2.1 Geometry optimizations and thermochemical and charge analysis

As the substrate in this study is a radical anion, all stationary points in the energy surface were optimized using density functional theory (DFT) based on the UM06-2X density functional (Zhao and Truhlar, 2008) and the aug-cc-pVTZ basis set (Dunning Jr. et al., 2001), setting the charge to −1 and the spin multiplicity to 2. The use of UM06-2X implies the use of unrestricted wave functions to describe the quantum state of the system. Spin contamination often arises from unrestricted DFT calculations and it is not guaranteed that the electronic states from these calculations are eigenstates of the ${\widehat{S}}^{\mathrm{2}}$ operator. The spin contamination was then evaluated for all electronic states as $\mathrm{\Delta }S=\langle {\widehat{S}}^{\mathrm{2}}\rangle -\langle {\widehat{S}}^{\mathrm{2}}{\rangle }_{\mathrm{ideal}}$, where $\langle {\widehat{S}}^{\mathrm{2}}\rangle$ is the actual value of the expectation value of the ${\widehat{S}}^{\mathrm{2}}$ operator from DFT calculations and $\langle {\widehat{S}}^{\mathrm{2}}{\rangle }_{\mathrm{ideal}}$ is the ideal expectation value. For systems explored in this study, $\langle {\widehat{S}}^{\mathrm{2}}{\rangle }_{\mathrm{ideal}}=\mathrm{0.75}$.

The UM06-2X functional has successfully proven to be adequate for reactions involving transition state (TS) configurations (Elm et al., 2012, 2013a, b). Harmonic vibrational frequency analyses on the optimized structures were performed (at 298 K and 1 atm) using the UM06-2X/aug-cc-pVTZ method under the harmonic oscillator–rigid rotor approximation. These calculations ensured that the stationary points obtained were minima or TS and, also, provided the thermal contributions to the Gibbs free energy and the enthalpy.

Transition-state structures were initially located by scanning the reactants configurations. The best TS guesses from the scans were then refined using the synchronous transit quasi-Newton method (Peng et al., 1996), and the final TS structures underwent internal reaction coordinate calculations (Fukui, 1981) to ensure they connected the reactants to desired products.

The electronic energies of the UM06-2X/aug-cc-pVTZ optimized geometries were corrected using the UCCSD(T) method (Purvis and Bartlett, 1982) in conjunction with the aug-cc-pVTZ basis set. The Gibbs free energies, G, of all relevant species were then calculated as

where E_UCCSD(T) is the electronic energy calculated using the UCCSD(T)/aug-cc-pVTZ method and G_therm is the thermal contribution to the Gibbs free energy, calculated at the UM06-2X/aug-cc-pVTZ level of theory. All geometry optimizations, harmonic vibrational frequency analysis, and electronic energy correction calculations were carried out in the Gaussian 09 package (Frisch et al., 2013).

To analyze the distribution of the excess electronic charge over different species and fragments in the optimized systems, we used the atoms-in-molecules charge partitioning method presented by Bader (Bader, 1998). This is an intuitive way of dividing the molecules of a system into atoms, which are purely defined in terms of electronic charge density. The Bader charge partitioning assumes that the charge density between atoms of a molecular system reaches a minimum, which is an ideal place to separate atoms from each other. As input, this method requires electronic density and nuclear coordinates from electronic structure calculations. We used the approach implemented in the algorithm developed by Henkelman and co-workers, which has been shown to work well both for charged and water-containing systems (Tang et al., 2009; Bork et al., 2011; Henkelman et al., 2006).

