the energy-vector diagram or ‘hedgehog’ of intermolecular interactions reflecting spatial distribution of intermolecular interactions of the basic molecule with the molecules belonging to its first coordination sphere. This diagram or ‘Hedgehog’ represents image of the molecule in terms of intermolecular interactions in the crystal and it may be multiplied by all symmetry operations of the crystal structure giving general picture of the topology of intermolecular interactions in the crystal. In the case of two molecules in asymmetric part of unit cell the hedgehogs of intermolecular interactions were constructed for each molecule separately.

Distribution of the electron density was investigated using cluster of molecules modelling the structure of the benzene–acetylene co-crystal. Wavefunction was obtained at the M06-2X/6-311G(d,p) and B3LYP/6-311G(d,p) levels of theory for the fragment of the crystal containing all molecules belonging to the first coordination sphere of benzene and acetylene. In total this cluster contains 10 benzene and six acetylene molecules. Geometry of cluster was fixed as it is observed in the crystal. Energy of interaction between molecules was estimated as energy of contact according to Espinosa et al. [26]. Properties of electron density were studied within R. Bader’s ‘Atoms in Molecules’ theory [12] using AIM2000 [27] programme.

Results and discussion

According to results of X-ray diffraction study [11] the crystal structure of benzene–acetylene co-crystal consists infinite benzene–acetylene chains formed by the C–H…π hydrogen bonds with distance between the hydrogen atom of the acetylene and the centre of the benzene ring being 2.47 Å (Fig. 1). Crystal structure is considered as packing of such chains which are shifted relative to each other. Such arrangement of molecules is the most favourable for the benzene–acetylene clusters [28].

Fig. 1
figure 1

Structure of the hydrogen-bonded chain and packing of chains in the benzene–acetylene co-crystal using molecules and Hedgehogs of intermolecular interactions. Geometrical centres of benzene and acetylene molecules are denoted as XB and Xa

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Results of calculations demonstrated that the first coordination sphere of basic benzene molecule contains six neighbouring benzene molecules and eight acetylene molecules. Environment of basic acetylene molecule contains only benzene molecules (eight benzene molecules per acetylene). More detailed analysis of the structure of dimers formed by the basic molecules located in asymmetric part of unit cell indicates that the benzene–benzene and benzene–acetylene dimers are observed in the first coordination sphere of the benzene molecule and only the benzene–acetylene dimers are found in the first coordination sphere of the acetylene molecule. Moreover, it was found that many of these dimers have the same geometry and they differ only in location around basic molecule. There are only three different dimers D1–D3 (Fig. 2).

Fig. 2
figure 2

Structure of different dimers formed by the basic molecules in the benzene–acetylene co-crystals

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Recently suggested [8] approach to analysis of crystal structure includes three steps: (i) determination of two of the most strongly bonded dimers with the highest values of intermolecular interaction energy; (ii) recognition of infinite fragment of the crystal containing such dimers which should be considered as basic structural motif (BSM) of the crystal; and (iii) determination of energy of interaction of the basic molecule with the molecules of its first coordination sphere belonging to the same BSM and to neighbouring BSMs. Following this procedure it is possible to find infinite fragment of crystal containing strongly bonded molecules with significantly weaker interactions between molecules of different such fragments.

Results of calculations by CCSD(T)/CBS method demonstrated (Table 1) that the most strongly bonded dimer D2 is stabilized by the C–H…π hydrogen bond. The smallest interaction energy is observed for dimer D3 with lateral location of the acetylene with respect to the benzene molecule. Three D2 dimers determine infinite chain which should be considered as a basic structural motif of the crystal in agreement with previous geometric consideration. Energy of interactions of each molecule with two neighbours within the chain is −8.34 kcal/mol. Each chain is surrounded by six neighbouring chains. Energy of interaction between the basic molecules and the molecules belonging to neighbouring chain is −3.79 kcal/mol, i.e. 2.2 times smaller. Therefore, hydrogen-bonded chain also represents a basic structural motif of the benzene–acetylene co-crystal from viewpoint of intermolecular interaction energies.

Table 1 Energy of intermolecular interactions (kcal/mol) in dimers D1–D3, total energy of interactions of basic benzene and acetylene molecules with neighbouring molecules within hydrogen-bonded chain (Echain, kcal/mol), to the molecules belonging to neighbouring chain (Eneighb, kcal/mol) and ratio between these energies (ER = Echain/Eneighb) calculated by different methods
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The same conclusion may be made easily from the analysis of packing of energy-vector diagrams (hedgehogs of intermolecular interactions). Straight line (Fig. 1) indicates direction of the strongest interactions between the acetylene and benzene molecules forming infinite chains of hedgehogs. Breaks between vectors of interactions of the molecules belonging to neighbouring chains reflect weaker interactions in agreement with energetic analysis mentioned above. Thus, packing of energy-vector diagrams represents a very powerful tool for visualization of the topology of intermolecular interactions and recognition of preferred pattern of the crystal packing based on energies of interaction between molecules.

