Catalytic effect of (H₂O)_n (n = 1–3) clusters on the HO₂ + SO₂ → HOSO + ³O₂ reaction under tropospheric conditions

Abstract

The HO₂ + SO₂ → HOSO + ³O₂ reaction, both without a catalyst and with (H₂O)_n (n = 1–3) as a catalyst, has been investigated using CCSD(T)/CBS//M06-2X/aug-cc-pVTZ methods, and canonical variational transition state theory with small curvature tunneling (CVT/SCT). The calculated results show that H₂O exerts the strongest catalytic role in the hydrogen atom transfer processes of HO₂ + SO₂ → HOSO + ³O₂ as compared with (H₂O)₂ and (H₂O)₃. In the atmosphere at 0 km altitude within the temperature range of 280.0–320.0 K, the reaction with H₂O is dominant, compared with the reaction without a catalyst, with an effective rate constant 2–3 orders of magnitude larger. In addition, at 0 km, it is worth mentioning that the relevance of the HO₂ + SO₂ → HOSO + ³O₂ reaction with H₂O depends heavily on its ability to compete with the primary loss mechanism of HO₂ radicals (such as the HO₂ + HO₂ and HO₂ + NO₃ reactions) and SO₂ (such as the SO₂ + HO reaction). The calculated results show that the HO₂ + SO₂ → HOSO + ³O₂ reaction with H₂O cannot be neglected in the primary loss mechanism of the HO₂ radical and SO₂. The calculated results also show that for the formation of HOSO and ³O₂, the contribution of H₂O decreases from 99.98% to 27.27% with an increase in altitude from 0 km to 15 km, due to the lower relative concentration of water. With the altitude increase, the HO₂ + SO₂ → HOSO + ³O₂ reaction with H₂O cannot compete with the primary loss mechanism of HO₂ radicals. The present results provide new insight into (H₂O)_n (n = 1–3) catalysts, showing that they not only affect energy barriers, but also have an influence on loss mechanisms. The present findings should have broad implications in computational chemistry and atmospheric chemistry.

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1. Introduction

As a significant gaseous pollutant from both industrial and natural sources,¹ sulfur dioxide (SO₂)^2,3 can cause a wide range of issues for human health, agriculture and the global climate.⁴ It leads to acid rain that damages forest and crops, and plays a role in the formation of atmospheric aerosols.^5,6 Furthermore, short-term exposure to high levels of SO₂ in the air can be life-threatening, as it can cause breathing difficulties and obstruct airways, while long-term exposure to persistent levels of SO₂ can cause chronic bronchitis, emphysema and respiratory illness.⁷ Similar to SO₂, the hydroperoxyl radical (HO₂) is also an important atmospheric species, which plays important roles in both the troposphere and the stratosphere.⁸ In the troposphere, the HO₂ radical is central to the production of ozone and the generation of hydroxyl radicals, whereas in the stratosphere, it is involved in catalytic cycles. HO₂ also acts as a significant transient intermediate in the combustion of hydrocarbon fuels, atmospheric photolysis cycles, and biochemical processes.^9,10 The reaction between HO₂ and SO₂ has aroused great interest from many research groups due to its great importance to supply a reverse path between SO₂ and HOSO (or SO₃) in the sulfur cycle. This reaction may take place via a number of pathways including the three main product paths shown in eqn (1)–(3), namely, O₂ + HOSO formation, O₂ + HSO₂ formation, and OH + SO₃ formation.^3,11 It is noteworthy that, as one of the most important products of the HO₂ + SO₂ reaction, HOSO has been found to be relatively stable¹² and its subsequent reactions¹³ are important for atmospheric chemistry. Therefore, its structure,^14,15 harmonic vibrational frequencies,¹⁶ thermochemistry^12,17 and subsequent reactions^13,18 have been widely investigated.

HO₂ + SO₂ → HOSO + O₂ (1)

HO₂ + SO₂ → SO₃ + OH (2)

HO₂ + SO₂ → HSO₂ + O₂ (3)

The kinetics and mechanism of the HO₂ + SO₂ reaction have been studied experimentally and theoretically for a range of temperatures and pressures. In an experimental study, Payne et al.¹⁹ found that the rate constant of the HO₂ + SO₂ reaction was (8.7 ± 1.8) × 10⁻¹⁶ cm³ per molecule per s at 300 K and a pressure of 30 Torr. Meanwhile, the rate constant of the HO₂ + SO₂ reaction at 298 K was found to be 2.01 × 10⁻¹⁷ cm³ per molecule per s, as measured by Burrows et al.²⁰ at low pressure. The kinetics and mechanism for the reaction of HO₂ with SO₂ were investigated by Wang et al.¹¹ with ab initio molecular orbital and transition state theory calculations. Their results showed that the reaction firstly proceeds to form a five-member-ring complex SO₂⋯HO₂, and then the complex undergoes barrierless decomposition to produce O₂ + HOSO. These investigations provide meaningful information about the mechanisms and kinetics for the HO₂ + SO₂ reaction under atmospheric conditions. However, this effort has only focused on the non-catalytic process of the HO₂ + SO₂ reaction.

Water has long been considered as a subject of chemical interest due to its abundance and unique properties in atmospheric chemistry.²¹ It not only forms hydrogen-bonded complexes with HO₂,^22,23 OH,^24–26 formic acid²¹ and nitric acid,⁸ but also can actively participate in atmospheric reactions as a catalyst,^27,28 of which HO₂⋯H₂O is a well-studied example.^29–31 An interesting result²⁹ concerning this complex is that the HO₂ self-reaction can be up to three times faster in the presence of water, since 30% of HO₂ radicals may exist in the form of HO₂⋯H₂O.³² Moreover, many experimental and theoretical studies have been reported on the electronic structure of the clusters SO₂⋯H₂O^33–37 and the gaseous hydrolysis reaction of SO₂ + H₂O.^3,11 These situations stimulated our interest in modeling the gas-phase reaction of the H₂O⋯HO₂⋯SO₂ ternary system, in which a single water molecule serves as a catalyst.

Recently, Chen et al.³ employed the CCSD(T)/aug-cc-pVTZ//B3LYP/aug-cc-pVTZ method to investigate the reactions of HO₂ + SO₂ and HO₂⋯H₂O + SO₂. Their calculations showed that H₂O slightly accelerates the SO₂ + HO₂ reaction to form a HSO₄⋯H₂O radical complex. Although Chen et al.³ provided meaningful information on the mechanism and kinetics of HO₂⋯H₂O + SO₂ and HO₂ + SO₂ reactions under tropospheric conditions, the most favorable, water-assisted channel (HO₂ + SO₂ → HOSO + O₂) has not yet been investigated in this way. Thus, it is still difficult to judge whether water molecules have an obvious positive catalytic effect on the HO₂ + SO₂ reaction or not. Furthermore, many experimental and theoretical studies have appeared in the literature on the electronic structure of the clusters HO₂⋯(H₂O)_n (n = 2–3)^38,39 and SO₂⋯(H₂O)_n (n = 2–3).^40,41 On the other hand, some studies have shown that water dimers^38,42–49 and trimers^39,49–52 can also have a significant catalytic effect in hydrogen abstraction reactions and hydrolysis of sulfur dioxide.^53,54 Thus, the investigation of the effect of (H₂O)_n (n = 1–3) on the HO₂ + SO₂ → HOSO + ³O₂ reaction is a logical path to pursue.

In the present study, the detailed effects of (H₂O)_n (n = 1–3) on the hydrogen atom transfer processes of the HO₂ + SO₂ → HOSO + ³O₂ reaction have been investigated at the CCSD(T)/CBS//M06-2X/aug-cc-pVTZ level of theory, which is organized as follows. Firstly, incorporation of (H₂O)_n (n = 1–3) into the HO₂ + SO₂ → HOSO + O₂ reaction produced two different types reactions: HO₂⋯(H₂O)_n (n = 1–3) + SO₂ and SO₂⋯(H₂O)_n (n = 1–3) + HO₂. Then, these (H₂O)_n (n = 1–3)-catalyzed channels were evaluated by investigating the direct hydrogen abstraction process and the double hydrogen transfer mechanism. Secondly, the rate constants and effective rate constant of the hydrogen abstraction reaction of HO₂ + SO₂ → HOSO + O₂ without and with (H₂O)_n (n = 1–3) were calculated to identify the favorable routes. Finally, the atmospheric relevance of the effect of (H₂O)_n (n = 1–3) was investigated by their competition with the primary loss mechanism of HO₂ radicals (such as the HO₂ + HO₂ and HO₂ + NO₃ reactions) and SO₂ (such as the SO₂ + HO reaction). Overall, this work may lead to a better understanding of the effects of (H₂O)_n (n = 1–3) on gas-phase reactions under tropospheric conditions.

