The structure, stability, thermochemistry, and bonding in SO3-(H2O)n (n = 1–7) clusters: a computational analysis

The structure, stability, and intermolecular interactions in SO3-(H2O)n (n = 1–7) clusters were investigated using density functional and wave functional methods. The putative global minimum shows the SO3 molecule tends to be on the surface water clusters. The increase in the number of water molecules chalcogen bond distance between water molecules and SO3 decreases, while the maximum number of water molecules coordinated to the SO3 molecule remains at three. The calculated solvation energy increases with the increase in the number of water molecules, and it does not saturate, which indicates that the addition of water molecules can add up to the existing water cluster network. The interaction energy between water molecules and SO3 was less than the solvation energy conforming to the cluster forming of water molecules. The Gibbs free energy and entropy values decrease with the increase in cluster size, signifying the amount of water molecule decide the sequential hydration process. Thermochemistry data at various temperatures show that low-temperature regions found in the upper part of the troposphere favor hydration formation. Molecular electrostatic potentials (MESP) show reduced Vs,max value of π-hole on sulfur atom and increased value on hydrogens of water molecules which results in the addition of water which leads to the sequential addition of water molecules to the water network. The quantum theory of atoms in molecules (QTAIM) shows the presence of S···O, O···H interactions between SO3 and water molecules. Between water molecules O···H, H-bonding interactions were observed, and in larger clusters, O···O interaction was also noticed. QTAIM analysis shows that the water–water HBs in these clusters are weak H-bond, while the SO3-water interaction can be classified as medium H-bonds which was further supported by the NCI and 2D RDG plots.


Introduction
Sulfur is one of the most abundant elements found in the universe and the earth's crust. Sulfur from land and sea enters the atmosphere through emissions and is converted to sulfur oxides [1]. These sulfur oxides particularly SO 2 and SO 3 are hazardous when emitted into the atmosphere and affect air quality, the environment, climate, and human health [2]. Once into the atmosphere, the SO gets oxidized in SO 2 and finally to SO 3 which acts as a source of H 2 SO 4 . The formed H 2 SO 4 plays an important role in aerosol formation and is also one of the primary constituents of acid rain. In the stratosphere, UV photolysis of SO 2 generates SO 3 which rapidly reacts with water and gets converted into H 2 SO 4 [3,4]. It has been shown experimentally that the hydrolysis of SO 3 is catalyzed by the presence of a third particle and is secondorder with respect to the partial pressure of water [5]. It is believed that the reaction proceeds via an intermediate complex SO 3 .H 2 O. Furthermore, the conversion of the aqueous complex to sulfuric acid is fast compared to the reversion reaction. To account for the negative temperature dependence of the above reaction, it is essential to estimate the enthalpy of activation of the above reaction. To accurately estimate the enthalpy of formation, Klopper et al. carried out a quantum chemical study of SO 3 .H 2 O complex and estimated the enthalpy at 298 and 0 K as −8.3 and −7.7 kcal/ mol respectively [6].
Previous experimental and theoretical studies have shown SO 3 to be a planar molecule with D 3h symmetry in the ground state [7]. Studies using matrix isolation techniques have shown the presence of cis-OSOO, whose presence was confirmed by IR absorption and UV/Vis spectroscopy measurements [8,9]. Under cryogenic conditions in N 2 atmosphere and Ne/Ar matrix, Wu et al. isolated OSOO molecule, which gets converted into SO 3 when irradiated [10]. Very recently, the electronic structure and bond dissociation of SO 3 was determined by Wang et al. using high-resolution cryogenic photoelectron imaging and high-level multireference multiconfigurational methodology [11]. Density functional theory (DFT) studies have predicted the monomer SO 3 to occur in five possible isomeric geometries [12]. Recently, wavefunction analysis shows the existence of charge holes in the sulfur atom which produces electrophilic regions around it that attracts electrophile and can form new types of weak interactions including the chalcogen bonds [13].