Results of calculations confirm significant dependence of values of energy of each individual intermolecular interaction on method of calculation (Table 1). However, relative strength of interactions remains the same for all methods. The most strongly bonded dimer is the D2 and the most weakly bonded is the D3. Thus, all methods allow correctly recognizing the dimer with the highest intermolecular interaction energy providing background for construction of a basic structural motif of the crystal.

The same conclusion should be made about ratio of total interaction energies within the same BSM and to neighbouring BSM. According to all methods of calculations this ratio is not less than 1.9 (see ER value in Table 1). The best estimation using CCSD(T)/CBS method gives the value 2.2. It should be noted that the ER magnitude is more sensitive to size of basis set than to level of theory applied. Considerable increase of this ratio is observed in the case of application of relatively small double zeta basis sets for all methods (Table 1).

It is very interesting that computationally cheap density functional methods with empirical dispersion correction provide the same results as very expensive coupled clusters theory (Table 1). Recently [29] similar conclusion was made for values of lattice energies of set of crystal structures. In contrast to DFT-D methods, meta-GGA functional M06-2X considerably underestimates energy of weak interactions in dimers D1 and D3 leading to significantly higher ER value. Therefore, application of this functional may lead to some errors in determination of a basic structural motif due to overestimation of contribution of the strongest interactions into total energy of intermolecular interactions in the crystal.

During the last decade it was demonstrated that information about energy of intermolecular interaction may be also extracted from the analysis of electron density distribution using R. Bader’s ‘Atoms in Molecules’ (AIM) theory (see, for example, references [30–33]). This is caused by a fact that properties of intermolecular bond critical points contain information about intermolecular interactions rather than just interatomic ones. Therefore, it seems interesting to see how the AIM theory may be used for the analysis of supramolecular architecture of molecular crystals. In order to do this cluster of molecules including the basic molecules and the molecules belonging to their first coordination sphere with atomic coordinates taken from the crystal data was constructed. In total this cluster includes 10 benzene and six acetylene molecules.

Results of calculations demonstrate that the basic dimer is connected to all neighbouring molecules of its first coordination sphere by bond paths containing bond critical points (BCP). Characteristics of BCPs for equivalent dimers are the same. According to values of electron density and Laplacian of electron density, interaction in the D2 dimer is the strongest and the D3 dimer has the weakest binding (Table 2). This agrees well with results of quantum chemical calculations as well as with energies of contacts derived from values of potential energy density at BCP [26]. It should be also noted that ratio of the total interaction energies within the hydrogen-bonded chain and to neighbouring chain is 2.0 in good agreement with quantum chemical data. Thus, it is possible to suppose that characteristics of electron density distribution also can be used for the recognition of a basic structural motif of molecular crystals. It is interesting that this conclusion is also correct for electron density obtained by B3LYP/6-311G(d,p) method despite the known inability of this functional to describe dispersion interactions.

Table 2 Values of electron density ρ, Laplacian of electron density L for bond critical points located between the molecules forming dimers D1–D3 and the energy of contact E cont derived from characteristics of intermolecular BCP for the benzene–acetylene co-crystal
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Conclusions

Crystal structure of the benzene–acetylene co-crystal has been analysed in terms of energies of intermolecular interactions between the basic molecules located in asymmetric part of unit cell and the molecules belonging to their first coordination sphere. This approach includes three main steps namely (i) determination of the most strongly bonded dimers with two the highest interaction energy; (ii) construction of infinite fragment of crystal containing these dimers as a BSM of crystal packing, and (iii) determination of energy of interaction of the basic molecule with the molecules of its first coordination sphere belonging to the same BSM and to neighbouring BSMs. Results of calculations by CCSD(T)/CBS method demonstrate that the BSM of the crystal is represented by the C–H…π hydrogen-bonded chains. Energy of interaction within such chain is 2.2 times higher than the energy of interaction with the molecules belonging to neighbouring chain.

Comparison of results obtained using MP2, DFT, CCSD(T) and LPNO-CEPA/1 methods with different basis sets indicates that values of interaction energy for individual dimers strongly depend on method of calculations. However, the strongest dimer is the same at all levels of theory, and the ratio between energies of interaction of the basic benzene and acetylene molecules with their environment within the same BSM and to molecules belonging to neighbouring BSM mainly is within 1.9–2.3. Only in the case of application of double zeta basis sets and M06-2X functional this value is considerably higher (up to 3.5). This indicates that such approach to the analysis of the crystal structure almost does not depend on method of calculation of intermolecular interaction energy.

Similar results were obtained from analysis of electron density distribution using AIM theory. Results of calculations for cluster of molecules including first coordination sphere of basic benzene and acetylene molecules show that bond critical points and bond paths are observed for all dimers under consideration. Ratio between energies of interaction within BSM and between neighbouring BSMs is 2.0 in good agreement with quantum chemical data. Therefore, AIM-based analysis can also be used for recognition of basic structural motif in molecular crystals.