Chalcogen Noncovalent Interactions between Diazines and Sulfur Oxides in Supramolecular Circular Chains

The noncovalent chalcogen interaction between SO2/SO3 and diazines was studied through a dispersion-corrected DFT Kohn–Sham molecular orbital together with quantitative energy decomposition analyses. For this, supramolecular circular chains of up to 12 molecules were built with the aim of checking the capability of diazine molecules to detect SO2/SO3 compounds within the atmosphere. Trends in the interaction energies with the increasing number of molecules are mainly determined by the Pauli steric repulsion involved in these σ-hole/π-hole interactions. But more importantly, despite the assumed electrostatic nature of the involved interactions, the covalent component also plays a determinant role in its strength in the involved chalcogen bonds. Noticeably, π-hole interactions are supported by the charge transfer from diazines to SO2/SO3 molecules. Interaction energies in these supramolecular complexes are not only determined by the S···N bond lengths but attractive electrostatic and orbital interactions also determine the trends. These results should allow us to establish the fundamental characteristics of chalcogen bonding based on its strength and nature, which is of relevance for the capture of sulfur oxides.


Introduction
Noncovalent interactions play a crucial role in various scientific fields such as chemistry, biology, and material science [1,2].While hydrogen bonds are well-known [3], there are other significant noncovalent interactions including halogen [4][5][6][7][8], chalcogen [9,10], pnictogen [11,12], and tetrel bonds [13,14].These interactions are highly directional and form between the electron-deficient region of a covalently bonded atom from groups IV-VII of the periodic table and a negative site, such as an anion or Lewis base.These interactions are collectively known as σ-hole interactions because of the uneven distribution of atomic charge on the participating atom, resulting in an electron-deficient region termed the σ-hole [15,16].The nature of σ-hole interactions has been assigned to be primarily electrostatic, and they involve donor-acceptor orbital interactions [17,18].Noticeably, very recently, de Azevedo Santos et al. proved that HOMO-LUMO orbital interactions, i.e., covalence, provide a determinant contribution to the bond energies of these bonds, besides the electrostatic attraction [19].The significance of σ-hole bonding is evident in various applications such as molecular recognition [20,21], biological activity [22,23], crystal engineering [24,25], and the development of transportation and sensing technologies [26,27].These interactions contribute to the understanding and design of complex molecular systems and materials, highlighting their importance in advancing scientific and technological progress [28].
strength; (2) analyze the interaction energies with the aim of comparing the stability and strength of chalcogen bonds in different diazine-sulfur oxide complexes; and (3) analyze the MEP isosurfaces to identify the positive regions (π-holes) on sulfur atoms in SO 2 and SO 3 and their interaction sites on diazine molecules.This study is based on a dispersioncorrected DFT Kohn-Sham molecular orbital analysis together with quantitative energy decomposition analysis.
Int. J. Mol.Sci.2024, 25, x FOR PEER REVIEW 3 of 23 eliminating sulfur oxides through chalcogen bonding interactions.For this, our three main objectives are the following: (1) characterize the N•••S chalcogen bonds, formed via π-holes between diazine molecules and sulfur oxides, by investigating their nature and strength; (2) analyze the interaction energies with the aim of comparing the stability and strength of chalcogen bonds in different diazine-sulfur oxide complexes; and (3) analyze the MEP isosurfaces to identify the positive regions (π-holes) on sulfur atoms in SO2 and SO3 and their interaction sites on diazine molecules.This study is based on a dispersion-corrected DFT Kohn-Sham molecular orbital analysis together with quantitative energy decomposition analysis.

Interaction between Diazines and SO2
We first discuss the supramolecular assemblies between the three diazines and SO2.In the case of pyridazine, all circular molecular assemblies (n)pyridazine/SO2 with n = 2-6 adopt a quasi-planar conformation with Cnv symmetry (Figure 2).Importantly, sulfur atoms are in the same plane as the pyridazine rings, with a distance that increases from 2.718 to 2.931 Å from n = 3 to 6, respectively.The longer distance in the case of 2pyridazine/SO2 than for either n = 3 or 4 is due to its more strained structure.For the same reason, in the case of either pyrimidine or pyrazine, the species with n = 2 is not stabilized because of the too-close proximity and thus repulsion between the two diazine rings.The structures formed with pyrimidine also adopt Cnv symmetry, but the rings are more bent, escaping from the quasi-planar conformation (Figure 3).Opposite to pyridazine, the S•••N bond length decreases from 2.829 to 2.750 Å from n = 3 to 6, respectively.The same trend is also given by the third group with pyrazine, with the S•••N bond length decreasing from 2.814 to 2.720 Å from n = 3 to 6, respectively (Figure 4).The structures formed with pyrazine present Dnh symmetry, with a complete perpendicular conformation of the rings with respect to the formed circular geometry with SO2.