In the past, studies have shown that water can act as a catalyst during hydrolysis and significantly reduces the energy barrier [14]. Carmona-Garcia et al. have pointed out that the photolysis lifetime of SO 3 molecules in the stratosphere of the earth is roughly 579 days [15]. Hence, it is expected that the formation of sulfuric acid happens by the hydrolysis of SO 3 molecules by association with water molecules. To understand the role of other gas molecules that were present in the stratosphere during hydrolysis reaction, extensive theoretical and experimental studies were carried [16][17][18]. Charge transfer was found to stabilize the diad complex + H 3 N···SO 3 -as revealed by the gas phase pulsed nozzle Fourier transform microwave spectroscopic method [19]. The possible existence of 1:1:1 cyclic structured H 2 SO 4 -H 2 O-SO 3 complex was identified by FT-IR matrix isolation spectra and was found to be stabilized by hydrogen bondings (HBs) [20]. Scheiner et al. studied the noncovalent interactions between SO 3 and CO using ab initio method, and the interaction was a strong chalcogen bond [21]. The existence of a chalcogen bond was also observed in the interaction between SO 3 and formaldehyde [22]. Recently, Li et al. studied the interaction of SO 3 with pyridine and pyrazine and found the complex to be stabilized by covalent interactions [23]. Later, Chandra et al. studied the interaction in substituted pyridine and SO 3 and reported the nature and strength of π-hole chalcogen bonding [24]. Very recently, we have reported the nature of the interaction between SO 3 clusters and water molecules. Besides the HBs, other predominant interactions are S…O and O…O chalcogen bonds [25]. The association of water to SO 3 cluster shows a larger entropic penalty and entropy-driven.
This work aims to investigate the association of water molecules with sulfur trioxide. Herein, we report our density functional theory study on SO 3 -(H 2 O)n (n = 1-7). The structure, growth behavior, energetics, and thermochemistry in various stratospheric conditions and the nature of interaction that occurs between SO 3 molecule and water molecules are studied using DFT, quantitative molecular electrostatic potential (MESP), atoms in molecule (AIM) analysis, and noncovalent interaction analysis (NCI). We believe that the fundamental knowledge of intermolecular interactions and binding modes is important for revealing the initial process in the formation of aerosols.

Computational details
The initial geometries of SO 3 -(H 2 O)n (n = 1-7) are generated using the ABCluster code, which searches the global as well as the local minima of atomic and molecular clusters. For the theoretical basis of the ABCluster code, the reader is requested to refer to the work of Zhan and Dolg [26,27]. The coordinates obtained in the ABCluster codes are further optimized with the dispersion corrected semiempirical PM7 method [28]. The obtained structures in the semiempirical calculations were further optimized using the DFT method PW6B95-D3 and Def2TZVP basis set, without any symmetric and geometrical constraints. Previous benchmark studies on SO 2 dimer recommend that the PW6B95-D3/Def2TZVP method delivers the best agreement for binding energy with a mean absolute deviation of 0.3 kcal mol −1 [29]. Moreover, Malloum and Conradie observed the smallest mean absolute deviation for binding energies for pronated acetonitrile clusters when using the PW6B95-D3/Def2TZVP method [30]. The cluster size in the work has been limited to n ≤ 7, due to the computational demand. Vibrational frequency calculations were carried out on the least energy conformers to conform the reported minimum energy geometries in real minima on the potential energy surfaces and to arrive at the thermodynamical parameters.
All the reported DFT calculations were carried and performed using the Gaussian 16 suite of programs [31]. The PW6B95-D3/Def2TZVP method was used for the construction of electrostatic potential surfaces. The wave function analysis-surface analysis suite (WFA-SAS) program was used to calculate quantitative electrostatic potential at 0.001 electron/Bohr 2 isodensity and to visualize the 3D surface [32]. Atoms in molecule (AIM) analyses were carried out using the AIMALL program package [33]. NCI analyses were performed using MultiWFN software and visualized using the CHEMCRAFT program [34,35].