2.2 Reactions kinetics

Regardless of the presence of water, the reaction between O₂SOO⁻ and O₃ begins by forming different van der Waals pre-reactive intermediates, depending on the orientation of the reactants at impact. The pre-reactive intermediate could either decompose to different species or react further through a transition state configuration to form new products:

The traditional approach to determine the rate constant of Reaction (R1) relies on the steady-state approximation and leads to the following equation:

where k_coll is the collision frequency for O₂SOO⁻–O₃ collisions, k_evap is the rate constant for the evaporation of the pre-reactive intermediate back to initial reactants, and k_reac is the unimolecular rate constant for the reaction of the pre-reactive intermediate to the products. Moreover, assuming that k_evap≫k_reac, the rate constant of Reaction (R1) becomes k=K_eqk_reac over a range of temperatures, with K_eq being the equilibrium constant of formation of the pre-reactive intermediate from the reactants. This consideration is, however, not valid for reactions with a submerged barrier, since the pre-reactive intermediate seldom thermally equilibrates. For such reactions, a two-transition-state theory has been introduced, treating two distinct transition state bottlenecks that define the net reactive flux (Klippenstein et al., 1988; Georgievskii and Klippenstein, 2005; Greenwald et al., 2005). The first bottleneck, the “outer” transition state, occurs in the association of the initial reactants to form the pre-reactive intermediate, whereas the second bottleneck, the “inner” transition state, occurs in the transformation of the pre-reactive intermediate to the products. Based on this theory, the overall rate constant (k) for a reaction channel is expressed in terms of the outer (k_out) and inner (k_in) rate constants as

The outer transition state is treated by the long-range transition state theory approach (Georgievskii and Klippenstein, 2005), while the inner transition state is resolved by the transition state theory. It was shown that for interactions between ions and neutral molecules, due to their long-range attraction, the collision cross section is larger than would be measured from the physical dimensions of the colliding species (Kupiainen-Määttä et al., 2013). To account for this phenomenon, the outer rate constant was determined from the ion–dipole parametrization of Su and Chesnavich, who performed trajectory simulations of collisions between a point charge and a rigidly rotating molecule (Su and Chesnavich, 1982). This parametrization is equivalent to a Langevin capture rate constant (k_L) scaled by a temperature-dependent term and was found to provide good agreement with experiments (Kupiainen-Määttä et al., 2013). It is given as

where $k_{\mathrm{L}}=q{\mathit{\mu }}^{-\mathrm{1}/\mathrm{2}}(\mathit{\pi }\mathit{\alpha }/{\mathit{\epsilon }}_{\mathrm{0}}{)}^{\mathrm{1}/\mathrm{2}}$, $x={\mathit{\mu }}_{\mathrm{D}}/(\mathrm{8}\mathit{\pi }{\mathit{\epsilon }}_{\mathrm{0}}\mathit{\alpha }{k}_{\mathrm{B}}T{)}^{\mathrm{1}/\mathrm{2}}$, q is the charge of the ion, μ is the reduced mass of the colliding species, ε₀ is the vacuum permittivity, α and μ_D are the polarizability and dipole moment of the neutral molecule (ozone), k_B is Boltzmann's constant, and T is the absolute temperature. The inner rate constant can be written as

where ΔG^‡ is the Gibbs free energy barrier separating the pre-reactive intermediate and the products, h is the Planck's constant, R is the molar gas constant, and c⁰ is the standard gas-phase concentration. The constants and variables in Eqs. (4) and (5) are given in the centimeter-gram-second (CGS) system of units and International System (SI) units, respectively. Details on these units are given in the Supplement.

Figure 1Optimized structures of the most stable intermediates in the O₂SOO⁻ + O₃ reaction (a) in the absence and (b) in the presence of a single water molecule. Optimizations were performed at the UM06-2X/aug-cc-pVTZ level of theory. Lengths (in Å) of some descriptive bonds are indicated. The color coding is yellow for sulfur, red for oxygen, and white for hydrogen.

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3.1 Cluster formation and decomposition of O₂SOO⁻ ⋯ (H₂O)₀₋₁