2. Computational details

The geometries of all the reactants, pre-reactive intermediates, post-reactive intermediates, transition states and products were optimized at the M06-2X/aug-cc-pVTZ level of theory,⁵⁵ and frequency analysis at the same level was performed to study the stationary point as well as the transition states. Moreover, the minimum energy path (MEP) was achieved by the intrinsic reaction coordinate (IRC)⁵⁶ theory with a gradient step size of 0.01–0.05 (amu)^1/2 Bohr, to confirm that the TS connects to minima along the reaction path. To obtain more reliable energy information, single-point energy calculations for the stationary points were obtained using the CCSD(T)⁵⁷/CBS method at the M06-2X/aug-cc-pVTZ optimized geometries. It is worth pointing out that single point energy calculations of CCSD(T)/CBS were carried out for all species at the CCSD(T) level of theory using the aug-cc-pVDZ and aug-cc-pVTZ basis sets. The energy values obtained at the DZ and TZ levels were used to extrapolate the results to a complete basis set (CBS) limit. The method used here for CBS extrapolation was proposed and developed by Varandas and Pansini.⁵⁸ In this method, the energy at the CBS limit is obtained by extrapolating the correlation (Corr) energy and Hartree–Fock (HF) energy separately using two different equations. To extrapolate the Corr energy and HF energy, the following equations were used:

To estimate the effect of (H₂O)_n (n = 1–3), the theoretical rate constants of canonical variational transition (CVT)^59–61 state theory with small curvature tunneling (SCT)^62,63 correction for every reaction channel were calculated using the VKLab⁶⁴ program coupled with the steady state approximation.

As described in eqn (6), the HO₂ + SO₂ → HOSO + O₂ reaction without and with (H₂O)_n (n = 1–3) all began with the formation of pre-reactive intermediates before occurring through the transition state.

Similar to the water-catalyzed OH + HOCl reaction,²⁶ the free energy results displayed in Table S4† suggest that the intermediates are, in part, shifted to the reactants. Assuming that the pre-reactive intermediates of the HO₂ + SO₂ → HOSO + O₂ reaction without and with (H₂O)_n (n = 1–3) are in equilibrium with the corresponding reactants and are at steady state, the overall rate constant of the SO₂ + HO₂ reaction is expressed as:

k = k_eqk₂ (7)

Here, the various Q values denote the partition functions of the intermediates, reactants R1 and R2. E_R and E_IM stand for the total energies of the reactants and intermediate, respectively, while σ is the symmetry factor. The rate constant k₂ of the second step of eqn (7) was evaluated by the VKLab⁶⁴ program in the framework of the CVT. To include tunneling effects for motion along the reaction coordinate of the title reaction at the CCSD(T)/CBS//M06-2X/aug-cc-pVTZ level, the SCT tunneling approximation was adopted. The electronic structure calculations were performed using Gaussian09 (ref. 65) software.

3. Results and discussion

The transition state in each reaction channel was denoted by “TS” followed by a number, and each pre-reactive intermediate was denoted by “IM” followed by a number. The letter “a” was used to distinguish the transition states and pre-reactive intermediates which are conformers of each other and therefore have the same features; species in the presence of H₂O, (H₂O)₂, and (H₂O)₃ were respectively denoted by a “WM”, “WD”, and “WT” suffix.

3.1 Potential energy surfaces for the SO₂ + HO₂ → HOSO + O₂ reaction

Consistent with previous studies on the HO₂ + SO₂ reaction,^3,11 three kinds of product pairs, HOSO + O₂, HSO₂ + O₂, and SO₃ + HO could be found. Two H-abstraction channels and one O-abstraction route were modeled for the HO₂ + SO₂ reaction without water, as shown in Fig. S1.† From an energetic point of view, the channel for HOSO + O₂ formation was determined to be the major channel, and this is in good agreement with previous results reported by Wang et al.¹¹ Herein, as seen in Fig. 1, the SO₂ + HO₂ → HOSO + O₂ reaction was mainly discussed, in order to focus on the catalytic role of (H₂O)_n (n = 1–3) in the SO₂ + HO₂ → HOSO + O₂ reaction under atmospheric conditions. As shown in Fig. 1, for the formation of HOSO and O₂, the reaction begins with the formation of the pre-reactive intermediate IM1. IM1 exhibits a five-membered ring structure. The relative energy of IM1 to the reactants (HO₂ + SO₂) is −5.6 kcal mol⁻¹, which is in reasonable agreement with the previous value (−6.2 kcal mol⁻¹) of CR1b reported by Wang et al.¹¹ The pre-reactive intermediate IM1 goes through an H-abstraction transition state TS1 to form the post-reactive intermediate IMF1. From an energetic standpoint, the transition state TS1 has been predicted to be 0.3 kcal mol⁻¹ below the reactants, which is slightly different from the corresponding values (−0.5 kcal mol⁻¹) obtained by Wang et al.¹¹ Intermediate IMF1 retains a five-membered ring structure with a binding energy of 1.4 kcal mol⁻¹, and it dissociates to produce HOSO + O₂, which lies 3.6 kcal mol⁻¹ above the energy of the HO₂ + SO₂ reactants.

Fig. 1 Schematic energy diagrams of the SO₂ + HO₂ → HOSO + O₂ reaction without water at the CCSD(T)/CBS//M06-2X/aug-cc-pVTZ level.

3.2 Geometrical analysis of the HO₂⋯(H₂O)_n (n = 1–3) and SO₂⋯(H₂O)_n (n = 1–3) complexes

It is noteworthy that the rates of pure termolecular interactions are insignificant as compared to the rates of sequential bimolecular interactions.⁶⁷ In this sense, in the presence of (H₂O)_n (n = 1–3), both HO₂ and SO₂ mainly interact with (H₂O)_n (n = 1–3) via hydrogen bonding to form two-body complexes of HO₂⋯(H₂O)_n (n = 1–3) and SO₂⋯(H₂O)_n (n = 1–3), respectively, in the entrance channels, before interacting with the third body, SO₂ or HO₂. Therefore, it is highly necessary to first find stable configurations of the complexes HO₂⋯(H₂O)_n (n = 1–3) and SO₂⋯(H₂O)_n (n = 1–3). In order to find all possible stable configurations of the complexes HO₂⋯(H₂O)_n (n = 1–3) and SO₂⋯(H₂O)_n (n = 1–3), global minimum searching of geometric structures was carried out using Tsinghua Global Minimum (TGMin).^68,69 Then, the initial structures for HO₂⋯(H₂O)_n (n = 1–3) and SO₂⋯(H₂O)_n (n = 1–3) were selected for geometry optimization using the M06-2X/6-31G(d) method. The isomer structures within 6.0 kcal mol⁻¹ of the global minimum were re-optimized using the M06-2X/aug-cc-pVTZ method. Fig. 2 and S2† show the possible stable geometrical structures of HO₂⋯(H₂O)_n (n = 1–3) and SO₂⋯(H₂O)_n (n = 1–3), which are in good agreement with the available experimental results⁷⁰ and the previously calculated values.^33,42,43,47 Interestingly, each type of equilibrium structure of the HO₂⋯(H₂O)_n (n = 1–3) and SO₂⋯(H₂O)_n (n = 1–3) complexes mainly differs in the relative orientations of the dangling hydrogen atoms of (H₂O)_n (n = 1–3). Thus, for each type of equilibrium structure of HO₂⋯(H₂O)_n (n = 1–3) and SO₂⋯(H₂O)_n (n = 1–3), as shown in Fig. 2, we focus our attention on only one configuration, which has a larger stabilization energy than its isomers.

Fig. 2 The optimized geometrical reactants for the SO₂ + HO₂ reaction without and with catalyst X (X = H₂O, (H₂O)₂ and (H₂O)₃) at the M06-2X/aug-cc-PVTZ level of theory.