Interaction between Diazines and SO 2
We first discuss the supramolecular assemblies between the three diazines and SO 2 .In the case of pyridazine, all circular molecular assemblies (n)pyridazine/SO 2 with n = 2-6 adopt a quasi-planar conformation with C nv symmetry (Figure 2).Importantly, sulfur atoms are in the same plane as the pyridazine rings, with a distance that increases from 2.718 to 2.931 Å from n = 3 to 6, respectively.The longer distance in the case of 2pyridazine/SO 2 than for either n = 3 or 4 is due to its more strained structure.For the same reason, in the case of either pyrimidine or pyrazine, the species with n = 2 is not stabilized because of the too-close proximity and thus repulsion between the two diazine rings.The structures formed with pyrimidine also adopt C nv symmetry, but the rings are more bent, escaping from the quasi-planar conformation (Figure 3).Opposite to pyridazine, the S•••N bond length decreases from 2.829 to 2.750 Å from n = 3 to 6, respectively.The same trend is also given by the third group with pyrazine, with the S•••N bond length decreasing from 2.814 to 2.720 Å from n = 3 to 6, respectively (Figure 4).The structures formed with pyrazine present D nh symmetry, with a complete perpendicular conformation of the rings with respect to the formed circular geometry with SO 2 .
Having discussed the geometries, we now focus on the bonding energies of this series of supramolecular species.The bonding energies increase with the size of the species (Table 1).For each value of n, ∆E decrease from pyridazine to pyrimidine to pyrazine.For instance, in the case of n = 3, the value decreases from −46.1 to −37.4 to −33.4 kcal mol −1 , respectively.Based on these values, we could assign the stronger interaction for 3pyridazine/SO 2 to its shorter distance.However, in the case of n = 5, ∆E also decreases from −72.0 to −71.5 to −63.8 kcal mol −1 from pyridazine to pyrimidine to pyrazine, respectively, despite the S•••N bond length of 5pyridazine/SO 2 being longer.System 6pyridazine/SO 2 is the only exception.In addition, the trends are not determined by the strain energy of the fragments (∆E strain ), which also increases with larger n, but with a maximum value of 3.9 kcal mol −1 (Table 1).Thus, the distance between SO 2 and the diazine rings is not the only factor responsible for the computed bonding energies, and further insight into the interaction energies is needed.But first, to make an easier comparison, the trends in the bonding energies with n can be better seen if the ∆E values are divided by the number of interacting species in each case (2n, i.e., 6 for n = 3, 8 for n = 4, 10 for n = 5 and 12 for n = 6).Also in this case, the bonding energy per unit decreases from pyridazine to pyrimidine to pyrazine (Table 2).However, whereas those for pyridazine decrease from n = 3 to 6, the opposite is observed for both pyrimidine and pyrazine.These latter trends agree with the shortening of the S•••N bond length in the case of pyridazine and the lengthening in the case of the two other diazines.However, this S•••N bond length is not responsible for the difference between the different diazines, as stated above.With the aim of understanding the interaction in this series of systems, we performed a Kohn-Sham molecular orbital analysis together with a quantitative energy decomposition analysis (EDA) [50][51][52][53][54]. First, the interaction under analysis can be described as a covalent interaction, as the orbital interactions are more than half the magnitude of the electrostatic interactions [19].In particular, if we focus on the attractive interactions (∆V elstat + ∆E oi + ∆E disp ), the electrostatic interactions are in the order of 49-52%, whereas the orbital interactions are in the order of 30-34%.Next, in general, larger interaction energies go with larger repulsive Pauli and attractive electrostatic and orbital interactions that compensate for the latter and make the interaction attractive (Table 3).This is perfectly observed for n = 3 and 4 with pyridazine, which presents larger ∆E Pauli than pyrimidine or pyrazine.However, from n = 5, the shorter S•••N bond lengths of these two diazines cause an increase in repulsive ∆E Pauli that, together with more attractive ∆V elstat and ∆E oi terms, makes their interaction become stronger than with pyridazine.It can also be observed that the differences in the EDA terms between pyrimidine and pyrazine are small, and for this, further analysis is needed.
We can gain insight into the EDA analysis by focusing on the interaction between one molecule of diazine and SO 2 and on the same geometry as the whole circular system [55,56].The corresponding EDA values of these diazine•••SO 2 show the effect of either elongating the S•••N bond length in the case of pyridazine or shortening it in the case of both pyrimidine and pyrazine (Table 4).The systems with pyridazine show a stronger interaction in all cases except for 6pyrimidine.The main determinant term is the Pauli repulsion, followed by the electrostatic interaction and then the orbital interaction.Dispersion interaction remains constant for all systems.Again, with respect to the comparison of the interaction between pyrimidine and pyrazine, small differences in favor of the former arise from the combination of slightly more attractive electrostatic and orbital interactions.For completeness, the above discussion can be complemented by the EDA analysis with the same methodology, i.e., analyzing the terms divided by the number of molecules for each interacting system (Table S1), showing the same trends.Having discussed the role of the distance between the diazine rings and SO 2 , i.e., the S•••N bond length, and its effect on the interactions, we now focus on the electrostatic interaction that has also been observed to play a determinant role in the trends.For this, we computed the VDD charges and depicted the molecular electrostatic potential (MEP) isosurfaces of the involved systems (Figure 5).The chalcogen bond formed between diazines and SO 2 can be attributed to the interaction between the positive MEP region (i.e., the π-hole) on sulfur, depicted in blue, and the negative MEP region of the nitrogens, depicted in red, in diazines (Figure 5).Pyrimidine presents the most negatively charged nitrogen atoms (−0.193 au) among the diazines, which cause a more favorable interaction with the positively charged S atom of SO 2 (+0.478).This is the reason why pyrimidine systems present a more attractive interaction with SO 2 than those with pyrazine, as just discussed above.In addition, from the MEP isosurfaces of the whole systems (Figure S1), we can observe the charge transfer from the diazines to the SO 2 molecules, specifically from the lone pair of the nitrogen to the π-hole of SO 2 .Consequently, this results in a decrease in the negative charge on the nitrogen atoms and a simultaneous increase in the charge on SO 2 .For instance, in the case of 3pyridazine, the N charge is decreased from −0.115 to −0.058 e, for 3pyrimidine, from −0.193 to −0.157 e, and for 3pyrazine, from −0.166 to −0.134 e (Table 5).This charge transfer is reduced with larger systems (the N charge is −0.084, −0.143, and −0.114 e for 6pyridazine, 6pyrimidine, and 6pyrazine, respectively).In all cases, pyrimidine gives the most attractive electrostatic interaction.However, a direct comparison among the systems must also consider the S•••N bond length, as stated above.