Structural of SO 3 -(H 2 O)n (n = 1-7) clusters
Sulfur trioxide has a D 3h symmetry with an S-O bond length of 1.414 Å [25]. In the semiempirical PM7 optimization, the SO 3 -H 2 O molecule exists in three conformers. When the above structures are subjected to density functional optimization with PW6B95-D3/Def2TZVP methods, we noticed that the structure shown in Fig. 1a is a putative global minimal (PGM) structure. In the putative global minima, the water molecule is bound to SO 3 through an intermolecular S···O chalcogen bonding with a bond length of 2.771 Å [25]. Upon the complex formation, a change in planarity in the SO 3 molecule was noticed. The S atom is ascended toward the oxygen atom of the water molecule. This indicates the existence of strong interaction between the water and SO 3 molecule.
To locate the lowest energy structure of SO 3 -(H 2 O)n (n = 2-7) clusters, we used the artificial bee colony (ABC) algorithm, which is an unbiased, population-based, swarmintelligence global optimization algorithm. For each cluster size, more than 50 particles in the swarm were created with a box size of 5 Å 3 [36]. As a result, about 300 structures were generated and subjected to semiempirical PM7 optimization. The obtained unique structures were further subjected to DFT optimization. Furthermore, to locate the PGM, we have either added or removed one water molecule at various positions of the first three minimum energy geometries obtained in the DFT optimization process. The obtained new structure is optimized at the DFT level without any geometrical constraints. The above intuitive generation is also essential to arrive at the correct PGM geometries [37,38].
The PGM for SO 3 -(H 2 O)n (n = 2-7 water molecules), calculated at PW6B95-D3/Def2TZVP level of theory, are shown in Fig. 1b-g. The other low-lying isomers for various SO 3 -water cluster sizes are provided in the supporting information Figs. S1, S2, S3, S4, and S5. The number of water molecules coordinated to SO 3 , intramolecular bond length, mean hydrogen bond length, and the number of mean intermolecular hydrogen bond lengths between water molecules is provided in Table 1. The PGM of heterotrimer SO 3 has one chalcogen with a bond distance of 2.062 Å and one hydrogen bonding with the water molecules, while there exists an intermolecular hydrogen bonding among water molecules. The structure with a second water molecule linked to the unoccupied side of SO 3 -H 2 O to create a "double sandwich" with two chalcogen bonds is 8.64 kcal/mol less stable than the PGM. In the PGM of heterotetramer, SO 3 molecules have one chalcogen and two hydrogen bonds. The intramolecular S … O distance got reduced to 1.092 Å, while the number of intermolecular hydrogen bonds between water molecules increases to two. The water molecules, which are attached to the SO 3 molecule, exist with two intermolecular hydrogen bonding with two adjacent molecules. The PGM of SO 3 -(H 2 O) 4 preserves the heterotetramer geometry, with the addition of a new water molecule existing with intermolecular hydrogen bonding, while the chalcogen bond distance got reduced to 1.828 Å, and the number of intermolecular hydrogen bonds increases to three.
In the heterohexamer and heteroheptamer, the number of water molecules coordinated to SO 3 molecules remains 3, preserving the penultimate PGM geometry, while the intermolecular chalcogen distance between SO 3 and water molecules reaches a near saturation of 1.779 Å. In the PGM heterooctamer, the number of intermolecular hydrogen bonds increases to 5, while the mean hydrogen bond length decreases to 1.887 Å. It is interesting to note that in all the clusters, the SO 3 molecule tends to be on the surface water clusters. Furthermore, the topology of SO 3 -(H 2 O)n+1 retains the geometry of SO 3 -(H 2 O)n. The additional water molecule gets added up to the penultimate topology in such a way that one of the hydrogen atoms of the water molecule gets attached to the oxygen atom of SO 3 and the other hydrogen atom making HBs with the existing water cluster. The number of water molecules coordinated to the SO 3 reaches the maximum of 3, while the number of intramolecular hydrogen bonds between water molecules reaches 7 in number. Intriguingly, the mean hydrogen bond length between the water molecules reaches a near saturation of 1.756 Å, which is a little shorter than the bond distance observed in the pristine water clusters [39]. The saturation in the number of intermolecular bonds between water and SO 3 and the gradual increase in the number of hydrogen bonds between water molecules indicates that the clustering of water molecules among them is more feasible than the formation of SO 3 -water cluster formation.