As shown in Fig. 2, consistent with previous reports,^70,71 the stable complex between HO₂ and H₂O is H₂O⋯HO₂, which involves a five-membered ring structure. Its bonding energy, listed in Table S3,† is 6.9 kcal mol⁻¹, which is much more stable than hydrogen bonded (van der Waals interaction) complexes of HO₂⋯H₂O, SO₂⋯H₂O and H₂O⋯SO₂. Stable termolecular complexes of HO₂⋯(H₂O)₂ and SO₂⋯(H₂O)₂ are made up of HO₂ or SO₂ and a water dimer (H₂O⋯H₂O). Similar to a binary complex of H₂O⋯HO₂, both HO₂ (SO₂) and the water dimer in these two complexes act simultaneously as a hydrogen bond acceptor and a hydrogen bond donor. From an energetic standpoint, due to ring tension, the stabilization energy of the seven-membered-ring HO₂⋯(H₂O)₂ complex is 12.6 kcal mol⁻¹, whereas the bonding energy of the six-membered ring SO₂⋯(H₂O)₂ complex is 6.9 kcal mol⁻¹. With HO₂ or SO₂ inserted into (H₂O)₃, two stable geometries, HO₂⋯(H₂O)₃ and SO₂⋯(H₂O)₃, are obtained. Complex HO₂⋯(H₂O)₃ exhibits a nine-membered-ring structure with a bonding energy of 12.3 kcal mol⁻¹, which is less stable than that of complex HO₂⋯(H₂O)₂. Similarly, the binding energy of the eight-membered-ring SO₂⋯(H₂O)₃ complex is 6.5 kcal mol⁻¹, which is 0.4 kcal mol⁻¹ lower than that of SO₂⋯(H₂O)₂ due to the effect of ring strain. The present study mainly focuses on the catalytic roles of (H₂O)_n (n = 1–3) in the SO₂ + HO₂ → HOSO + O₂ reaction. Only the favorable reaction channels will be discussed according to their energy barriers and thermodynamic properties. Details on the remaining reaction channels are presented in the ESI.†

3.3 Mechanism for H₂O-assisted SO₂ + HO₂ → HOSO + O₂ reaction

The significance of the water-catalyzed SO₂ + HO₂ → HOSO + O₂ reaction depends on the extent to which the respective bimolecular complexes, H₂O⋯HO₂, HO₂⋯H₂O, SO₂⋯H₂O and H₂O⋯SO₂ exist, which can be determined by computing their stability, as above (Fig. 2), and their corresponding equilibrium constants, as shown in Table S2.† The equilibrium constants of these complexes at 298 K are 8.59 × 10⁻¹⁹, 6.56 × 10⁻²³, 3.08 × 10⁻²⁴ and 4.88 × 10⁻²⁴ cm³ per molecule, respectively (Table S2†). Taking into account typical tropospheric concentrations of 7.73 × 10¹⁷ molecules per cm³ of H₂O, 3 × 10⁸ molecules per cm³ of HO₂,⁷² and 1.0 × 10¹² molecules per cm³ of SO₂,³³ it is estimated that the atmospheric concentration of the H₂O⋯HO₂ complex is 1.99 × 10⁸ molecules per cm³. Meanwhile, the concentrations of the HO₂⋯H₂O, SO₂⋯H₂O and H₂O⋯SO₂ complexes are estimated to be 1.52 × 10⁴, 2.38 × 10⁶, and 3.77 × 10⁶ molecules per cm³. Thus, as shown in Fig. 3 and 4, when one water molecule is introduced into the SO₂ + HO₂ → HOSO + O₂ reaction, there are four possible types of bimolecular reactions, labeled as Channels WM1, WM2, WM3, and WM4.

H₂O⋯HO₂ + SO₂ → H₂O⋯HOSO + O₂ (9)

HO₂⋯H₂O + SO₂ → H₂O + HOSO + O₂ (10)

SO₂⋯H₂O + HO₂ → HOSO⋯H₂O + O₂ (11)

H₂O⋯SO₂ + HO₂ → HOSO⋯H₂O + O₂ (12)

Fig. 3 Schematic energy diagrams of the water-assisted channel of HOSO + O₂ formation, occurring through H₂O⋯HO₂ + SO₂ and HO₂⋯H₂O + SO₂ at the CCSD(T)/CBS//M06-2X/aug-cc-pVTZ level.

Fig. 4 Schematic energy diagrams of the water-assisted channel of HOSO + O₂ formation, occurring through SO₂⋯H₂O + HO₂ and H₂O⋯SO₂ + HO₂ at the CCSD(T)/CBS//M06-2X/aug-cc-pVTZ level.

3.3.1 Mechanism of H₂O⋯HO₂ + SO₂ and HO₂⋯H₂O + SO₂ reactions. For the formation of HOSO⋯H₂O + O₂ occurring through the H₂O⋯HO₂ + SO₂ reaction, Fig. 3 displays two Channels, namely Channel WM1a and Channel WM1b. Similar to other water-catalyzed reactions reported before,^26,33,73 the influence of the water monomer in the H₂O⋯HO₂ + SO₂ reaction was investigated by a stepwise mechanism, where the reaction occurs via a ring enlargement at first, and then proceeds through a hydrogen abstraction mechanism. However, the stepwise mechanism of Channel WM1a is different from Channel WM1b in three aspects. First, for the ring enlargement, the form of the S⋯O in complex IM_WM3 is different from that in complex IM_WM1. In complex IM_WM3, the S⋯O bond is formed by the S atom of SO₂ and the O atom of the H₂O moiety in the HO₂⋯H₂O complex, whereas the S⋯O bond in the IM_WM1 complex is formed by the S atom of SO₂ and the terminal O atom of the HO₂ moiety in the HO₂⋯H₂O complex. However, the stabilization energies of complexes IM_WM3 and IM_WM1 are close to each other, but with a difference of 0.5 kcal mol⁻¹. Secondly, the complex IM_WM4 has similar seven-membered cyclic structures as IM_WM2, except that SO₂ and H₂O have exchanged their positions. This is possibly the reason that the stabilization energy of complex IM_WM4 is 1.8 kcal mol⁻¹ lower than that of complex IM_WM2. The third one is that, for the rate-determining step of hydrogen abstraction, the barrier height of Channel WM1b is 2.1 kcal mol⁻¹, which is 7.3 kcal mol⁻¹ lower than the process of IM_WM2 → TS_WM2 → H₂O⋯HOSO + O₂ in Channel WM1a. This indicates that the double hydrogen transfer mechanism in Channel WM1a is less favorable than the direct hydrogen abstraction involved in Channel WM1b. This conclusion is consistent with our previous reports.^71,73,74 As a result, in the following section, the reaction type of the multiple hydrogen transfer mechanism, where water molecules serve as a “bridge”, will be neglected in the stepwise reaction between SO₂ and HO₂⋯(H₂O)₂ (or HO₂⋯(H₂O)₃).

As for Channel WM2, it can be seen in Fig. 3c that a hydrogen bond complex, IM_WM5, has been identified, with a stabilization energy of 5.8 kcal mol⁻¹. From a geometrical point of view, complex IM_WM5 exhibits a five-membered ring structure, which is similar to the five-membered ring structure of complex IM1 in the reaction without a catalyst, as shown in Fig. 1. Compared with complex IM1, an additional O⋯H bond (O2⋯H2, 2.14 Å) is involved in IM_WM5. This leads to the stabilization energy of complex IM_WM5 being enhanced by 0.2 kcal mol⁻¹. Starting from complex IM_WM5 seen in Fig. 3c, the products HOSO + H₂O + O₂ can be obtained via transition state TS_WM5. From a geometrical point of view, similar to the naked transition state TS1, TS_WM5 follows direct hydrogen abstraction. Compared with TS1, the additional water molecule in the transition state TS_WM5 is bonded to the terminal O atom of the HO₂ radical. Such a weak hydrogen bonding (O2⋯H2, 2.23 Å) interaction may lead to the energy of the TS_WM5 to HO₂⋯H₂O + SO₂ reactants being 0.3 kcal mol⁻¹ higher than that of the TS1 to the HO₂ + SO₂ reactants.

3.3.2 Potential energy surfaces for SO₂⋯H₂O + HO₂ and H₂O⋯SO₂ + HO₂ reactions. Beyond the water-assisted reaction channels described above, two additional water-assisted channels for HOSO + O₂ formation were found by taking into account the bimolecular reactions of SO₂⋯H₂O + HO₂ (Channel WM3) and H₂O⋯SO₂ + HO₂ (Channel WM4).

As for Channel WM3, the reaction of the SO₂⋯H₂O + HO₂ entry channel proceeds through the formation of complex IM_WM6 before the transition state TS_WM6 and complex IMF_WM6. From a geometric point of view, complex IM_WM6 has one five-membered ring, H(1)–O(2)–O(1)⋯S–O(3), and one four-membered ring, O(4)⋯H(2)–O(5)⋯S, with a computed relative energy of 5.3 kcal mol⁻¹ below the reactants of SO₂⋯H₂O + HO₂, as shown in Table S4† and Fig. 4. Starting from complex IM_WM6, with the H atom of HO₂ migrating to one O atom of the SO₂ moiety in the SO₂⋯H₂O complex, the reaction can proceed via the transition state TS_WM6 to form complex IMF_WM6. As shown in Fig. 4, only the five-membered ring was retained in both the transition state TS_WM6 and complex IMF_WM6 with the elongation of the S⋯O5 bond. Compared with the transition state TS1 in Fig. 1, the additional water molecule in TS_WM6 did not change the active features of the five-membered ring. Furthermore, with the water molecule added, the relative energy of TS_WM6 to SO₂⋯H₂O + HO₂ is 0.7 kcal mol⁻¹ less than the corresponding value of TS1 (−0.3 kcal mol⁻¹) to SO₂ + HO₂.

