Hydration increases the lifetime of HSO 5 and enhances its ability to act as a nucleation precursor – a computational study

Introduction Conclusions References Tables Figures


Introduction
The formation of atmospheric aerosol particles by nucleation from gas-phase molecules has recently received growing experimental and theoretical interest due to the climate and health -related effects of fine particles (see e.g.Kulmala, 2003;Kulmala et al., 2004).Despite a large number of experimental and modeling studies, little is known about the detailed molecular -scale processes behind the particle formation events observed in the atmosphere.Based on observed correlations between particle formation rates and trace gas concentrations (see e.g.Weber et al., 1995;Kulmala et al., 2006;Riipinen et al., 2007), and also on thermodynamic classical nucleation theory calculations, nucleation in the lower troposphere has been thought to involve mainly water and sulfuric acid, with possible contributions from ions, ammonia or various organic molecules.A comparison of field measurements with laboratory experiments on sulfuric acid -ammonia -water mixtures reveals some curious discrepancies.For example, the nucleation rate typically depends on the 7th or 8th power of the sulfuric acid concentration in the laboratory experiments (Ball et al., 1999), but only on the 1st or 2nd power in the field experiments (Weber et al., 1995;Kulmala et al., 2006;Riipinen et al., 2007).This indicates that the critical cluster (the smallest cluster for which growth is thermodynamically more favorable than decay) in field conditions contains only 1-2 sulfuric acid molecules, as opposed to 7-8 in laboratory conditions.Also, sulfuric acid concentrations in the atmosphere during nucleation events are typically around 10 6 -10 7 molecules cm −3 , while the threshold concentration for nucleation in laboratory experiments is around 10 10 molecules cm −3 .
Recently, Berndt et al. (2005Berndt et al. ( , 2006Berndt et al. ( , 2007) compared the nucleation rates of sulfuric acid formed from SO 2 oxidation by OH radicals to that of sulfuric acid taken from a liquid reservoir.They found that the threshold H 2 SO 4 concentration for particle formation was 10 7 molecules cm −3 if H 2 SO 4 was produced in situ via the reaction of OH radicals with SO 2 in the presence of water vapor, and 10 10 molecules cm −3 if the H 2 SO 4 was taken from a liquid reservoir or produced in situ via the reaction of SO 3 with water vapor.Other experimental findings confirm a H 2 SO 4 threshold concentration of about 10 7 -10 8 molecules cm −3 (Burkholder et al., 2007;Young et al., 2008;Benson et al., 2008) when produced in situ from SO 2 .It should be noted that the differences regarding the stated threshold H 2 SO 4 concentrations reported from various studies are probably caused by the different experimental conditions used.
Sulfuric acid production from SO 2 oxidation proceeds via a series of radical reactions.First, SO 2 reacts with an OH radical, which is normally produced from water vapor via reaction with electronically excited atomic oxygen O( 1 D), which in turn is formed from ozone photolysis.(Alternatively, the OH radical can be formed from ozonolysis of an alkene in a dark reaction.) Where M denotes a collision partner (typically N 2 or O 2 ).
The HSO 3 radical then rapidly reacts with molecular oxygen to yield either SO 3 and HO 2 or an intermediate complex HSO 5 : SO 3 reacts with water (catalyzed by another water molecule) to yield sulfuric acid: The HSO 5 radical may also decompose to SO 3 and HO 2 , or it may react with other compounds.For example, self-reaction would form peroxodisulfuric acid, H 2 S 2 O 8 : HSO  Various authors (Davis et al., 1979;Wayne, 2000) have suggested that HSO 5 in atmospheric conditions is likely to be hydrated, but to our knowledge no experimental or computational evidence of this has been presented prior to this study.The HSO 3 +O 2 reaction has been investigated by several groups.In a computational study, Majumdar et al. (2000) proposed that the reaction proceeds via three steps: HSO 5 (TS) → HSO 5 (P) (R2e) HSO 5 (P) → SO 3 + HO 2 (R2f) Where HSO 5 (R) is a reactant complex corresponding to HSO 3 •O 2 (HOSO 2 •O 2 ), HSO 5 (P) is a product complex corresponding to SO 3 •HO 2 , and HSO 5 (TS) is a transition state connecting the two, where the H-atom is shared by the S-O-O and S-O groups.It should be noted that R2d and R2e do not correspond to real, individual chemical reactions, but to elementary steps of the same reaction.