As O₃ approaches O₂SOO⁻ ⋯ (H₂O)₀₋₁ towards the oxygen atoms of the peroxy fragment or the oxygen atom of the SO₂ moiety, the immediate outcome of O₂SOO⁻ ⋯ (H₂O)₀₋₁ and O₃ collisions is the formation of the van der Waals O₃ ⋯ O₂SOO⁻ ⋯ (H₂O)₀₋₁ complex in which O₃ interacts with O₂SOO⁻. Among the different stable configurations found upon optimization, we solely report the most stable one with respect to the Gibbs free energy, which is henceforth denoted RC1 and RCW1 for the unhydrated and monohydrated reactions, respectively, shown in Fig. 1. Exploration of the RC1 and RCW1 structures reveals that O₂SOO⁻ ⋯ (H₂O)₀₋₁ basically keeps its configuration upon clustering with O₃. The spin contaminations for RC1 and RCW1 are negligible, being 0.0086 and 0.0081, respectively. The electronic energies of formation of RC1 and RCW1 are −5.1 and −4.6 kcal mol⁻¹, respectively. Despite the fact that these complexes may form at 0 K, the Gibbs free energies of their formation under atmospheric pressure and 298 K (4.5 and 4.7 kcal mol⁻¹, respectively) indicate that their formation is endergonic under atmospherically relevant conditions. These Gibbs free energy values indicate that, if formed, these complexes would not live long and will rather decompose either to initial reactants or to different species. Hence, the Gibbs free energies of formation of RC1 and RCW1 define the lowest states at which O₂SOO⁻ can interact with O₃ to allow electron transfer and O₂S–OO decomposition, and thus represent the energy barrier towards O₂ + SO₂ + ${\mathrm{O}}_{\mathrm{3}}^{-}$ formation. Inspecting the vibrational modes of RC1 and RCW1, two vibrations are found that would clearly lead to the dissociation of O₂SOO⁻ within the complex. The analysis of the charge distribution over O₃ ⋯ O₂SOO⁻ ⋯ (H₂O)₀₋₁ shows that the extra electron initially located on O₂SOO in the reactants has migrated to the O₃ molecule in the van der Waals product complex, as can be observed in Fig. S2. This is as expected, given the high electronegativity of O₃ relative to that of O₂ and SO₂ (Rothe et al., 1975). The charge distribution over the different atoms of the optimized complex is weakly affected by the presence of water, as previously demonstrated by Bork and co-workers (Bork et al., 2011).

Table 1Electronic energies (ΔE), enthalpies (ΔH_298 K), and Gibbs free energies (ΔG_298 K) of the different states in the O₂SOO⁻ + O₃ reaction both in the absence and in the presence of water, calculated relative to the energy of initial reactants at the UCCSD(T)/aug-cc-pVTZ//UM06-2X/aug-cc-pVTZ level of theory.

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The most likely fates of RC1 and RCW1 are, therefore, decomposition into O₂, SO₂, and ${\mathrm{O}}_{\mathrm{3}}^{-}$ as follows:

The numerical values of the formation energies of all intermediate species in Reaction (R2) are given in Table 1, and the energy surfaces are plotted in Fig. 2. RC1 and RCW1 decompositions are highly exergonic at 298 K, occurring with −18.1 and −16.7 kcal mol⁻¹ Gibbs free energy changes, respectively. These processes are, therefore, likely to occur in the atmosphere upon formation of O₃ ⋯ O₂SOO⁻ ⋯ (H₂O)₀₋₁.

The limiting step in Reaction (R2) is the formation of RC1 and RCW1, whose formation energies indicated above can then be considered as the only barrier to the formation of O₂ + SO₂ + ${\mathrm{O}}_{\mathrm{3}}^{-}$. This leads to overall rate constants (according to Eq. 5) of $\mathrm{1.4}\times {\mathrm{10}}^{-\mathrm{10}}$ and $\mathrm{9.9}\times {\mathrm{10}}^{-\mathrm{11}}$ cm³ molecule⁻¹ s⁻¹ at 298 K for the unhydrated and monohydrated reactions, respectively. Both reactions are, in principle, collision-limited and the effect of hydration on the kinetics is found to be negligible. The atmospheric relevance of Reaction (R2) has been determined already (Bork et al., 2012, 2013; Enghoff et al., 2012).