The fourth channel of the water-assisted SO₂ + HO₂ → HOSO + O₂ reaction (Channel WM4, Fig. 4b) begins with complex IM_WM7 via the interaction between the O atom of H₂O and the S atom of the SO₂ moiety in H₂O⋯SO₂. The relative energy of complex IM_WM7 to the H₂O⋯SO₂ + HO₂ reactants is 6.1 kcal mol⁻¹. Complex IM_WM7 then decomposes via transition state TS_WM7 to form complex IMF_WM7. As shown in Fig. 4, similar to the structure of TS_WM6 and complex IMF_WM6, a five-membered ring is still involved in both TS_WM7 and complex IMF_WM7. Compared with TS1 in Fig. 1, the additional water molecule in TS_WM7 leads to the relative energy of TS_WM7 to H₂O⋯SO₂ + HO₂ being 0.7 kcal mol⁻¹, which is 0.4 kcal mol⁻¹ less than the corresponding value of TS1 (−0.3 kcal mol⁻¹) to SO₂ + HO₂.

3.4 Mechanism for (H₂O)₂-catalyzed the SO₂ + HO₂ → HOSO + O₂ reaction

As shown in Fig. 5, (H₂O)₂ catalyzed channels were determined, starting from HO₂⋯(H₂O)₂ + SO₂ and SO₂⋯(H₂O)₂ + HO₂ reactants, which are labeled as Channel WD1 and Channel WD2.

Fig. 5 Schematic energy diagrams of the dimer water-assisted channel of HOSO + O₂ formation, occurring through HO₂⋯(H₂O)₂ + SO₂ and SO₂⋯(H₂O)₂ + HO₂ at the CCSD(T)/CBS//M06-2X/aug-cc-pVTZ level.

For the HO₂⋯(H₂O)₂ + SO₂ reaction (Channel WD1), two reaction types, namely Channel WD1a and Channel WD1b were found, depending on how the S atom of SO₂ approaches the complex of HO₂⋯(H₂O)₂. In the same way as the H₂O⋯HO₂ + SO₂ reaction above (Fig. 3), both Channels WD1a and WD1b occur via a stepwise mechanism, which involves a direct hydrogen abstraction with one O atom of SO₂ abstracting the H atom of a HO₂ moiety in the HO₂⋯(H₂O)₂ complex. However, as shown in Fig. 5, unlike the hydrogen abstraction in Channel WD1a, where the number of rings increases with decreasing ring size, in Channel WD1b the hydrogen abstraction does not change the ring size or the ring number. This discrepancy of hydrogen abstraction between Channels WD1a and WD1b may mean that, in Channel WD1a, the barrier height of the rate determining step (IM_WD2 → TS_WD2 → (H₂O)₂⋯HOSO + O₂) is 8.4 kcal mol⁻¹ higher than the corresponding barrier height of the rate determining step (IM_WD4 → TS_WD4 → (H₂O)₂⋯HOSO + O₂) in Channel WD1b. As a result, for the HO₂⋯(H₂O)₂ + SO₂ reaction, Channel WD1b is mainly taken into account here.

For Channel WD1b, with the interaction between the S atom of SO₂ and O(5) of the H₂O moiety in the HO₂⋯(H₂O)₂ complex, the reaction begins with the seven-membered cyclic complex IM_WD3, whose stability has been computed to be 2.7 kcal mol⁻¹. Then, the reaction proceeds through the ring enlargement from IM_WD3 to IM_WD4 via the transition state TS_WD3, with an energy barrier of 1.5 kcal mol⁻¹. Complex IM_WD4 has a similar structure to IM_WM4 (Fig. 3), but with a water monomer substituted by a water dimer. The binding energy of IM_WD4 is 1.8 kcal mol⁻¹, which is 0.9 kcal mol⁻¹ less stable than the complex IM_WD3. Starting from the complex IM_WD4, the reaction can proceed via the transition state TS_WD4 to form the products of (H₂O)₂⋯HOSO + O₂. Similar to complex IM_WD4, TS_WD4 also shows a nine-membered ring structure with three components of (H₂O)₂, the SO moiety of SO₂ and the HO₂ radical. The relative energy of TS_WD4 to HO₂⋯(H₂O)₂ + SO₂ is 0.4 kcal mol⁻¹, which is 2.5 kcal mol⁻¹ lower than that of the H₂O-assisted transition state TS_WM4 to H₂O⋯HO₂ + SO₂ reactants. Meanwhile, from the viewpoint of ring sizes, for TSW2 and TSWD, with the ring size increase, the rate constant of the rate-determining step increases when one water molecule has been replaced by a water dimer. This is possibly because when the seven-membered ring has been replaced by a nine-membered ring, the number of hydrogen bonds increases, leading to the stabilization of the ring increasing. These facts indicate that the second water molecule plays a positive catalytic role by significantly reducing the barrier height of the SO₂ + HO₂ → HOSO + O₂ reaction.

As for Channel WD2, starting from SO₂⋯(H₂O)₂ + HO₂ reactants, the reaction begins with the formation of complex IM_WD5. As shown in Fig. 5c, complex IM_WD5 exhibits a double-ring structure with a five-membered ring, O2⋯H1–O3–S⋯O1, and a six-membered ring, O4⋯H2–O5⋯H4–O6⋯S. Aside from these characteristics, complex IM_WD5 has a similar structure to that of IM_WM5, but with an additional water molecule inserted at the site between H₂O and SO₂. This leads to the stabilization energy of complex IM_WD5 (Table S4† and Fig. 5) being enhanced by 0.3 kcal mol⁻¹ compared with complex IM_WM5. After complex IM_WD5, with the H atom of HO₂ migrating to one O atom of the SO₂ moiety in the SO₂⋯(H₂O)₂ complex, the products of HOSO⋯(H₂O)₂ + O₂ can be obtained via transition state TS_WD5 and complex IMF_WD5. As shown in Fig. 5, the double-ring structure is retained in transition state TS_WD5 and complex IMF_WD5. Compared with TS_WM5 in Fig. 4, with the introduction of the second water molecule, the active features of the five-membered ring in TS_WD5 exhibit no change. The relative energy of TS_WD5 to SO₂⋯(H₂O)₂ + HO₂ is 0.9 kcal mol⁻¹, which is only changed by 0.6–0.9 kcal mol⁻¹ compared with the corresponding values of TS_WM5 with H₂O and TS1 without a catalyst. This implies that Channel WD2 proceeds easily.

3.5 Mechanism for (H₂O)₃-catalyzed the SO₂ + HO₂ → HOSO + O₂ reaction

Similar to the instance of the (H₂O)₂-assisted reaction that occurs through the HO₂⋯(H₂O)₂ + SO₂ and SO₂⋯(H₂O)₂ + HO₂ reactions discussed above, the (H₂O)₃-assisted SO₂ + HO₂ → HOSO + O₂ reaction can be obtained both by the insertion of SO₂ into the HO₂⋯(H₂O)₃ complex, and by the collision of the HO₂ radical with the SO₂⋯(H₂O)₃ complex. Through the SO₂⋯(H₂O)₃ + HO₂ reaction (which proceeds easily, as shown in Table S6,† with an energy barrier of 1.3 kcal mol⁻¹, as shown in Table S2†), the concentration of SO₂⋯(H₂O)₃ at 298 K is 4 orders of magnitude smaller than that of HO₂⋯(H₂O)₃. Therefore, the SO₂⋯(H₂O)₃ + HO₂ reaction can be neglected under tropospheric conditions, and herein, we only focus our attention on the HO₂⋯(H₂O)₃ + SO₂ reaction. For comparison, a schematic energy diagram of HO₂ + SO₂⋯(H₂O)₃ reaction is displayed in Fig. S6.†

Similar to H₂O⋯HO₂ + SO₂ and HO₂⋯(H₂O)₂ + SO₂ reactions above, HO₂⋯(H₂O)₂ + SO₂ reaction (Channel WT1) shown in Fig. 6

Fig. 6 Schematic energy diagrams of the trimer water-assisted channel of HOSO + O₂ formation, occurring through HO₂⋯(H₂O)₃ + SO₂ at the CCSD(T)/CBS//M06-2X/aug-cc-pVTZ level.

also proceeds through a stepwise mechanism to form HOSO⋯(H₂O)₃ + O₂. In the first step, the reaction starts with complex IM_WT1, which has a binding energy of 3.0 kcal mol⁻¹. From a geometrical point of view, complex IM_WT1 has one nine-membered ring-like structure. After complex IM_WT1, the ring enlargement from complex IM_WT1 to the eleven-membered ring complex IM_WT2 occurs via the double-ring transition state TS_WT1 (nine-membered ring, right; four-membered ring, left) with a barrier height of 0.7 kcal mol⁻¹. Complex IM_WT2 has similar quasi-planar structures to IM_WD4, with an additional water molecule inserted between SO₂ and HO₂. Complex IM_WT2 is 1.0 kcal mol⁻¹ less stable than IM_WT1, and is also 0.8 kcal mol⁻¹ less stable than IM_WD4. In the second step, following complex IM_WT2, Channel WT1 proceeds through transition state TS_WT2 to produce the product of HOSO⋯(H₂O)₃ + O₂. At transition state TS_WT2, the eleven-membered ring structure is still conserved, and the structure is similar to TS_WD4, but with an additional water molecule inserted between SO₂ and HO₂. From the viewpoint of ring sizes, for TS_WT2 and TS_WD4, although the hydrogen bonds increase in the ring when the nine-membered ring becomes an eleven-membered ring, the eleven-membered ring is unstable, and the tension is large, so TS_WT2 lies 16.0 kcal mol⁻¹ above the HO₂⋯(H₂O)₃ + SO₂ reactants, which is 16.4 kcal mol⁻¹ higher in energy than the relative energy of TS_WD4 to HO₂⋯(H₂O)₂ + SO₂ reactants. Meanwhile, the rate constant of the rate-determining step decreases when the water dimer has been replaced by the water trimer (Fig. 6).