Solvent stabilization and interaction energy
The solvation stabilization energy induced by solvent water molecules for the formation of SO 3 -(H 2 O)n cluster can be expressed by the relation given by Eq. (1).
where E SO3−(H2O)n is the total energy of SO 3 -water complexes, E H2O is the total energy of water in a molecule, and E SO3 is the total energy of sulfur trioxide in a molecule. The calculated solvation energy and the interaction energy for the SO 3 -(H 2 O)n cluster n = 1-7) are provided in Table 2.
The E solv energy of the clusters increases with the increase in the number of water molecules. The plot of the variation of E solv vs the number of water molecules "n" is shown in Fig. S6. The E solv value varies linearly with the number of solvent water molecules with a correlation coefficient greater than 0.999. The slope of the best fitter equation suggests that the solvent stabilization energy per solvent water molecule is 13.80 kcal/mol. If one assumes that each water molecule's hydrogen atom makes hydrogen bonds, then the stabilization energy per hydrogen bond would be 6.90 kcal/ mol. It is interesting to note that the E solv does not saturate which indicates that the addition of water molecules can add up to the existing water cluster network. Besides, the existence of a linear correlation between the solvation energy vs the size of the clusters proves the presence of cooperativity in clusters [40][41][42]. The interaction energy (E int ) between SO 3 molecule and the water clusters in their putative global minimum energy are calculated by the following Eq. (2).
where E (H2O)n is the total energy of the water clusters in their putative global minimum energy state [38]. The E int increases with the water cluster size except for the heterotetramer. In the heterotetramer, only three water molecules are bonded to the SO 3 molecule, while in the pristine water cluster, four HBs exist between water molecules. Thus, the reduced number of HBs results in the lower stability of SO 3 -(H 2 O) 3 heterotetramer. In the larger clusters, the computed E int energies are lower than the solvation energies, because additional solvent water molecules interact with the water network of forming larger water clusters.
To understand the cooperativity in hetero clusters, we have computed the pairwise energies for the heterotrimer to heterooctamer clusters [42,43]. The pairwise interaction energies for the studied clusters are provided in Table 3, and the numbering pattern to identify the pair is provided in Fig. 1.
In the heterotimer and heterotetramer, the chalcogen bonding between the SO 3 and water molecule shows a large stabilization, while the interaction between SO 3 and water molecules with pair numbering ((I, III) and (I, IV)) shows a destabilizing effect. Intriguingly, the interaction between water molecules which are in bonding ((II, III) in heterotrimer and (II, III and (II, IV) in heterotetramer) show a stabilization effect. However, these intermolecular HBs have smaller interaction energy compared to the chalcogen bonding interactions [25].
In the heteropentamer to heterooctamer clusters, we observe the chalcogen bonding turns to be a destabilization term. Similarly, all the molecular interactions between SO 3 and water were destabilization in nature, while the intermolecular interaction between close water pairs has stabilization interactions. All these findings support the addition of new solvent water molecules interacting with the existing water network of the clusters in the SO 3 -(H 2 O)n clusters. In order to understand the nature of interactions between pairs in molecules, we have computed the many body interaction energy (DE). The total energy of the n-body cluster can be written as the sum of the one-, two-, three-, …. n body terms. For detailed methodology, the readers are requested to refer to the previous works [44]. The results obtained for the SO 3 -(H 2 O) 2 , SO 3 -(H 2 O) 3 , and SO 3 -(H 2 O) 4 are provided in the supporting information Table S1. According to the results, the main interactions are due to the two-and threebody terms for SO 3 -(H 2 O) 2 and four and three-body interaction terms for the SO 3 -(H 2 O) 3

and SO 3 -(H 2 O) 4 clusters.