Figure 2Formation Gibbs free energies of the most stable intermediate species in the O₂SOO⁻ + O₃ reaction in the absence and in the presence of water. “W” is the shorthand notation for water. RC1, RC2, TS2, RCW1, RCW2, and TSW2 structures are shown in Fig. 1. P1 = O₂ + SO₂ + ${\mathrm{O}}_{\mathrm{3}}^{-}$, P2 = ${\mathrm{SO}}_{\mathrm{3}}^{-}$ + 2O₂, PW1 = O₂ + SO₂ + ${\mathrm{O}}_{\mathrm{3}}^{-}$ + H₂O, and PW2 = ${\mathrm{SO}}_{\mathrm{3}}^{-}$ ⋯ H₂O + 2O₂. Calculations were performed at the UCCSD(T)/aug-cc-pVTZ//UM06-2X/aug-cc-pVTZ level of theory.

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Figure 3Representation of the spin density (in blue color) on intermediate structures in the O₂SOO⁻ + O₃ reaction. The spin density clearly indicates that the extra electron is progressively distributed over all the atoms from (a) the pre-reactive complex through (b) the transition state to (c) the product complex.

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3.2 O₂SOO⁻ ⋯ (H₂O)₀₋₁ reaction with O₃

When O₃ approaches O₂SOO⁻ ⋯ (H₂O)₀₋₁ from the sulfur atom side, the formation of a more stable cluster than found above prevails. The incoming O₃ molecule strongly interacts with O₂SOO⁻ ⋯ (H₂O)₀₋₁ by forming a coordination bond with the sulfur atom and hereby inducing the ejection of the O₂ molecule that remains in interaction with the remainder of the system. This process leads to the formation of the O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ ⋯ (H₂O)₀₋₁ complex which further transforms, through an intramolecular SO₂ oxidation, into ${\mathrm{SO}}_{\mathrm{3}}^{-}$ ⋯ (H₂O)₀₋₁ + 2O₂ according to the following equation:

The configurations of the most stable intermediate structures in Reaction (R3) are given in Fig. 1. The charge analysis on this system indicates that the electronic charge on the pre-reactive complex is essentially on two oxygen atoms of the O₃ moiety that is coordinated to SO₂ as can be seen in Fig. 3a. The net charge on these two oxygen atoms is 1.04e, whereas the net charge on the free O₂ molecule is 0.01e. The latter value shows that the O₂ molecule formed in the pre-reactive complex has no unpaired electrons, and hence is a singlet. Although the charge is still on the O₃ moiety in the transition state configuration, it is mostly located on the oxygen atom bound to sulfur (Fig. 3b). The net charge on the two outer oxygen atoms of the O₃ moiety in the transition state has substantially decreased to 0.30e, while the charge on the free O₂ molecule has slightly increased to 0.04e. The strong decrease in the charge of the two outer oxygen atoms of O₃ from the pre-reactive complex to the transition state suggests that the O₂ molecule to form in the product will likely be a singlet. In the products (Fig. 3c), the old free O₂ molecule has a net charge of 1.99e, whereas the charge on the newly formed O₂ is 0.06e. The 1.99e charge on the old free O₂ molecule indicates the presence of unpaired electrons in its configuration, meaning that the singlet O₂ has been transformed into a triplet. This clearly shows that a spin flip has occurred in O₂ during further optimization of the products. The newly formed O₂ with 0.06e charge is obviously a singlet. This analysis shows for the unhydrated system that the singlet O₂ initially formed in the pre-reactive complex transforms into a triplet in the product state, while a new singlet O₂ is also formed. In the monohydrated system, the singlet O₂ initially formed in the pre-reactive complex remains as a singlet in the product state and a triplet O₂ is also released. Overall, the two forms of O₂ (singlet and triplet) are formed in the studied reaction, despite following different mechanisms. Although the water molecule in the monohydrated system does not retain any electric charge, it most likely stabilizes the initially formed singlet O₂ and prevents the spin flip.