3.6 Reaction kinetics

With regard to the reactions involving (H₂O)_n (n = 1–3), we consider HO₂⋯(H₂O)_n (n = 1–3) and SO₂ or SO₂⋯(H₂O)_n (n = 1–3) and HO₂. When HO₂⋯(H₂O)_n (n = 1–3) and SO₂ act as reactants, the reactions occur via the following reaction mechanism:

HO₂ + (H₂O)_n (n = 1–3) ↔ HO₂⋯(H₂O)_n (n = 1–3) (13)

HO₂⋯(H₂O)_n (n = 1–3) + SO₂ → HOSO⋯(H₂O)_n (n = 1–3) + O₂ (14)

Meanwhile when SO₂⋯(H₂O)_n (n = 1–3) and HO₂ are considered to be the reactants, the reaction mechanism is as follows:

SO₂ + (H₂O)_n (n = 1–3) ↔ SO₂⋯(H₂O)_n (n = 1–3) (15)

SO₂⋯(H₂O)_n (n = 1–3) + HO₂ → HOSO⋯(H₂O)_n (n = 1–3) + O₂ (16)

The rate via the reaction processes (13) and (14) is expressed in eqn (17).

Here, K_eq(HO₂⋯(H₂O)_n (n = 1–3)) is the equilibrium constant for the formation of the HO₂⋯(H₂O)_n (n = 1–3) complex from isolated HO₂ and (H₂O)_n (n = 1–3) and k_a represents the rate constant of eqn (14). The rate via the reaction processes (15) and (16) is expressed in eqn (18).Here, K_eq(SO₂⋯(H₂O)_n (n = 1–3)) is the equilibrium constant for the formation of the SO₂⋯(H₂O)_n (n = 1–3) complex from isolated SO₂ and (H₂O)_n (n = 1–3) and k_b represents the rate constant of eqn (16). It is worth noting that, herein, we do not consider how pressure effects influence the formation of these complexes, because there are no experimental results showing that the equilibrium constants of these complexes depend on pressure.

The computed rate constants are provided in Table 1. For the reaction without a catalyst, the computed rate constant (k_(R1)) is 9.09 × 10⁻¹⁸ to 5.22 × 10⁻¹⁷ cm³ per molecule per s in the temperature range of 218.6–320.0 K. At 298 K, the calculated value of k_(R1) is 3.87 × 10⁻¹⁷ cm³ per molecule per s, which is close to the experimental value (1.0 × 10⁻¹⁸ cm³ per molecule per s).⁶⁶ For the reaction with (H₂O)_n (n = 1–3), the rate ratio v_a/v_b of the HO₂ + SO₂ + (H₂O)_n (n = 1–3) reaction shows that the entrance of HO₂⋯(H₂O)_n (n = 1–3) and SO₂ is more important than that of SO₂⋯(H₂O)_n (n = 1–3) and HO₂ because the rate ratios v_a1/v_b1, v_a2/v_b2, and v_a3/v_b3 are 2.15 × 10⁹ to 1.27 × 10⁷, 5.00 × 10⁸ to 4.95 × 10⁶, 4.03 × 10¹³ to 4.12 × 10⁹, respectively, between 275.0 and 320.0 K (Table 1).

Table 1 Ratio of reaction rate and effective rate constants (cm³ per molecules per s) for HOSO + O₂ formation from the SO₂ + HO₂ reaction without and with (H₂O)_n (n = 1–3) within the temperature range of 275.0–320.0 K^a

T (K)	kR	va1/vb1	va2/vb2	va3/vb3	k′a(WM1)	k′a(WD1)	k′a(WT1)
a The letters “a” and “b” are used to distinguish the complexes: “a” is the complex of HO2⋯(H2O)n (n = 1–3), while “b” is the complex of SO2⋯(H2O)n (n = 1–3). va1, va2 and va3 are the reaction rates of HO2⋯(H2O)n (n = 1–3), vb1, vb2 and vb3 are the equilibrium constants of SO2⋯(H2O)n (n = 1–3). Species in the presence of a water molecule, water dimer and water trimer are respectively denoted by “1”, “2”, and “3”. k′a(WM1) is the effective rate constant for the process of HO2⋯H2O + SO2 → HOSO⋯H2O + O2. k′a(WD1) is the rate constant for the process of HO2⋯(H2O)2 + SO2 → HOSO⋯(H2O)2 + O2. k′a(WT1) is the rate constant for the process of HO2⋯(H2O)3 + SO2 → HOSO⋯(H2O)3 + O2.
218.6	9.09 × 10^−18	2.15 × 10^9	5.00 × 10^8	4.03 × 10^13	5.84 × 10^−13	2.99 × 10^−23	—
223.7	1.02 × 10^−17	1.48 × 10^9	3.69 × 10^8	1.73 × 10^14	5.38 × 10^−13	3.57 × 10^−22	—
229.7	1.08 × 10^−17	9.78 × 10^7	3.96 × 10^8	1.78 × 10^12	4.93 × 10^−13	5.07 × 10^−18	1.68 × 10^−21
235.1	1.31 × 10^−17	6.82 × 10^8	1.97 × 10^8	9.96 × 10^11	4.57 × 10^−13	1.81 × 10^−18	—
249.9	1.75 × 10^−17	2.78 × 10^9	9.59 × 10^7	1.58 × 10^12	3.83 × 10^−13	5.73 × 10^−18	—
259.3	1.76 × 10^−17	1.66 × 10^8	4.07 × 10^5	3.24 × 10^11	3.47 × 10^−13	3.09 × 10^−18	5.95 × 10^−21
275.0	2.72 × 10^−17	7.64 × 10^7	3.47 × 10^7	8.65 × 10^10	3.02 × 10^−13	3.90 × 10^−17	1.63 × 10^−19
280.0	2.95 × 10^−17	6.07 × 10^7	2.92 × 10^7	3.10 × 10^10	2.90 × 10^−13	4.57 × 10^−17	1.55 × 10^−18
290.0	3.44 × 10^−17	3.92 × 10^7	2.09 × 10^7	4.12 × 10^9	2.71 × 10^−13	6.58 × 10^−17	1.54 × 10^−19
298.2	3.87 × 10^−17	2.82 × 10^8	1.63 × 10^8	7.88 × 10^13	2.57 × 10^−13	8.95 × 10^−17	1.59 × 10^−18
300.0	3.98 × 10^−17	2.62 × 10^8	1.54 × 10^7	3.19 × 10^13	2.54 × 10^−13	9.50 × 10^−17	1.58 × 10^−18
310.0	4.57 × 10^−17	1.80 × 10^8	1.16 × 10^7	4.87 × 10^10	2.41 × 10^−13	1.32 × 10^−16	1.57 × 10^−19
320.0	5.22 × 10^−17	1.27 × 10^7	4.95 × 10^6	2.72 × 10^11	2.30 × 10^−13	1.72 × 10^−16	1.48 × 10^−18

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In the HO₂⋯(H₂O)_n (n = 1–3) + SO₂ reaction, it is necessary to calculate its effective rate constant k′, which can be considered as a measure of the efficiencies of the different catalysts of (H₂O)_n (n = 1–3) under atmospheric conditions, as it includes the concentration of (H₂O)_n (n = 1–3), equilibrium constants of HO₂⋯(H₂O)_n (n = 1–3), and the rate constant from eqn (14). Based on this, the rates for the channels of the H₂O⋯HO₂ + SO₂ reaction (k_a(WM1b)), the HO₂⋯(H₂O)₂ + SO₂ reaction (k_a(WD1b)) and the HO₂⋯(H₂O)₃ + SO₂ reaction (k_a(WT1)) are respectively written as follows.