However, in all cases, the many-body interaction energy is always negative, and it decreases with the increase in cluster size. It is worth pointing out that the use of dispersion correction in DFT calculations struggles to predict the threebody interaction energies accurately [45]. The change in sign in the three-and four-body interaction terms in our calculations can be attributed to the use of DFT-D method in our calculations.

Thermodynamics of the hydration of SO 3
We now explore the thermodynamics of the hydration of SO 3 molecules at three different temperatures observed at the tropospheric temperatures (216, 273, and 298 K) [46]. In the troposphere, the temperature at the top layer will be close to 216 K and that at the bottom will have a temperature of 298 K. Since the studied system involves the HBs, it is critical to perform scaling on harmonic/anharmonic frequencies.
In the present work, we have used the scaling factor 0.973 used by Truhular in their study of small molecules [47]. Previous studies on the hydration of atmospheric trace gasses have proved that vibration frequency scaling provides a better comparison of experimental with computed spectra of hydrated clusters [48]. The change in enthalpy, free energy, and absolute TDS values for the association reaction in the formation of hydrated clusters at T = 298 K are reported in Table 2. These thermodynamic parameters are computed by the sequential addition of water molecules to the SO 3 molecule in the gas phase incorporating the zero-point vibration corrections. The influence of entropy on the complex formation is negative and very small as anticipated due to the small size and rigidity of the SO 3 and water molecule [35]. The enthalpy values for all the clusters were negative indicating the association process is exothermic, and the exothermicity increases almost linearly with the cluster size signifying the number of water molecules decides the thermodynamic process. Also, the Gibbs free energy change was negative indicating the hydration process is spontaneous and thermodynamically favorable. Furthermore, from Table 2, one may notice ΔG° and ΔH° values decrease with an increase in the cluster size, signifying the amount of water molecule decides the sequential hydration process. The computed sequential free energy change for the hydration process at 216 and 278 K are also presented in Table 2, and the comparative diagram at the various studied temperature at 1 am pressure is provided in Fig. 2. It is evident Δfrom Fig. 2 that the ΔG o energies lower sequentially with the addition of water and temperature. Furthermore, its values reach a near-maximum at six water molecules. It is evident from Fig. 2 that hydration is more facile at low temperatures than at room temperature. Thus, the hydration process is more facile in the low-temperature regions found in the upper part of the troposphere than in the lower part.

Nature of intermolecular interaction
Molecular electrostatic potentials (MESP) assist to know the growth pattern of noncovalent and charged complexes [48,49]. The calculated isosurface on a total electron density of 0.001 a.u., along with the quantitative values designated as V s,max , and V s,min for the positive potential and negative potentials respectively. In the picture, the electron-rich regions are shown in red color and the electron-deficient regions in blue, while the green color regions are the regions that have almost near zero potential. The MESP diagram mapped onto the isosurface for the SO 3 -(H 2 O)n (n = 1-7) is provided in Fig. 3a-g, while the pristine SO 3 and H 2 O are shown in Fig. 3h, i.  On the SO 3 -H 2 O complex, only one π-hole was observed on the sulfur and is localized on the opposite side of the water molecule with a V s,max value of 30.5 kcal mol −1 which is far less than observed on the pristine SO 3 molecule. On the contrary, the V s,max value on hydrogen atom of water molecule was 64.0 kcal mol −1 , which is more than observed on a pristine water molecule. Thus, the increased V s,max value on hydrogen atom of water molecule helps for the further addition of water molecules to the hydrogen atom site. In the SO 3 -(H 2 O) 2 π-hole on the sulfur atom is further reduced to 18.1 kcal mol −1 , while the V s,max value on hydrogen atom is 60.9 and 62.9 kcal mol −1 . The reduced V s,max value on π-hole and increased value on hydrogen result in the addition of water to the water network rather than to the SO 3 molecule. A similar trend has been observed in larger clusters, wherein the π-hole helps in the stabilization of the clusters. The presence of such enhanced V s,max has been used to comprehend the sequential addition of monomers in HCN, OCS, OCSe, and SO 2 monomers in homo-oligomers and in SO 3 addition to formaldehyde in the hetero-oligomers [50][51][52].