Very recently, Berndt et al. (2008) and Laaksonen et al. (2008) have discussed the implications of the experimental results in terms of these mechanisms, and concluded that a nucleation mechanism involving HSO 5 or its reaction products is likely to explain nucleation starting from OH+SO 2 observed in atmosphere as well as in the laboratory.Salonen et al. (2009) have investigated the stability of clusters of sulfuric acid together with various intermediate and alternative products of the SO 2 oxidation chain, and concluded that while HSO 5 itself is unlikely to nucleate very effectively, a mixture of sulfuric acid and HSO 5 reaction products containing more than one sulfur atom (such as peroxodisulfuric acid) are likely to nucleate more effectively than sulfuric acid on its own.
A crucial uncertainty in the proposed nucleation mechanism is the lifetime of HSO 5 especially with respect to dissociation via path (4) and (2f).Based on computational and experimental studies (Stockwell and Calvert, 1983;Li and McKee, 1997), the dissociation of HSO 5 into SO 3 and HO 2 could be very rapid.Thus, the fraction of HSO 5 molecules that live long enough to collide with other sulfur-containing molecules would be small, and any nucleation mechanism starting from HSO 5 would be inefficient.However, previous computational studies have neglected the effect of water vapor, which is always present in both laboratory and field studies in concentrations far exceeding those of all sulfur -containing molecules combined.In principle, hydration could increase the steady-state concentration of HSO 5 , and thus the probability of bimolecular reactions involving HSO 5 , by four different mechanisms: -Thermodynamic stabilization of HSO 5 with respect to the products SO 3 and HO 2 .
-Increasing the fraction of HSO 3 +O 2 collisions that lead to HSO 5 formation as opposed to direct SO 3 +HO 2 formation (if the latter is at all possible).E.g. if HSO 3 were hydrated in Reactions (2a/2b), there would be more vibrational degrees of freedom to accommodate the excess energy liberated in the reaction, and thus a greater degree of energy accommodation, and a higher probability for the transiently formed reaction complex to stay together.
Mechanism number 3 is not very likely, as water typically tends to catalyze proton transfer reactions rather than hindering them.In this study, we have assessed the effect of mechanisms 1 and 2 above using advanced quantum chemical methods, and found them both to be significant.

Computational details
We have computed free energies of formation for SO 2 , OH, HSO 3 , O 2 , HSO 5 , SO 3 and HO 2 molecules and most of their mono-and dihydrates.(The hydrates of O 2 were not studied, as they are likely to be extremely weakly bound due to the nonpolarity of the O 2 molecule.Also, only the monohydrate of SO 2 was considered, as experimental evidence (Dermota et al., 2005) indicates that SO 2 in water clusters only interacts with one water molecule).Free energies have been computed using the G3B3 combination method (Baboul et al., 1999) which involves B3LYP/6-31G(d) geometry optimizations and frequency calculations (using a scaling factor of 0.96) together with a series of higher-level (MP2(full), MP4 and QCSID(T)) single-point energy calculations and empirical corrections.Assuming certain additivity rules, the G3B3 energy is a reasonable estimate of the QCISD(T)(full) energy with a large triple-zeta basis set.For a sample set of 299 data points (Baboul et al., 1999), the mean absolute deviation of the energies predicted by the G3B3 method is 0.99 kcal/mol.Some calculations have also been performed with the G2 combination method (Curtiss et al., 1991), which is similar to G3B3 except that frequencies are computed at the HF/6-31G(d) level and geometries at the MP2/6-31G(d) level.Unfortunately, most of the G2 calculations on hydrated HSO 5 failed as the MP2 geometry optimizations did not converge.This may be related to the finding by Majumdar et al. (2000) that the MP2 method does not always yield reliable results (especially structures or vibrational frequencies) for all configurations of the HSO 5 system.To assess the reliability of the G3B3 values (and also to compute more reliable estimates for the reaction energetics than published in earlier studies), we further optimized the structures of the free molecules, monohydrates and selected dihydrates at the UB3LYP/6-311++G(3df,3pd) level (Becke, 1993;Lee et al., 1988) and performed UCCSD(T)/6-311++G(3df,3pd) and UCCSD(T)/aug-cc-pV(T+d)Z (Knowles et al., 1993;Deegan and Knowles, 1994) single-point energy calculations at these geometries.(The aug-cc-pV(T+d)Z basis set is identical to the standard aug-cc-pVTZ basis set for first-and second-row atoms, and contains an extra d -orbital for third-row atoms, as this has been shown to yield more accurate atomization energies, see Dunning et al., 2001).Due to the extreme computational cost and prohibitive scaling of the UCCSD(T) method, we were unable to treat the HSO 5 dihydrate clusters at this level.