v_a(WM1b) = k_a(WM1b)[H₂O⋯HO₂][SO₂] = k′_a(WM1b)[SO₂][HO₂] (19)

v_a(WD1b) = k_a(WD1b)[HO₂⋯(H₂O)₂][SO₂] = k′_a(WD1b)[HO₂][SO₂] (20)

v_a(WT1) = k_a(WT1)[HO₂⋯(H₂O)₃][SO₂] = k′_a(WT1)[HO₂][SO₂] (21)

In these equations, k′_a(WM1b) = k_a(WM1b)K_eq(H₂O⋯HO₂)[H₂O]; k′_a(WD1b) = k_a(WD1b)K_eq(HO₂⋯(H₂O)₂)[(H₂O)₂]; and k′_a(WT1) = k_a(WT1)K_eq(HO₂⋯(H₂O)₃)[(H₂O)₃]. K_eq(H₂O⋯HO₂), K_eq(HO₂⋯(H₂O)₂) and K_eq(HO₂⋯(H₂O)₃) are the equilibrium constants for the formation of the H₂O⋯HO₂, HO₂⋯(H₂O)₂ and HO₂⋯(H₂O)₃ complexes, respectively. [H₂O], [(H₂O)₂] and [(H₂O)₃] are the concentrations of H₂O, (H₂O)₂, and (H₂O)₃, respectively. As shown in Table 1, within the temperature range of 275.0–320.0 K, the effective rate constant of k′_a(WM1b) is 1.32 × 10⁻¹⁴ to 2.01 × 10⁻¹⁴ cm³ per molecule per s, which is 2–3 and 3–6 orders of magnitude larger than the corresponding values of k′_a(WD1b) and k′_a(WT1), respectively, indicating that the catalytic effect of H₂O is the largest among the (H₂O)_n (n = 1–3) catalysts, and the catalytic effect of (H₂O)₂, and (H₂O)₃ is negligible. Compared with the rate constant of the naked reaction of SO₂ + HO₂ → HOSO + O₂, the value of k′_a(WM1b) is 2–3 orders of magnitude larger within the temperature range of 275.0–320.0 K, indicating that at 0 km altitude within this temperature range, the positive effect of water is significant under atmospheric conditions.

The average concentrations of water at 5, 10 and 15 km altitudes are known to be 2.41 × 10¹⁶, 4.92 × 10¹⁵ and 1.96 × 10¹³ molecules per cm³, respectively.⁷⁵ Considering the average concentrations of water at 5, 10 and 15 km altitudes in the troposphere, the calculated effective rate constants of the H₂O-assisted Channel WM1, the (H₂O)₂-assisted Channel WD1 and the (H₂O)₃-assisted Channel WT1 are listed in Table 2. Compared with an altitude of 0 km, the dominant channel does not change in the three different cases, which is similar to the dominant channel in Table 1. Meanwhile, the values of k′(WM1) are 7.49 × 10⁻¹⁷, 2.68 × 10⁻¹⁷ and 2.95 × 10⁻¹⁸ at altitudes of 5, 10 and 15 km, respectively. Compared with the value of k′(WM1) at 0 km, we found that the values of k′(WM1) show a declining trend, decreasing by about 4–5 orders of magnitude, as we move towards higher altitude. Furthermore, as shown in Table 2, it has been estimated that the enhancement factors of water vapor are 99.98%, 78.23%, 69.63% and 27.27% at altitudes of 0, 5, 10 and 15 km, respectively. This implies that, as we move towards higher altitudes, the contribution of water on increasing the reaction rate becomes gradually lessened.

Table 2 Effective rate constants (cm³ per molecules per s) for HOSO + O₂ formation from the SO₂ + HO₂ reaction without and with (H₂O)_n (n = 1–3) within the altitude range of 0–15 km^a

Alt. (km)	T (K)	k′a(WM1)	k′a(WD1)	k′b(WD2)	k′a(WT1)	k′a(WM)/ktot
a k′a(WM1) is the effective rate constant occurring through Channel WM1. k′a(WD1) and k′b(WD2) are the effective rate constants of Channel WD1 and WD2, respectively; k′a(WT1) is the effective rate constant of Channel WT1. ktot = kR1 + k′a(WM)(100% RH).
0	298.2	1.67 × 10^−13	1.15 × 10^−21	4.03 × 10^−17	1.59 × 10^−18	99.98%
5	259.3	7.49 × 10^−17	3.09 × 10^−18	1.10 × 10^−18	3.53 × 10^−20	78.23%
10	229.7	2.68 × 10^−17	5.07 × 10^−18	1.40 × 10^−18	1.68 × 10^−21	69.63%
15	212.6	2.95 × 10^−18	8.95 × 10^−17	2.63 × 10^−22	2.42 × 10^−26	27.27%

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3.7 Atmospheric implications

In gas-phase reactions of the atmosphere, previous investigations have shown that the dominant sink of HO₂ is its reaction with HO₂ (ref. 71) and NO₃.⁷⁶ Therefore, it is of great importance to discuss the effective rate constant of the H₂O⋯HO₂ + SO₂ reaction (k′_a(WM1b)) with the rate constant of these dominant sink processes of HO₂. To meet this goal, the reaction mechanisms of HO₂ + HO₂ and HO₂ + NO₃ at the CCSD(T)/CBS//M06-2X/aug-cc-pVTZ level are shown in Fig. S7,† while their rate constants within the temperature range of 218.6–320.0 K are listed in Table 3, with rate ratios between H₂O⋯HO₂ + SO₂ and HO₂ + HO₂ (HO₂ + NO₃) as shown in eqn (22). The calculated rate constants are in good agreement with the available experimental values.^77–79Here, k_R1 and k_R2 are the rate constants of the HO₂ + HO₂ reaction and the HO₂ + NO₃ reaction, respectively, obtained from Table 3. The rate ratios v_a1/v_R1 and v_a1/v_R2 depend on the H₂O, SO₂, NO₃ and HO₂ concentrations in the atmosphere. At 298 K, when the concentration of water is at a relative humidity of 100%, SO₂ and NO₃ are 7.73 × 10¹⁷, 1 × 10¹² and 2.46 × 10⁷ molecules per cm³, respectively, and the gas-phase concentration of HO₂ is 3 × 10⁸ molecules per cm³, the H₂O⋯HO₂ + SO₂ reaction can compete with the HO₂ + HO₂ reaction, because the rate ratio v_a1/v_R1 is about 4.35 × 10² at 298 K. Meanwhile, the H₂O⋯HO₂ + SO₂ reaction can compete well with the HO₂ + NO₃ reaction because the rate ratio v_a1/v_R2 is about 6.83 × 10³ at 298 K. Thus, compared with the primary loss mechanism of HO₂ radicals, the HO₂ + SO₂ → HOSO + ³O₂ reaction with H₂O cannot be neglected. Meanwhile, with increasing altitude, the values of v_a1/v_R1 and v_a1/v_R2 obviously decrease. At 218.6 K, the values of v_a1/v_R1 and v_a1/v_R2 are 5.22 × 10⁻⁴ and 1.73 × 10⁻³, showing that the HO₂ + SO₂ → HOSO + ³O₂ reaction with H₂O cannot compete with the primary loss mechanism of HO₂ radicals.

Table 3 Rate constants (cm³ per molecules per s) for H₂O₂ + O₂ formation from the HO₂ + HO₂ reaction (R1), HNO₃ + O₂ formation from the NO₃ + HO₂ reaction (R2) and HSO₃ formation from the SO₂ + OH reaction (R3) within the temperature range of 218.6–320.0 K