Though the necessity to determine the electronic structure of the O₂ molecule in the singlet state (¹Δ_g) has been demonstrated to be useful (Buttar and Hirst, 1994), obtaining a reliable electronic energy for O₂(¹Δ_g) is difficult (Drougas and Kosmas, 2005). An alternative approach to determine this energy is to add the experimental energy spacing (22.5 kcal mol⁻¹) between triplet and singlet states of O₂ to the computed electronic energy of the triplet O₂ (Schweitzer and Schmidt, 2003; Drougas and Kosmas, 2005). We used this approach to determine the energies of the products of Reaction (R3). The numerical values of the formation energies of all intermediate species in Reaction (R3) are thus given in Table 1 and the energy surfaces are plotted in Fig. 2. The most stable optimized structures of O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ ⋯ (H₂O)₀₋₁ according to our calculations are denoted as RC2 and RCW2 for the unhydrated and monohydrated systems, respectively, and are shown in Fig. 1. Regardless of the presence of water, the O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ configuration results from O₂ being switched by O₃ in the O₂SOO⁻ molecular ion. In the optimized O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ structure, the O3 atom (labeled in Fig. 1) of O₃ points towards the S atom of O₂SOO⁻, forming S–O3 bonds at distances of 1.90 and 1.87 Å in the absence and in the presence of water, respectively. These bonds are coordination bonds in nature since the S–O covalent bond is, e.g., 1.43 Å in SO₃ and 1.4 Å in H₂SO₄. The S–O3 coordination bond distances in RC2 and RCW2 are shorter by 0.04 and 0.03 Å than O₂S–OO⁻ bond distances in unhydrated and monohydrated O₂SOO⁻ forms. This indicates stronger interaction between O₃ and SO₂, and hence higher stability of O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ relative to O₂SOO⁻.

The formations of RC2 and RCW2 are highly exergonic, with Gibbs free energy changes at 298 K of −14.7 and −12.4 kcal mol⁻¹, respectively. These values, with corresponding electronic energies and enthalpies, are shown in Table 1. These Gibbs free energy changes for the formation of RC2 and RCW2 are about 19 kcal mol⁻¹ lower than those of RC1 and RCW1 at similar conditions, indicating the higher stability of RC2 and RCW2 and the highly favorable switching reaction at ambient conditions. SO₂ oxidation can readily occur within the O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ ⋯ (H₂O)₀₋₁ cluster and lead to ${\mathrm{SO}}_{\mathrm{3}}^{-}$ formation. In principle, to form the products of Reaction (R3), the O3 atom of the O₃ moiety in O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ ⋯ (H₂O)₀₋₁ is transferred through transition state configurations, to SO₂ and forms ${\mathrm{SO}}_{\mathrm{3}}^{-}$ followed by the ejection of the O₂ molecule. The transition states are denoted TS2 and TSW2 for the unhydrated and monohydrated systems, respectively, and their structures are presented in Fig. 1. While RC2 and RCW2 are formed with similar Gibbs free energies within 2 kcal mol⁻¹ difference, the formation Gibbs free energies of their transition states at similar conditions greatly differ. TS2 is located at 10 kcal mol⁻¹ Gibbs free energy above RC2, while TSW2 is located at −4 kcal mol⁻¹ below RCW2. It is speculated that the low energy barrier in the monohydrated reaction is the result of a strong stabilization of the transition state due to hydration, with the S–O3 bonds in RCW2 and TSW2 shorter by ∼0.03 Å than in RC2 and TS2. Another reason for this substantial drop in the energy barrier is the difference in the electronic configurations of the two outer oxygen atoms of the O₃ moiety in the two transition states that form O₂ with different multiplicities in the products.

Based on the TS2 energy, the unimolecular decomposition of O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ at 298 K in the absence of water was found to occur at a rate constant of 3.1×10⁵ s⁻¹, corresponding to an atmospheric lifetime of 3.3 µs. Both this short lifetime and the negative energy barrier of the monohydrated reaction indicate that O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ would not live long enough to experience collisions with other atmospheric oxidants. It should be noted that few to no collisions with nitrogen can, however, be achieved. It follows that the most likely outcome of O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ is decomposition to the products of Reaction (R3), which are formed with about −46 kcal mol⁻¹ overall Gibbs free energy at 298 K. The net reaction is an ${\mathrm{O}}_{\mathrm{2}}^{-}$-initiated SO₂ oxidation to ${\mathrm{SO}}_{\mathrm{3}}^{-}$ by O₃.