with the positively charged S atom of SO2 (+0.478).This is the reason why pyrimidine systems present a more attractive interaction with SO2 than those with pyrazine, as just discussed above.In addition, from the MEP isosurfaces of the whole systems (Figure S1), we can observe the charge transfer from the diazines to the SO2 molecules, specifically from the lone pair of the nitrogen to the π-hole of SO2.Consequently, this results in a decrease in the negative charge on the nitrogen atoms and a simultaneous increase in the charge on SO2.For instance, in the case of 3pyridazine, the N charge is decreased from −0.115 to −0.058 e, for 3pyrimidine, from −0.193 to −0.157 e, and for 3pyrazine, from −0.166 to −0.134 e (Table 5).This charge transfer is reduced with larger systems (the N charge is −0.084, −0.143, and −0.114 e for 6pyridazine, 6pyrimidine, and 6pyrazine, respectively).In all cases, pyrimidine gives the most attractive electrostatic interaction.However, a direct comparison among the systems must also consider the S•••N bond length, as stated above.The other attractive interaction that plays a role is the orbital interaction, as discussed above.The main interaction is found between the HOMO of the diazine ring and the LUMO of the SO 2 molecule (Figure 6).The former mainly involves the sigma lone pairs of the N atoms of the rings (pyr HOMO ), whereas the latter mainly involves the π lone pairs of S and O atoms (SO 2 LUMO ), both being antibonding.Thus, these supramolecular circular systems under analysis are mainly determined by a donor-acceptor interaction between the diazine ring and the SO 2 molecule.The overlaps between these two orbitals (<pyr HOMO |SO 2 LUMO >, Table 6) clearly correlate with the above-discussed ∆E oi terms [57].In particular, while ∆E oi decreases for (n)pyridazine systems and increases for (n)pyrimidine and (n)pyrazine with increasing n, the same trend is given by the corresponding overlaps.In addition, to complete the comparison between the close values between the pyrimidine and pyrazine systems, the slightly more favorable ∆E oi for the former is also supported by its larger overlap values and, at the same time, larger charge transfer from the HOMO to the LUMO (Figure 7).It is also important to notice that in all cases, pyrimidine gives a larger <pyr HOMO | SO 2 LUMO > than either pyridazine or pyrazine.So, this bent geometry that these circular molecular assemblies adopt with pyrimidine gives rise to the best donor-acceptor interaction between the ring and the SO 2 molecule.Computed noncovalent interaction plots (NCI) help to further understand the stronger interaction in the case of pyridazine and pyrimidine than pyrazine when interacting with SO2.In particular, noncovalent interactions can be revealed from the electron density as they are highly nonlocal and manifest in real space as low-gradient isosurfaces with low densities.For such, we use the sign of the second Hessian eigenvalue to determine the kind of interaction, and its strength can be derived from the density on the noncovalent interaction surface [58,59].In our case, it can be observed that in addition to the  Computed noncovalent interaction plots (NCI) help to further understand the stronger interaction in the case of pyridazine and pyrimidine than pyrazine when interacting with SO 2 .In particular, noncovalent interactions can be revealed from the electron density as they are highly nonlocal and manifest in real space as low-gradient isosurfaces with low densities.For such, we use the sign of the second Hessian eigenvalue to determine the kind of interaction, and its strength can be derived from the density on the noncovalent interaction surface [58,59].In our case, it can be observed that in addition to the N•••S interaction for the three pyrazines, in the case of both pyridazine and pyrimidine, there is a weak interaction between one oxygen of SO 2 and one H of the diazine (Figure 8).This interaction is not present in the case of pyrazine because of its perpendicular geometry.Computed noncovalent interaction plots (NCI) help to further understand the stronger interaction in the case of pyridazine and pyrimidine than pyrazine when interacting with SO2.In particular, noncovalent interactions can be revealed from the electron density as they are highly nonlocal and manifest in real space as low-gradient isosurfaces with low densities.For such, we use the sign of the second Hessian eigenvalue to determine the kind of interaction, and its strength can be derived from the density on the noncovalent interaction surface [58,59].In our case, it can be observed that in addition to the N•••S interaction for the three pyrazines, in the case of both pyridazine and pyrimidine, there is a weak interaction between one oxygen of SO2 and one H of the diazine (Figure 8).This interaction is not present in the case of pyrazine because of its perpendicular geometry.
Overall, the total bonding energies of the supramolecular assemblies decrease in the order of pyridazine, pyrimidine, and pyrazine.Although the N•••S bond lengths partially explain these trends, a quantitative energy decomposition analysis (EDA) was necessary to understand the strength of these chalcogen interactions fully.The trends are primarily determined by the attractive electrostatic and orbital interactions, with the strongest interactions observed in pyridazine and weaker interactions in both pyrimidine and pyrazine as the size of the complex increases.In the case of pyrimidine, the most favorable electrostatic interaction is attributed to its most negatively charged nitrogen.Additionally, the orbital interaction, driven by the donor-acceptor interaction between the HOMO of the diazine ring and the LUMO of the SO2 molecule, is also most favorable for pyrimidine because of the stronger overlap.