T (K)	kR1	va1/vR1	kR2	va1/vR2	kR3	va1/vR3	va1/vR3(night)
218.6	1.38 × 10^−10	5.22 × 10^−4	5.06 × 10^−10	1.74 × 10^−3	2.06 × 10^−12	3.15 × 10^−3	3.15 × 10^−1
223.7	8.29 × 10^−11	8.24 × 10^−4	2.55 × 10^−10	1.09 × 10^−3	2.09 × 10^−12	3.27 × 10^−4	3.27 × 10^−2
229.7	4.69 × 10^−11	1.63 × 10^−1	1.22 × 10^−10	2.55 × 10^−1	2.14 × 10^−12	3.58 × 10^−2	3.58 × 10^0
235.1	2.88 × 10^−11	4.54 × 10^−4	6.72 × 10^−11	7.92 × 10^−4	2.20 × 10^−12	5.95 × 10^−5	5.95 × 10^−3
249.9	8.50 × 10^−11	2.53 × 10^−3	1.68 × 10^−11	5.20 × 10^−2	2.32 × 10^−12	9.26 × 10^−4	9.26 × 10^−2
259.3	4.22 × 10^−11	1.02 × 10^0	8.20 × 10^−12	2.13 × 10^1	2.37 × 10^−12	1.82 × 10^−1	1.82 × 10^1
280.0	1.08 × 10^−11	1.27 × 10^1	2.32 × 10^−12	2.41 × 10^2	2.53 × 10^−12	5.43 × 10^−1	5.43 × 10^1
290.0	6.01 × 10^−12	2.53 × 10^1	1.42 × 10^−12	4.35 × 10^2	2.64 × 10^−12	5.76 × 10^−1	5.76 × 10^1
298.2	3.85 × 10^−12	4.35 × 10^2	9.96 × 10^−13	6.83 × 10^3	2.70 × 10^−12	6.20 × 10^0	6.20 × 10^2
300.0	3.49 × 10^−12	4.88 × 10^2	9.23 × 10^−13	7.51 × 10^3	2.71 × 10^−12	6.29 × 10^0	6.29 × 10^2
310.0	2.11 × 10^−12	8.88 × 10^2	6.29 × 10^−13	1.21 × 10^4	2.79 × 10^−12	6.72 × 10^0	6.72 × 10^2
320.0	1.32 × 10^−12	1.53 × 10^2	4.46 × 10^−13	1.84 × 10^3	2.90 × 10^−12	6.95 × 10^−1	6.95 × 10^1

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In order to test the competition of the H₂O⋯HO₂ + SO₂ reaction with the primary loss mechanism of SO₂ (such as the SO₂ + HO reaction), it is of great importance to discuss the rate ratio between H₂O⋯HO₂ + SO₂ and SO₂ + OH, as shown in eqn (24).

Here, k_R3 is the rate constant of the SO₂ + OH reaction, which is obtained from Table 3. The rate ratio v_a1/v_R3 depends on the H₂O, OH, and HO₂ concentrations in the atmosphere, At 298 K, when the concentration of water is at a relative humidity of 100%, OH is 7.73 × 10¹⁷ and 1 × 10⁶ molecules per cm³, respectively, and the gas-phase concentration of HO₂ is 3 × 10⁸ molecules per cm³, the H₂O⋯HO₂ + SO₂ reaction can compete with the SO₂ + HO reaction, because the rate ratio v_a1/v_R3 is about 6.20 at 298 K. Consequently, the hydrogen atom transfer processes of the H₂O⋯HO₂ + SO₂ reaction are more obvious in the atmosphere during the day than those of the SO₂ + HO reaction. Meanwhile, when the OH concentration decreases to 1 × 10⁴ molecules per cm³ during the night,⁸⁰ the H₂O⋯HO₂ + SO₂ reaction can compete well with the SO₂ + HO reaction because the rate ratio v_a1/v_R3 is about 6.20 × 10² at 298 K. Thus, the H₂O⋯HO₂ + SO₂ reaction can also make a contribution to the sink of SO₂ during the night under the conditions of OH (10⁴ molecules per cm³), HO₂ (molecules per cm³), and H₂O (molecules per cm³).

4. Summary and conclusions

In this article, the hydrogen atom transfer processes of the HO₂ + SO₂ → HOSO + ³O₂ reaction without and with (H₂O)_n (n = 1–3) have been investigated using the CCSD(T)/CBS//M06-2X/aug-cc-pVTZ method and canonical variational transition state theory with small curvature tunneling (CVT/SCT). The calculated results show that, for the (H₂O)_n (n = 1–3) + HO₂ + SO₂ reaction, the main entrance channel is the reaction of the HO₂⋯(H₂O)_n (n = 1–3) complex with SO₂. Additionally, H₂O exerts the strongest catalytic influence in the hydrogen atom transfer processes of the HO₂ + SO₂ → HOSO + ³O₂ reaction compared with (H₂O)₂ and (H₂O)₃, which is due to the fact that the effective rate constant of the HO₂⋯H₂O + SO₂ reaction is 2–3 and 4–6 orders of magnitude larger than the corresponding rate constants of the HO₂⋯(H₂O)₂ + SO₂ and HO₂⋯(H₂O)₃ + SO₂ reactions, respectively.

In the gas-phase reactions of the atmosphere, the importance of the HO₂⋯H₂O + SO₂ reaction depends on its competition with the HO₂ + SO₂ → HOSO + ³O₂ reaction. We show that, at 0 km altitude within the temperature range of 275–320 K, the effective rate constant of the HO₂⋯H₂O + SO₂ reaction was 2–3 orders of magnitude larger than that of the HO₂ + SO₂ → HOSO + ³O₂ reaction without a catalyst, indicating that the water monomer plays an obvious positive role in increasing the rate of the HO₂ + SO₂ → HOSO + ³O₂ reaction. Compared with the primary loss mechanism of HO₂ radicals and SO₂, the HO₂ + SO₂ → HOSO + ³O₂ reaction with H₂O cannot be neglected with its rate constant close to the rate constant of those reactions of HO₂ + HO₂, HO₂ + NO₃ and SO₂ + HO. In addition, with the altitude increase, for the formation of HOSO + ³O₂, the contribution of H₂O decreases from 99.98% to 27.27% at the lower relative concentration of water, which indicates that the HO₂⋯H₂O + SO₂ reaction cannot compete with the primary loss processes of HO₂ radicals and SO₂ at higher altitudes.

The findings of the present work not only show a specific (H₂O)_n (n = 1–3) catalyzed reaction for its reaction mechanism and kinetics, but also show that H₂O can obviously promote the hydrogen atom transfer processes of the HO₂ + SO₂ → HOSO + ³O₂ reaction. Thus, the present investigation has wide applications in the hydrogen atom transfer processes of atmospheric processes, such as HO₂ + HO₂ and HO₂ + NO₃ reactions.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 21603132, 41805107), the Project of Education Department in Shaanxi (18JK0147) and the Funds of Research Programs of Shaanxi University of Technology (No. SLGQD13(2)-3, SLGQD13(2)-4).