The spin contamination for electronic states in Reaction (R3) is quite significant, being 1.0122, 1.4666, and 2.0374 for the pre-reactive complex, transition state, and product, respectively, and is almost insensitive to the presence of water. The actual values of the expectation values of the ${\widehat{S}}^{\mathrm{2}}$ operator for all electronic states obtained from our calculations are given in the Supplement, along with their Cartesian coordinates. The high values of spin contamination likely reflect the formation of O₂ with different multiplicities within the system. As the charge analysis indicates, starting with singlet O₂ in the pre-reactive complexes of Reaction (R3), both singlet and triplet O₂ are formed in the final products.

The overall rate constants of Reaction (R3), determined at 298 K using Eq. (3), are $\mathrm{1.3}\times {\mathrm{10}}^{-\mathrm{14}}$ and $\mathrm{8.0}\times {\mathrm{10}}^{-\mathrm{10}}$ cm³ molecule⁻¹ s⁻¹ for the unhydrated and monohydrated reactions, respectively. The values of the different components (k_out and k_in) of these rate constants are listed in Table S1 of the Supplement. It is observed from Table S1 that the inner transition state provides the dominant bottleneck to the rate constant of the unhydrated reaction, whereas the outer transition state provides the dominant bottleneck to the rate constant of the monohydrated reaction.

The effective effect of water on the rate constant can be evaluated by taking into account the stability of O₂SOO⁻ ⋯ H₂O (which is formed at the entrance channel of the reaction in the presence of water before colliding with O₃) and the equilibrium vapor pressure of water. Starting from the definition of the reaction rate for the hydrated reaction,

where k_(R3w) is the overall rate constant for the hydrated reaction, determined using Eq. (3), and $k_{\left(\mathrm{R}\mathrm{3}\mathrm{w}\right)}^{\mathrm{eff}}$ is the effective reaction rate constant calculated as $k_{\left(\mathrm{R}\mathrm{3}\mathrm{w}\right)}^{\mathrm{eff}}=k_{\left(\mathrm{R}\mathrm{3}\mathrm{w}\right)}\times K_{\mathrm{eq}}\times p_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$. K_eq is the equilibrium constant for the O₂SOO⁻ + H₂O ↔ O₂SOO⁻ ⋯ H₂O reaction and $p_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ the actual water vapor pressure. Details on the determination of K_eq and $p_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ are given in the Supplement.

At 298 K and 50 % relative humidity, the effective rate constant of the monohydrated reaction is $\mathrm{1.7}\times {\mathrm{10}}^{-\mathrm{10}}$ cm³ molecule⁻¹ s⁻¹, 4 orders of magnitude higher than the rate constant of the unhydrated reaction. Therefore, water plays a catalytic role in the kinetics of Reaction (R3). The net rate constant of Reaction (R3) can be obtained by weighing the rate constants of the unhydrated and monohydrated reactions to corresponding equilibrium concentrations of O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ hydrates. Using the law of mass action, we find that O₂ ⋯ O₂S–O${}_{\mathrm{3}}^{-}$ mostly exists as a dry species, constituting 77 % of the total population, whereas the monohydrated species forms 23 % of the total population. The net rate constant of Reaction (R3) is then determined to be $\mathrm{4.0}\times {\mathrm{10}}^{-\mathrm{11}}$ cm³ molecule⁻¹ s⁻¹ at 298 K.

Considering only the unhydrated process of Reaction (R3), the rate constant is 4–5 orders of magnitude lower than the rate constant obtained for the SO₂ + ${\mathrm{O}}_{\mathrm{3}}^{-}$ → ${\mathrm{SO}}_{\mathrm{3}}^{-}$ + O₂ reaction (Fehsenfeld and Ferguson, 1974; Bork et al., 2012). Despite this difference, the oxidation process follows a similar mechanism to the one presented by Bork et al. for the SO₂ + ${\mathrm{O}}_{\mathrm{3}}^{-}$ → ${\mathrm{SO}}_{\mathrm{3}}^{-}$ + O₂ reaction, consisting of the oxygen transfer from O₃ to SO₂ (Bork et al., 2012). The discrepancy between the two results is associated with the effect of the presence of the O–O fragment initially coordinated to SO₂ in the current study, which tends to stabilize the O₂ ⋯ O₂S–${\mathrm{O}}_{\mathrm{3}}^{-}$ pre-reactive complex. The presence of the O–O fragment seemingly deactivates SO₂ for the upcoming O transfer from O₃ to form ${\mathrm{SO}}_{\mathrm{3}}^{-}$. However, this situation is rapidly reversed with the presence of water as the reaction becomes much faster, proceeding nearly at collision rate.