Interaction between Diazines and SO3
In this second section, we substitute SO2 with SO3 and analyze how the interaction with diazines changes.We do not have a symmetric interaction of SO3 with the two diazine rings, but it gets closer to one of them, thus approaching the formation of a covalent N-S bond.If we focus on the shorter S•••N bond length, with the increase in n, it becomes shorter in the case of pyridazine (Figure 9), whereas it becomes longer in the case of both pyrimidine (Figure 10) and pyrazine (Figure 11), which is a completely opposite trend Overall, the total bonding energies of the supramolecular assemblies decrease in the order of pyridazine, pyrimidine, and pyrazine.Although the N•••S bond lengths partially explain these trends, a quantitative energy decomposition analysis (EDA) was necessary to understand the strength of these chalcogen interactions fully.The trends are primarily determined by the attractive electrostatic and orbital interactions, with the strongest interactions observed in pyridazine and weaker interactions in both pyrimidine and pyrazine as the size of the complex increases.In the case of pyrimidine, the most favorable electrostatic interaction is attributed to its most negatively charged nitrogen.Additionally, the orbital interaction, driven by the donor-acceptor interaction between the HOMO of the diazine ring and the LUMO of the SO 2 molecule, is also most favorable for pyrimidine because of the stronger overlap.

Interaction between Diazines and SO 3
In this second section, we substitute SO 2 with SO 3 and analyze how the interaction with diazines changes.We do not have a symmetric interaction of SO 3 with the two diazine rings, but it gets closer to one of them, thus approaching the formation of a covalent N-S bond.If we focus on the shorter S•••N bond length, with the increase in n, it becomes shorter in the case of pyridazine (Figure 9), whereas it becomes longer in the case of both pyrimidine (Figure 10) and pyrazine (Figure 11), which is a completely opposite trend compared with SO 2 .The only exception is 3pyrimidine/SO 3 because of its more constrained conformation.The circular systems with pyridazine adopt a planar conformation with D nh symmetry, whereas those with either pyrimidine or pyrazine adopt an almost perpendicular conformation of the rings with C n symmetry.Importantly, for these systems, the sulfur of SO 3 only directly interacts with one N atom of the ring, as on the other side, we do not have an S•••N interaction any longer, and instead, the oxygen atoms are closer to the diazine ring, especially for the more bent pyrimidine and diazine systems.With respect to the bonding interaction between the diazines and SO 3 , we observe an important change in SO 2 because now, in all cases, pyrimidine interacts the strongest, with the only exception of 3pyrimidine/SO 3 mentioned above (Table 7).The other important change is the strain energy, which, in this case, is really large because of the deformation of SO 3 , which loses its planarity and becomes pyramidalized when it interacts with diazine rings by forming the N-S bond.Nonetheless, its magnitude is not determinant in the interaction energies of these systems, which will be further discussed below.With respect to the bonding interaction between the diazines and SO3, we observe an important change in SO2 because now, in all cases, pyrimidine interacts the strongest, with the only exception of 3pyrimidine/SO3 mentioned above (Table 7).The other important change is the strain energy, which, in this case, is really large because of the deformation of SO3, which loses its planarity and becomes pyramidalized when it interacts with dia-  These interaction energies were again analyzed by means of a Kohn-Sham molecular orbital analysis together with a quantitative EDA analysis (Table 8).The pyrimidine systems present the largest ∆E int despite Pauli repulsion also being the largest because of the shorter S•••N bond lengths compared with either circular systems with pyridazines or pyrazines.However, such large steric repulsion is compensated for by more attractive electrostatic and orbital interactions and even dispersion ones.Importantly, these systems with SO 3 involve a more covalent interaction than the above ones, with much closer values of electrostatic (47-49%) and orbital interactions (42-47%).However, in this case, as we do not have a symmetrical system with SO 3 equally interacting with the diazine on one side as the one on the other, we do not only have one diazine•••SO 3 interaction but two.Therefore, we also analyzed both interactions individually (Tables 9 and 10).The shorter S•••N bond implies a much stronger interaction because a covalent N-S bond is formed.For instance, in the case of 3pyridazine/SO 3 , ∆E int amounts to −27.2 and −6.9 kcal mol −1 for the shorter and longer S•••N interactions.The former has a stronger covalent character compared with the latter, which has a predominant electrostatic character.Importantly, there is a good correlation between ∆E int and the S•••N distances discussed above.For instance, for n = 6, the S•••N goes from 2.084 to 2.002 to 2.037 Å from pyridazine to pyrimidine to pyrazine, whereas ∆E int goes from −28.8 to −30.6 to −29.2 kcal mol −1 , respectively.Such trends are determined by both attractive ∆V elstat and ∆E oi terms, where those for (n)pyrimidine are the most favorable, despite the repulsive ∆E Pauli also being the largest.In contrast, in the case of the longer S•••N bond (Table 10), for all diazines, the interaction energy decreases with larger n, thus correlating with the opposite behavior given by the shorter bond.Nonetheless, the fact that their strengths are almost one order of magnitude smaller, their contribution to the overall interaction of the supramolecular cluster is less relevant.With respect to the electrostatic interaction, in the case of pyrimidine, the charge transfer from the ring to the SO 3 is larger because of its more favorable interaction, as mentioned above.Once again, pyrimidine is the diazine that shows the most negatively charged N atoms (Table 11), thus giving rise to a more attractive electrostatic interaction with the even more positively charged S atoms of SO 3 (+0.625e).The MEP isosurfaces (Figure S2) show the charge transfer from the diazines to the SO 3 molecules, with an important decrease in the negative charge on the N atoms, with pyrimidine showing the largest.Next, we move to the ∆E oi term, which is also responsible for the more attractive interaction.The most determinant orbital interaction is between the HOMO (pyr HOMO ) of the diazine and the LUMO of SO 3 (SO 3 LUMO , Figure 12).In contrast to SO 2 , the latter involves sigma S-O bonds.Thus, we observe a donor-acceptor interaction between the diazine ring and the SO 3 molecule.As mentioned above, the interaction orbital increases with larger n, and, in general, this is also the trend supported by the charge transfer from the diazines to the SO 3 as well as the corresponding overlap between pyr HOMO and SO 3 LUMO (Table 12).However, in this case, we must keep in mind that although this discussion is based on the shorter S•••N interaction, that of the longer one also plays a role, thus affecting the trends.Importantly, in line with the more attractive electrostatic interaction, pyrimidine also gives rise to the most attractive orbital interactions.Next, we move to the ∆Eoi term, which is also responsible for the more attractive interaction.The most determinant orbital interaction is between the HOMO (pyr HOMO ) of the diazine and the LUMO of SO3 (SO3 LUMO , Figure 12).In contrast to SO2, the latter involves sigma S-O bonds.Thus, we observe a donor-acceptor interaction between the diazine ring and the SO3 molecule.As mentioned above, the interaction orbital increases with larger n, and, in general, this is also the trend supported by the charge transfer from the diazines to the SO3 as well as the corresponding overlap between pyr HOMO and SO3 LUMO (Table 12).However, in this case, we must keep in mind that although this discussion is based on the shorter S•••N interaction, that of the longer one also plays a role, thus affecting the trends.Importantly, in line with the more attractive electrostatic interaction, pyrimidine also gives rise to the most attractive orbital interactions.Finally, and for completeness, NCI plots were also computed for the diazines interacting with SO 3 (Figure 13), which confirmed the covalent character of the shorter N-S bond formed between the diazines and the SO 3 .This N-S interaction is complemented by weaker H•••O noncovalent interactions for the three diazines.On the other hand, we also confirmed that we do not have a longer S•••N interaction, but for the three diazines, their oxygen atoms interact with the diazine rings, i.e., a lateral interaction.Finally, and for completeness, NCI plots were also computed for the diazines interacting with SO3 (Figure 13), which confirmed the covalent character of the shorter N-S bond formed between the diazines and the SO3.This N-S interaction is complemented by weaker H•••O noncovalent interactions for the three diazines.On the other hand, we also confirmed that we do not have a longer S•••N interaction, but for the three diazines, their oxygen atoms interact with the diazine rings, i.e., a lateral interaction.For completeness, nucleophilic and electrophilic Fukui functions were computed, together with the dual descriptor [60][61][62].This latter is defined as the difference between the Fukui functions for nucleophilic and electrophilic attacks.Thus, the dual descriptor gives a combination of both Fukui functions, positive for locations where a nucleophilic attack is more probable than an electrophilic one, and negative where the electrophilic attack is more probable.Figure 14 shows the dual descriptor plots for the diazines, further supporting the conclusions given above that were obtained from the EDA analysis.It can also be observed how the SO2/SO3 can perfectly interact with the diazine through the sulfur atom.This study can be further complemented by the electrophilic and nucleophilic Fukui functions (Figure S3).For completeness, nucleophilic and electrophilic Fukui functions were computed, together with the dual descriptor [60][61][62].This latter is defined as the difference between the Fukui functions for nucleophilic and electrophilic attacks.Thus, the dual descriptor gives a combination of both Fukui functions, positive for locations where a nucleophilic attack is more probable than an electrophilic one, and negative where the electrophilic attack is more probable.Figure 14 shows the dual descriptor plots for the diazines, further supporting the conclusions given above that were obtained from the EDA analysis.It can also be observed how the SO 2 /SO 3 can perfectly interact with the diazine through the