In the heterodimer SO 3 -H 2 O, we noticed the presence of S···O intermolecular chalcogen bonding. In the heterotrimer SO 3 -(H 2 O) 2 , in addition to the S···O intermolecular chalcogen bond, an H···O interaction between water and SO 3 molecule exists. Besides, intramolecular bonding exists between water molecules. In the heterotetramer and heteropentamer clusters, we noticed the existence of one S···O intermolecular chalcogen bond and two H···O interactions between water and SO 3 molecule, while the number of HBs between water molecules was 2 and 3 respectively. In the hetrohexamer, heteroheptamer, and heterooctamer, the HBs between water and SO 3 molecule remain the same at three, while the H-bonds between water increase with the cluster size. Besides the above bonding, we also observe the presence of O···O bonding in clusters with five and above water molecules.
Atom in molecules is widely used to analyze bond types via electron density (ρ(r)), Laplacian of the electron density ( ∇ 2 (r) ) at the bond critical point (BCP). The ρ(r) values at various BCP are an important measure of the strength of the bonds [57]. Koch and Popelier suggest that hydrogen bonds have ρ(r) in the range of 0.001-0.100 a.u [58]. The sign of the ∇ 2 (r) at BCPs helps to identify the covalent and ionic nature of interactions. If the BPC has ∇ 2 (r) < 0, it is termed covalent, and if ∇ 2 (r) > 0, then it is a charge depletion and can be accounted for as a closed shell (electrostatic) interaction [59]. Besides, the ratio of eigenvalues of the Hessian matrix also helps to understand the change concentration/depletion which corresponds to covalent/ closed shells (hydrogen, van der Waals, and ionic bonds) interaction respectively [60]. Furthermore, the strength of hydrogen bonds can be assessed as below using the total electron density H(r) along with ∇ 2 (r) sign. In case of a strong hydrogen bond ∇ 2 (r) < 0 and H(r) < 0; for a medium H-bond ∇ 2 (r) > 0 and H(r) < 0; and for a weak H-bond ∇ 2 (r) > 0 and H(r) > 0.
The average topological parameters obtained at various BCPs between water and SO 3 for all the studied clusters are provided in Table 4. In the supporting information in Tables S2, S3, S4, S5, S6, and S7, we have provided topological parameters for all the intermolecular BCPs that exist between water molecules and SO 3 respectively. The ρ(r) value at the intermolecular S···O bond is in the range of 0.049-0.162 a.u. With the increase in the number of water molecules, the ρ(r) value at the intermolecular S···O increases and reaches a near maximum, which signifies that the addition of water increases the strength. The water-water BCPs have ρ(r) values in the range of 0.082-0.029 a.u. It is interesting to observe that the ρ(r) values for the water-water interactions are higher for the molecules which are coordinating with the water molecule bound to SO 3 by a covalent bond. Furthermore, the HBs between water have a higher ρ(r) value than the SO3-water interactions, which implies that water clustering is more facile than the SO 3 solvation. The sign of ∇ 2 (r) value for the intermolecular chalcogen S···O bond alone was found to be negative, while other intermolecular SO 3 -water and intramolecular water-water interactions were all positive. This implies that chalcogen S···O observed has a covalent nature, while other BCPs are of electrostatic or closed-shell interactions [61]. The ratio of eigenvalues of the Hessian matrix was all positive indicating the existence of weak interaction [62]. The sign of the total electron density H(r) was found to depend on the nature of interactions observed. In all the studied clusters, the chalcogen bonds H(r) > 0, while the sign for the hydrogen bonds varies. In general, the water-water interaction has H(r) > 0, and SO 3 − water interactions have H(r) < 0. Thus, the water-water HBs interactions in these clusters are of weak hydrogen bonding in nature, while the SO 3 -water interaction can be classified as medium HBs. Furthermore, HBs surrounding the water molecule interconnected to the SO 3 are stronger than other HBs. Thus, the presence of chalcogen bonding between SO 3 and water varied the HBs strength.