3.3 Further chemistry

In real atmospheric and ionized conditions, despite O₂ having lower electron affinity than O₃, it would likely ionize faster than O₃, owing to its much higher concentration. Considering, for example, chamber experiments, upon interaction of ionizing particles with the gas, electrons can transfer from one species to another and, e.g., ${\mathrm{O}}_{\mathrm{2}}^{-}$ can form and rapidly hydrate within 1 ns (Svensmark et al., 2007; Fahey et al., 1982). Furthermore, Fahey et al. (1982) showed that the ${\mathrm{O}}_{\mathrm{2}}^{-}$ ⋯ (H₂O)₀₋₁ association reaction with SO₂ is faster than the electron transfer from ${\mathrm{O}}_{\mathrm{2}}^{-}$ ⋯ (H₂O)₀₋₁ to O₃ (Fahey et al., 1982). This means that in an ionized environment containing O₂, O₃, and SO₂, the formation of O₂SOO⁻ resulting from SO₂ and ${\mathrm{O}}_{\mathrm{2}}^{-}$ association will happen faster than ${\mathrm{O}}_{\mathrm{3}}^{-}$ formation. O₂SOO⁻ would react thereafter with O₃ and the following stepwise process could take place:

The Gibbs free energy change of this net reaction at 298 K is about −40 kcal mol⁻¹ more negative than that of the SO₂ + ${\mathrm{O}}_{\mathrm{3}}^{-}$ →${\mathrm{SO}}_{\mathrm{3}}^{-}$ + O₂ reaction at similar conditions. Given that the intermediate steps of Reaction (R4) are significantly fast, this reaction is believed to be an important process in most environments of SO₂ ion-induced oxidation to ${\mathrm{SO}}_{\mathrm{3}}^{-}$ or more oxidized species. The limiting step in the process of Reaction (R4) is Reaction (R4c) for which an energy barrier has to be overcome before the products are released.

${\mathrm{SO}}_{\mathrm{3}}^{-}$ is an identified stable ion detected in the atmosphere and in experiments (Ehn et al., 2010; Kirkby et al., 2011, 2016). The chemical fate of ${\mathrm{SO}}_{\mathrm{3}}^{-}$ is fundamentally different from that of SO₃ that forms H₂SO₄ by hydration. Likely outcomes of ${\mathrm{SO}}_{\mathrm{3}}^{-}$ are hydrolysis, electron transfer by collision with O₃, reaction with O₂ and H₂O, and, possibly, radicals, according to the following equations:

Fehsenfeld and Ferguson showed that H₂SO₄ formation could occur in the ${\mathrm{SO}}_{\mathrm{3}}^{-}$ ⋯ H₂O cluster, releasing a free electron (Reaction R5) (Fehsenfeld and Ferguson, 1974). Owing to the high electron affinity of O₃ relative to SO₃ (Rothe et al., 1975), the electron can transfer from ${\mathrm{SO}}_{\mathrm{3}}^{-}$ to O₃ and lead to the formation of SO₃, the precursor for sulfuric acid in the atmosphere. Moreover, the free electron released and the ${\mathrm{O}}_{\mathrm{3}}^{-}$ formed in Reactions (R5) and (R6), respectively, are potential triggers of new SO₂ oxidations with implication for aerosol formation (Svensmark et al., 2007; Enghoff and Svensmark, 2008; Bork et al., 2013). Reactions (R7) and (R8) are potential outcomes for ${\mathrm{SO}}_{\mathrm{3}}^{-}$ as well, forming the highly stable ${\mathrm{HSO}}_{\mathrm{4}}^{-}$ species that would terminate the oxidation process of SO₂ initiated by a free electron. Reactions (R5)–(R8) are likely competitive processes upon ${\mathrm{SO}}_{\mathrm{3}}^{-}$ formation in the gas phase, and their different rates would determine the number of SO₂ oxidations induced by a free electron. However, they have no other fate than ${\mathrm{HSO}}_{\mathrm{4}}^{-}$ or H₂SO₄, the most oxidized sulfur species in the atmosphere, which both share many properties and play a central role in atmospheric particle formation.