Computational Details
All calculations were carried out using the Amsterdam Density Functional (ADF) program [63].All stationary points and energies were calculated at the BLYP level of the generalized gradient approximation (GGA) using the exchange functional developed by Becke (B) and the GGA correlation functional developed by Lee, Yang, and Parr (LYP) [64,65].The DFT-D3(BJ) method developed by Grimme and coworkers [66,67], which contains the damping function proposed by Becke and Johnson [68], was used to describe non-local dispersion interactions.Scalar relativistic effects were accounted for using the zeroth-order regular approximation (ZORA) [69][70][71].This level is referred to as BLYP-D3(BJ)/TZ2P and has been proven to accurately describe weak interactions [72][73][74].A large uncontracted optimized TZ2P Slater-type orbitals (STOs) basis set containing diffuse functions were used.The TZ2P all-electron basis set [69], with no frozen-core approximation, is of triple-ζ quality for all atoms and has been augmented with two sets of polarization functions on each atom.The accuracies of the integration grid (Becke grid) and the fit scheme (Zlm fit) were set to VERYGOOD [75,76].
The total bonding energy ∆E of the circular supramolecular systems (n)diazine/SOx is defined as [Equation ( 1)]: where E(n)diazine/SO2 is the energy of the optimized circular system and Ediazine and ESO2 are the energies of the optimized diazine and SOx.∆E can be divided into two components by means of the activation strain model (ASM) [77][78][79] [Equation (2)]: where the strain energy ∆Estrain is the amount of energy required to deform the individual monomers from their equilibrium structure to the geometry that they acquire in the circular system.The interaction energy ∆Eint corresponds to the actual energy change when the prepared monomers are combined to form the whole system.∆Eint can be further analyzed through a quantitative energy decomposition analysis (EDA) [80,81], which decomposes ∆Eint into the classical electrostatic interaction (∆Velstat) among the unperturbed charge distributions of the deformed monomers, the Pauli