To maintain consistency with G3B3 thermochemical parameters, a scaling factor of 0.967 was used in calculating the thermal entropy and enthalpy contributions at the UB3LYP/6-311++G(3df,3pd) level.This was based on data for the 6-311+G(3df,2p) basis set from the Computational Chemistry Comparison and Benchmark Database (http:// cccbdb.nist.gov/default.htm).This scaling factor was based on 3448 individual vibrational frequencies from 308 different molecules.A slightly larger scaling factor of 0.970 was given for the basis set 6-311+G(3df,2pd), which is closer in size to the basis used in this work.However, as this value was not based on direct calculations, but on comparisons with B3LYP/cc-pVTZ calculations, we chose the former value as somewhat more reliable.This is partly supported by a recent study by Anderson and Uvdal (2005), in which a scaling factor of 0.9679 was recommended for B3LYP calculations with all Pople basis sets larger than 6-311++G(d,p).In any case, the difference between results obtained with the three scaling factors (0.967, 0.9679 and 0.970) are minimal.All thermochemical parameters presented here have been computed using the rigid rotor and (scaled) harmonic oscillator approximations.
with n=1, 2, . ..) clusters have been computationally studied previously (Li and McKee, 1997;Du et al., 2006;Aaltonen and Fransisco, 2003;Fliegl et al., 2006;Larson et al., 2000;Alongi et al., 2006), and initial geometries for these systems were taken from the published literature.To our knowledge, there are no previous studies on HSO 3 •(H 2 O) 2 or HSO 5 •(H 2 O) 1...2 clusters.For these systems, initial guesses were generated using the Spartan program (Wavefunction Inc, 2002), and pre-optimized at the B3LYP/6-31G(d) level (and for some clusters also at the PBE/6-31+G(2d,p) level with density fitting).For the HSO 5 mono-and dihydrates, several of the initial guess geometries failed to converge at the B3LYP and PBE levels, and only a few structures could finally be used for the higher-level calculations.UCCSD(T) calculations were performed using the Molpro 2006.1 program (Werner et al., 2006), some RI-MP2 test calculations (Weigend and Häser, 1997;Weigend et al., 1998) were performed on Turbomole 5. 10 (Alrichs et al., 1989), and all other calculations were carried out using the Gaussian 03 program suite (Frisch et al., 2004).The default energy and geometry convergence criteria for each program were used in all calculations.It should be noted that for the HSO 5 (TS) and HO 2 systems, both of which belong to the point group Cs, using the default symmetry settings of the Molpro program causes severe errors in the computed reaction energies, as the calculation then corresponds to an excited electronic state (with symmetry 2 A ).To compute the energy for the ground state (with symmetry 2 A ), the Based on data by Benson, 1978; d Based on data by Atkinson et al. (1997); given by Li and McKee (1997); e Klopper et al. (2008); advanced multistep calculations involving MP2-F12/aug-cc-pwCV5Z and UCCSD(T)/cc-pV(T+d)Z calculations, together with anharmonic, hindered rotor and relativistic corrections.f Li and McKee (1997); g This study symmetry of the ground-state wavefunction must either be explicitly specified (using the wf and occ keywords), or the calculation must be performed without using symmetry (keyword nosym).A similar phenomenon occurs for the OH radical.