Experimental studies have shown that in atmospheres heavily enriched in SO₂ and O₃, a free electron could initiate SO₂ oxidation and induce the formation of ∼10⁷ cm⁻³ sulfates in the absence of UV light, clearly indicating the importance of other ionic SO₂ oxidation mechanisms than UV-induced oxidation (Enghoff and Svensmark, 2008). To evaluate the importance of the mechanism presented in this study in the formation of sulfate, it is necessary to identify the scavengers that terminate the SO₂ oxidation initiated by ${\mathrm{O}}_{\mathrm{2}}^{-}$. Possible scavengers include radicals, NO_x, acids, cations, and other particles. The main ones are likely NO_x, OH, HO₂, and organic acids, which lead to the formation of the stable ${\mathrm{NO}}_{\mathrm{3}}^{-}$, ${\mathrm{HSO}}_{\mathrm{4}}^{-}$, and ${\mathrm{CO}}_{\mathrm{3}}^{-}$ species. If the ion concentration was known, the contribution of Reaction (R4) to H₂SO₄ formation could be determined by comparing its formation rates from ionic and electrically neutral mechanisms. Alternatively, it can be assumed that Reaction (R4) is terminated when the ion cluster hits a scavenger. The free electron which acts as a catalyst is then scavenged. The average catalytic turnover number (TON) is defined as follows (Kozuch and Martin, 2012):

The concentration of the catalyst can be approximated to the concentration of the scavengers and, considering that at most atmospheric conditions [O₃] > [SO₂], SO₂ is the limiting species in Reaction (R4). Equation (8) can then be re-written as

The catalytic efficiency of SO₂ ion-induced oxidation is then given as

where k_ion is the ion production rate. Depending on the tropospheric temperature and altitude, measurements at the Cosmics Leaving Outdoor Droplets chamber at CERN found k_ion=2–100 cm⁻³ s⁻¹, covering the typical ionization range in the troposphere (Franchin et al., 2015). We assume nearly pristine conditions with [SO₂] = 5 ppb = 1.2×10¹¹ molecule cm⁻³, [NO_x] = 200 ppt = 4.9×10⁹ molecule cm⁻³, [OH] = 5.0×10⁵ molecule cm⁻³ (day and night average), and [HO₂] = 10⁸ molecule cm⁻³ (Dusanter et al., 2009; Holland et al., 2003). Noting that formic acid and acetic acid are the most abundant organic acids in the atmosphere, their concentrations are considered in Eq. (9) as representative examples for organic acids, [organic acids] = 110 ppt = 2.7×10⁹ molecule cm⁻³ (Le Breton et al., 2012; Baasandorj et al., 2015). We then determine J_ion in the range 3.2×10¹–1.6×10³ cm⁻³ s⁻¹. The rate of the UV-induced SO₂ oxidation by OH is

With $k_{\mathrm{UV}}=\mathrm{1.3}\times {\mathrm{10}}^{-\mathrm{12}}$ cm³ molecule⁻¹ s⁻¹ (Atkinson et al., 2004), $J_{\mathrm{UV}}=\mathrm{7.9}\times {\mathrm{10}}^{\mathrm{4}}$ molecule cm⁻³ s⁻¹, the proportion of H₂SO₄ formed from ion-induced oxidation can be estimated from the following equation:

We find that the contribution of ion-induced SO₂ oxidation to H₂SO₄ formation can range from 0.1 % to 2.0 % of the total formation rate. This estimate could be improved by also considering the rate of SO₂ oxidation by Criegee intermediates, another important channel for H₂SO₄ formation.