Computational Details
All calculations were carried out using the Amsterdam Density Functional (ADF) program [63].All stationary points and energies were calculated at the BLYP level of the generalized gradient approximation (GGA) using the exchange functional developed by Becke (B) and the GGA correlation functional developed by Lee, Yang, and Parr (LYP) [64,65].The DFT-D3(BJ) method developed by Grimme and coworkers [66,67], which contains the damping function proposed by Becke and Johnson [68], was used to describe non-local dispersion interactions.Scalar relativistic effects were accounted for using the zeroth-order regular approximation (ZORA) [69][70][71].This level is referred to as BLYP-D3(BJ)/TZ2P and has been proven to accurately describe weak interactions [72][73][74].A large uncontracted optimized TZ2P Slater-type orbitals (STOs) basis set containing diffuse functions were used.The TZ2P all-electron basis set [69], with no frozen-core approximation, is of triple-ζ quality for all atoms and has been augmented with two sets of polarization functions on each atom.The accuracies of the integration grid (Becke grid) and the fit scheme (Zlm fit) were set to VERYGOOD [75,76].
The total bonding energy ∆E of the circular supramolecular systems (n)diazine/SO x is defined as [Equation ( 1)]: where E (n)diazine/SO2 is the energy of the optimized circular system and E diazine and E SO2 are the energies of the optimized diazine and SO x .∆E can be divided into two components by means of the activation strain model (ASM) [77][78][79] [Equation (2)]: where the strain energy ∆E strain is the amount of energy required to deform the individual monomers from their equilibrium structure to the geometry that they acquire in the circular system.The interaction energy ∆E int corresponds to the actual energy change when the prepared monomers are combined to form the whole system.∆E int can be further analyzed through a quantitative energy decomposition analysis (EDA) [80,81], which decomposes ∆E int into the classical electrostatic interaction (∆V elstat ) among the unperturbed charge distributions of the deformed monomers, the Pauli repulsion among occupied orbitals (∆E Pauli ), the stabilizing orbital interactions term (∆E oi ) that accounts for charge transfer (i.e., donor-acceptor interactions between occupied orbitals on one moiety and unoccupied orbitals on the other, including the HOMO-LUMO interactions) and polarization (i.e., empty-occupied orbital mixing on one fragment due to the presence of another fragment), and the dispersion correction ∆E disp [Equation (3)]: The orbital interaction energy can be further decomposed into the contributions from each irreducible representation Γ of the interacting system [82].In our planar model systems, the C S symmetry partitioning allows us to distinguish between σ and π interactions [Equation ( 4)]: The electron density distribution is analyzed using the Voronoi deformation density (VDD) method [83] for computing atomic charges.The VDD atomic charge on atom A (Q A VDD ) is computed as the (numerical) integral of the deformation density in the volume of the Voronoi cell of atom A [Equation (5)].The Voronoi cell of atom A is defined as the compartment of space bounded by the bond midplanes on and perpendicular to all bond axes between nucleus A and its neighboring nuclei.
where ρ(r) is the electron density of the molecule and ∑ B ρ B (r) is the superposition of atomic densities ρ B of a fictitious promolecule without chemical interactions that is associated with the situation in which all atoms are neutral.The interpretation of the VDD charge, Q A VDD is rather straightforward and transparent: instead of measuring the amount of charge associated with a particular atom A, Q A VDD directly monitors how much charge flows, because of chemical interactions, out of (Q A VDD > 0) or into (Q A VDD < 0) the Voronoi cell of atom A.