To comprehend the above findings, we examined the NCI isosurface and 2D RDG graphs for the hydrated sulfur trioxide clusters [63]. The NCI isosurface and 2D RDG graphs for the heterodimer and heterotrimer are provided in Fig. 5a, b, while for all the PGM clusters, the same is provided in the supporting information Figs. S6, S7, S8, S9, and S10.
In general, in the 2D RDG graphs, van der Waal's regions appear as spikes in the region 0.000-0.010 a.u., while the H-bonds appear in the region above −0.01 a.u. In the heterodimer, we did not notice any spikes in the van der Waal's regions as well as the HBs regions. However, we notice a broad spike close to 0.040 a.u. presumable due to the chalcogen bonding present in the system, which allies with the QTAIM and MESP findings. In the heterotrimer, we notice the presence of a spike near 0.090 a.u. which is due to the chalcogen bonds. Furthermore, spikes near −0.020 a.u. are noticed due to the weak H-bond between the water molecule and the SO 3 molecule. In the larger clusters, we notice the presence of several spikes in the −0.010 to −0.030 a.u. regions which can be attributed to the HBs between water molecules. Besides, we also notice several spikes in the van der Waal's regions due to the weak interaction between water molecules.

Conclusion
In conclusion, structure, stability, and intermolecular interactions in SO 3 -(H 2 O)n (n = 1-7) clusters were investigated using density functional and wave functional methods. The putative global minimum shows the SO 3 molecule tends to be on the surface water clusters. The heterodimer is formed by the dominant intermolecular S···O chalcogen bonding. Table 4 The average of QTAIM parameters (in a.u.) of bonding between the SO 3 molecule and water and among water molecules for SO 3 -(H 2 O)n (n = 1-7) clusters  With the increase in the number of water molecules, the chalcogen bond distance between water molecules and SO 3 decreases, while the maximum number of water molecules coordinated to the SO 3 molecule remains at three. The number of HBs between water molecules increases with the increase in the number of water molecules, and the mean hydrogen bond length increases with the cluster size and reaches a near saturation, but has a shorter bond length than observed in the pristine water clusters. The calculated solvation energy increases with the increase in the number of water molecules, and it does not saturate which indicates that the addition of water molecules can add up to the existing water cluster network. The interaction energy between water molecules and SO 3 was less than the solvation energy conforming cluster forming of water molecules. The pairwise energy for the molecular interactions between SO 3 and water in larger clusters has destabilization in nature, while the interaction between close water pairs has stabilization interactions. The Gibbs free energy and entropy values decrease with the increase in cluster size, signifying the amount of water molecule decide the sequential hydration process. Thermochemistry data at various temperatures show that low-temperature regions found in the upper part of the troposphere favor the hydration process.
Molecular electrostatic potentials (MESP) show reduced V s,max value of π-hole on sulfur atom and increased value on hydrogens of water molecules which results in the addition of water which leads to the sequential addition of water molecules to the water network. The quantum theory of atoms in molecules (QTAIM) shows the presence of S···O, O···H interactions between SO 3 and water molecules. Between water molecules O···H, H-bonding interactions were observed, and in larger clusters, O···O interaction was also noticed. The S···O chalcogen bonding has covalent nature which is evident from the negative ∇ 2 (r) value. The water-water interaction has H(r) > 0, and SO 3 -water interactions have H(r) < 0. Thus, the water-water HBs interactions in these clusters are weak HBs, while the SO 3 -water interaction can be classified as medium HBs which was further supported by the NCI and 2D RDG plots. Thus, the presence of S···O chalcogen bonding and weak HBs between water clusters act cooperatively and stabilize the SO 3 -(H 2 O)n clusters.