References

1. C. N. Hewitt, Atmos. Environ., 2011, 35, 1155–1170.
2. B. Long, J. L. Bao and D. G. Truhlar, Phys. Chem. Chem. Phys., 2017, 19, 8091–8100.
3. X. Chen, C. Tao, L. Zhong, Y. Gao, W. Yao and S. Li, Chem. Phys. Lett., 2014, 608, 272–276.
4. J. Liu, S. Fang, Z. Wang, W. Yi, F. M. Tao and J. Liu, Environ. Sci. Technol., 2015, 49, 13112–13120.
5. M. R. Rd, T. Berndt, M. Sipilä, P. Paasonen, T. Petäjä, S. Kim, T. Kurtén, F. Stratmann, V. M. Kerminen and M. Kulmala, Nature, 2012, 488, 193.
6. R. Zhang, A. Khalizov, L. Wang, M. Hu and W. Xu, Chem. Rev., 2011, 112, 1957–2011.
7. N. Sang, Y. Yun and G. Y. Yao, et al., Toxicol. Sci., 2011, 124, 400–413.
8. Y. Tang, G. S. Tyndall and J. J. Orlando, J. Phys. Chem. A, 2010, 114, 369–378.
9. P. D. Lightfoot, R. A. Cox, J. N. Crowley, M. Destriau, G. D. Hayman, M. E. Jenkin, G. K. Moortgat and F. Zabel, Atmos. Environ., 1992, 26, 1805–1961.
10. D. Stone, L. K. Whalley and D. E. Heard, Chem. Soc. Rev., 2012, 41, 6348–6404.
11. B. Wang and H. Hou, Phys. Lett., 2005, 410, 235–241.
12. S. E. Wheeler, J. Phys. Chem. A, 2009, 113, 6779–6788.
13. A. Lesar and A. Tavčar, J. Phys. Chem. A, 2011, 115, 11008–11015.
14. A. J. Frank, M. Sadílek, J. G. Ferrier and F. E. Tureček, J. Am. Chem. Soc., 1996, 118, 11321–11322.
15. E. Isoniemi, L. Khriachtchev, J. Lundell and M. Räsänen, J. Mol. Struct., 2001, 563, 261–265.
16. R. J. Boyd, A. Gupta, R. F. Langler, S. P. Lownie and J. A. Pincock, Can. J. Chem., 1980, 58, 331–338.
17. M. C. Mccarthy, V. Lattanzi, J. Oscar Martinez and F. E. Tureček, J. Phys. Chem. Lett., 2013, 4, 4074–4079.
18. Y. Q. Sun, X. Wang, F. Y. Bai and X. M. Pan, Environ. Chem., 2017, 14, 19–30.
19. W. A. Payne, L. J. Stief and D. D. Davis, J. Am. Chem. Soc., 1973, 95, 7614–7619.
20. J. P. Burrows, D. I. Cliff, G. W. Harris, B. A. Thrush and J. P. T. Wilkinson, Proc. R. Soc. London, Ser. A, 1979, 368, 463–481.
21. R. J. Buszek, J. S. Francisco and J. M. Anglada, Int. Rev. Phys. Chem., 2011, 30, 335–369.
22. T. Zhang, W. Wang, P. Zhang, J. Lü and Y. Zhang, Phys. Chem. Chem. Phys., 2011, 13, 20794–20805.
23. R. J. Buszek, M. Torrentsucarrat, J. M. Anglada and J. S. Francisco, J. Phys. Chem. A, 2012, 116, 5821–5829.
24. J. M. Anglada and J. Gonzalez, ChemPhysChem, 2009, 10, 3034–3045.
25. J. Gonzalez and J. M. Anglada, J. Phys. Chem. A, 2010, 114, 9151–9162.
26. J. Gonzalez, J. M. Anglada, R. J. Buszek and J. S. Francisco, J. Am. Chem. Soc., 2011, 133, 3345–3353.
27. J. C. Hansen and J. S. Francisco, ChemPhysChem, 2002, 3, 833–840.
28. V. Vaida, H. G. Kjaergaard, P. E. Hintze and D. J. Donaldson, Science, 2003, 299, 1566–1568.
29. N. Kanno, K. Tonokura, A. Tezaki and M. Koshi, J. Phys. Chem. A, 2005, 109, 3153–3158.
30. K. Suma, Y. Sumiyoshi and Y. Endo, Science, 2006, 311, 1278–1281.
31. N. Butkovskaya, M. T. Rayez, J. C. Rayez, A. Kukui and G. L. Bras, J. Phys. Chem. A, 2009, 113, 11327–11342.
32. S. Aloisio, J. S. Francisco and R. R. Friedl, J. Phys. Chem. A, 2000, 104, 6597–6601.
33. J. Liu, S. Fang, W. Liu, A. Kukui and G. L. Bras, J. Phys. Chem. A, 2015, 119, 102–111.
34. T. L. Tarbuck and G. L. Richmond, J. Am. Chem. Soc., 2005, 127, 16806–16807.
35. T. L. Tarbuck and G. L. Richmond, J. Am. Chem. Soc., 2006, 128, 3256–3267.
36. S. T. Ota and G. L. Richmond, J. Am. Chem. Soc., 2011, 133, 7497–7508.
37. K. Matsumura, F. J. Lovas and R. D. Suenram, J. Chem. Phys., 1989, 91, 5887–5894.
38. T. Zhang, C. Yang, X. Feng, J. Kang, L. Song, Y. Lu, Z. Wang, Q. Xu, W. Wang and Z. Wang, Phys. Chem. Chem. Phys., 2016, 18, 17414–17427.
39. L. P. Viegas and A. J. C. Varandas, Eur. Phys. J. D, 2016, 70, 48.
40. Y. P. Zhao and Y. P. Zeng, Chin. J. Struct. Chem., 2010, 29, 499–508.
41. R. Steudel and Y. Steudel, Eur. J. Inorg. Chem., 2009, 2009, 1393–1405.
42. B. Du and W. Zhang, Comput. Theor. Chem., 2014, 1049, 90–96.
43. E. Vöhringermartinez, B. Hansmann and H. Hernandezsoto, Science, 2007, 315, 497–501.
44. R. Wang, J. Kang and S. Zhang, Comput. Theor. Chem., 2017, 1110, 25–34.
45. P. Kumar, P. Biswas and B. Bandyopadhyay, Phys. Chem. Chem. Phys., 2016, 18, 27728–27732.
46. B. Du and W. Zhang, Comput. Theor. Chem., 2015, 1069, 77–85.
47. L. P. Viegas and A. J. Varandas, J. Phys. Chem. B, 2016, 120, 1560–1568.
48. J. M. Anglada, G. J. Hoffman, L. V. Slipchenko, M. M. Costa, M. F. Ruizlópez and J. S. Francisco, J. Phys. Chem. A, 2013, 117, 10381–10396.
49. P. Miró and C. J. Cramer, Phys. Chem. Chem. Phys., 2013, 15, 1837–1843.
50. J. Hernández-Rojas and D. J. Wales, Chem. Phys., 2014, 444, 23–29.
51. E. A. Cobar, P. R. Horn, R. G. Bergman and M. Headgordon, Phys. Chem. Chem. Phys., 2012, 14, 15328–15339.
52. J. A. Anderson, K. Crager, L. Fedoroff and G. S. Tschumper, J. Chem. Phys., 2004, 121, 11023–11029.
53. M. E. Dunn, E. K. Pokon and G. C. Shields, J. Am. Chem. Soc., 2004, 126, 2647–2653.
54. N. Goldman, R. S. Fellers and C. Leforestier, J. Phys. Chem. A, 2001, 105, 515–519.
55. Y. S. Lee, S. A. Kucharski and R. J. Bartlett, J. Chem. Phys., 1985, 82, 5761.
56. C. Gonzalez and H. B. Schlegel, J. Chem. Phys., 1989, 90, 2154–2161.
57. Y. S. Lee, S. A. Kucharski and R. J. Bartlett, J. Chem. Phys., 1984, 81, 5906–5912.
58. A. J. C. Varandas and F. N. N. Pansini, J. Chem. Phys., 2014, 141, 224113.
59. B. C. Garrett and D. G. Truhlar, J. Chem. Phys., 1979, 70, 1593–1598.
60. B. C. Garrett and D. G. Truhlar, Chem. Informationsdienst, 1979, 10, 4534–4548.
61. B. C. Garrett, D. G. Truhlar, R. S. Grev and A. W. Magnuson, J. Chem. Phys., 1980, 84, 1730–1748.
62. D. H. Lu, T. N. Truong, S. G. Melissas, C. Lynch, Y. P. Liu and B. C. Garrett, Comput. Phys. Commun., 1992, 71, 235–262.
63. Y. P. Liu, G. C. Lynch, T. N. Truong, D. H. Lu, D. G. Truhlar and B. C. Garrettet, J. Am. Chem. Soc., 1993, 6, 2408–2415.
64. S. W. Zhang and N. T. Truong, VKLab, version 1.0, University of Utah, Salt Lake City, 2001.
65. M. J. Frisch, G. W. Trucks and J. A. Pople, Gaussian 09, Revision A.01, Gaussian Inc., Pittsburgh, PA, 2009.
66. R. Atkinson, D. L. Baulch and R. A. Cox, Atmos. Chem. Phys., 2004, 3, 1461–1738.
67. R. J. Buszek, J. R. Barker and J. S. Francisco, J. Phys. Chem. A, 2012, 116, 4712–4719.
68. X. Chen, Y. F. Zhao, L. S. Wang and J. Li, Comput. Theor. Chem., 2017, 1107, 57–65.
69. Y. Zhao, X. Chen and J. Li, Nano Res., 2017, 10, 3407–3420.
70. B. Long, X. F. Tan, Z. W. Long, Y. B. Wang, D. S. Ren and W. J. Zhang, J. Phys. Chem. A, 2011, 115, 6559–6567.
71. T. Zhang, R. Wang, W. Wang, S. Min, Q. Xu, Z. Wang, C. Zhao and Z. Wang, Comput. Theor. Chem., 2014, 1045, 135–144.
72. J. Gonzalez, M. Torrent-Sucarrat and J. M. Anglada, Phys. Chem. Chem. Phys., 2010, 12, 2116–2125.
73. T. Zhang, W. Wang, C. Li and Y. Du, RSC Adv., 2013, 3, 7381–7391.
74. T. Zhang, R. Wang, H. Chen, S. Min, Z. Wang, C. Zhao, Q. Xu, L. Jin, W. Wang and Z. Wang, Phys. Chem. Chem. Phys., 2015, 17, 15046–15055.
75. S. Mallick, S. Sarkar, P. Kumar and B. Bandyopadhyay, J. Phys. Chem. A, 2018, 122, 350–363.
76. A. Mellouki, R. K. Talukdar and A. M. R. P. Bopegedera, Int. J. Chem. Kinet., 1993, 25, 25–39.
77. D. Stone and D. M. Rowley, Phys. Chem. Chem. Phys., 2005, 7, 2156–2163.
78. E. Becker, M. M. Rahman and R. N. Schindler, Ber. Bunsenges. Phys. Chem., 1992, 96, 776–783.
79. D. Fulle, H. F. Hamann and H. Hippler, Phys. Chem. Chem. Phys., 1999, 1, 2695–2702.
80. M. A. H. Khan, M. J. Ashfold, G. Nickless, D. Martin, L. A. Watson, P. D. Hamer, R. P. Wayne, C. E. Canosa-Mas and D. E. Shallcross, Atmos. Sci. Lett., 2008, 9, 140–146.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9ra00169g

‡ Qiuyue Yao, Mingjie Wen, Shaobo Tian, and Yan Wang contributed equally to this work.

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