Conclusions
Overall, in our study, we strategically employed organic molecules, specifically diazines, interacting with sulfur oxides via chalcogen bonding giving rise to supramolecular circular structures.This innovative approach is aimed at allowing a better comprehension of the capability of diazine molecules to detect SO 2 /SO 3 compounds within the atmosphere.With their two nitrogen atoms, diazines exhibit the unique potential to interact with two SO 2 /SO 3 molecules simultaneously, making them an ideal choice for our circular molecular assembly.The dispersion-corrected DFT Kohn-Sham molecular analysis together with a quantitative energy decomposition analysis highlights the significant importance of these noncovalent interactions.We believe that this newfound knowledge can play a pivotal role in advancing the development of novel materials designed to capture polluting gases, leveraging the unique properties of our molecules.
In particular, we characterized in detail the nature of the S•••N interactions in this series of diazines interacting with SO 2 /SO 3 molecules.For this purpose, we expanded our systems to chains consisting of six to twelve monomers to explore the impact of these larger molecular structures on chalcogen interactions.Both σ-hole and π-hole interactions are further supported by the computed molecular electrostatic potential isosurfaces depending on the interaction between either the lone pair of nitrogen or the ring of the diazines.
Noticeably, the trends in the interaction energies of these supramolecular systems when going from pyridazine to pyrimidine to pyrazine are partially determined by the S•••N bond lengths and further supported by the attractive electrostatic and orbital interactions.As a whole, the findings of this paper contribute significantly to a better understanding of chalcogen bonding interactions.

Figure 2 .
Figure 2. Geometries of complexes with SO2 with pyridazine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.Figure 2. Geometries of complexes with SO 2 with pyridazine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.

Figure 2 .
Figure 2. Geometries of complexes with SO2 with pyridazine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.Figure 2. Geometries of complexes with SO 2 with pyridazine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.

Figure 3 .
Figure 3. Geometries of complexes with SO2 with pyrimidine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.

Figure 4 .
Figure 4. Geometries of complexes with SO2 with pyrazine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.

Figure 4 .
Figure 4. Geometries of complexes with SO2 with pyrazine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.Figure 4. Geometries of complexes with SO 2 with pyrazine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.

Figure 4 .
Figure 4. Geometries of complexes with SO2 with pyrazine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.Figure 4. Geometries of complexes with SO 2 with pyrazine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.

Figure 7 .
Figure 7. HOMO of ring (bottom) and LUMO of SO2 (top) fragments in complexes with pyridazine, pyrimidine, and pyrazine with 3 units.Red and blue isosurfaces represent positive and negative phases for HOMO, whereas orange and turquoise represent these phases for LUMO.

Figure 7 .
Figure 7. HOMO of ring (bottom) and LUMO of SO2 (top) fragments in complexes with pyridazine, pyrimidine, and pyrazine with 3 units.Red and blue isosurfaces represent positive and negative phases for HOMO, whereas orange and turquoise represent these phases for LUMO.

Figure 7 .
Figure 7. HOMO of ring (bottom) and LUMO of SO 2 (top) fragments in complexes with pyridazine, pyrimidine, and pyrazine with 3 units.Red and blue isosurfaces represent positive and negative phases for HOMO, whereas orange and turquoise represent these phases for LUMO.

Figure 9 .
Figure 9. Geometries of complexes with SO 3 with pyridazine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.

Figure 10 .
Figure 10.Geometries of complexes with SO3 with pyrimidine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.Figure 10.Geometries of complexes with SO 3 with pyrimidine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.

Figure 11 .
Figure 11.Geometries of complexes with SO 3 with pyrazine.Bond lengths in Å. Atom colors for N: blue, C: grey, S: yellow, O: red, H: white.

Table 2 .
Total bonding energies (∆E, in kcal mol −1 ) together with its value divided by the number of molecules for each system (2n) for the (n)pyridazine/SO 2 complexes with pyridazine, pyrimidine and pyrazine a .∆E = ∆E int + ∆E strain .

Table 4 .
Energy decomposition analysis (in kcal mol −1 ) of the interaction between one molecule of pyridazine, pyrimidine, and pyrazine and SO 2 at the same geometry as the circular systems a .

Table 6 .
Energies of fragment molecular orbitals (in eV), their overlap, and their Gross Mulliken populations (in au) of complexes of SO 2 with pyridazine, pyrimidine, and pyrazine.

Table 9 .
Energy decomposition analysis (in kcal mol −1 ) of the interaction between one molecule of diazine and SO 3 (shorter S•••N distance) at the same geometry as the circular systems a .∆E int = ∆E Pauli + ∆V elstat + ∆E oi +∆E disp .

Table 10 .
Energy decomposition analysis (in kcal mol −1 ) of the interaction between one molecule of diazine and SO 3 (longer S•••N distance) at the same geometry as the circular systems a .Computed at BLYP-D3(BJ)/TZ2P level of theory.∆E int = ∆E Pauli + ∆V elstat + ∆E oi +∆E disp . a

Table 12 .
Energies of fragment molecular orbitals (in eV), their overlap, and their Gross Mulliken populations (in au) of complexes of SO 3 with pyridazine, pyrimidine, and pyrazine.