HSO 3 +O 2 reaction energetics
Previous studies have found the net reaction HSO 3 +O 2 →SO 3 +HO 2 (Reaction 2a) to be either exother-mic (Li and McKee, 1997;Klopper et al., 2008), weakly endothermic (Majumdar et al., 2000) or strongly endothermic (Benson, 1978;Nagase et al., 1988).Experimentally, the reaction is known to be the exclusive fate of HSO 3 in the atmosphere with a rate constant of 4.3×10 −13 cm 3 molecule −1 s −1 at 293 K (Gleason et al., 1987;Gleason and Howard, 1988).The experimental results thus indicate that the studies predicting a strong endothermicity are likely to be in error.The reaction energetics predicted for this reaction at various levels of theory are given in Table 1.(Note that several more basis sets were used in the study by Majumdar et al., 2000; the rest of their values lie between those presented here.) It can be seen from Table 1 that while B3LYP and MP2 calculations predict the reaction to be strongly endothermic unless very large basis sets are used, the more Table 2. Electronic energy change ( E elec ), enthalphy change ( H) and Gibbs free energy change ( G) for the four individual steps of the overall reaction HSO 3 +O 2 → SO 3 +HO 2 , computed at various levels of theory.A scaling factor of 0.967 has been used in the UB3LYP/6-311++G(3df,3pd) frequency calculations.All values in kcal/mol.Enthalpies and Gibbs free energies correspond to 298 K and 1 atm reference pressure.HSO 5 (R), HSO 5 (TS) and HSO 5 (P) correspond to the reactant, transition state and product complexes, respectively.(See the text and Fig. 1 2004); e This study advanced methods including high-level correlation (G2, G3B3, UCCSD(T)) predict the reaction energy and enthalpy to be slightly below, though quite close to, zero.The reaction free energies predicted by the more advanced methods lie on both sides of zero, and are all quite small.The large differences between MP2 or DFT energies on one hand and G2, G3B3 or coupled-cluster energies on the other indicates that high-level electron correlation plays a central role in the energetic of the reaction, as noted by Majumdar et al. (2000).The difference between the G3B3, UCCSD and UCCSD(T) energies (as well as the UCCD and UCCSD(T) energies using two different basis sets) are of the same size as the predicted endo-or exothermicity, indicating that a quantitatively reliable prediction of the sign of the energy change of the reaction likely requires even more advanced methods than those used here.In a very recent state-of-the-art computational study, Klopper et al. (2008) 2004); e This study best results are in fairly good agreement with their enthalpy value of −2.3±0.7 kcal/mol.Comparing the free energy and enthalpy values, we can conclude that also the free energy change of the reaction is very likely to be slightly below zero.
To our knowledge, only Majumdar et al. (2000) have previously addressed the detailed energetics of all the elementary reaction steps 2c-2f.The study by Nagase et al. (1988) contains data for Reaction (2c), while that of Solimannejad et al. (2004) contains data for Reaction (2f).The energetics of these individual steps are reported in Table 2.The structure of the molecules HSO 3 , HSO 5 (R), HSO 5 (TS) and HSO 5 (P) are shown in Fig. 1.The structures are drawn using the MOLEKEL 4.3 visualization package (Portmann, 2002).
As for the net Reaction (2a), there are considerable differences between the different methods.Compared to the more advanced G2, G3, G3B3 and coupled-cluster methods, MP2 and B3LYP significantly underestimate the binding energy of HSO 5 (R) and the barrier height for the HSO 5 (R)→HSO 5 (P) conversion.However, the energetics for Reaction (2f) predicted by the different methods are relatively similar.
Regardless of the method used, the HSO 5 (TS) transition state is predicted to be lower in electronic energy and enthalpy than the reactants HSO 3 +O 2 .
On the other hand, all methods except G2 predict the transition state to lie higher in free energy than the free reactant molecules.The highest-level (UCCSD(T)/augcc-pV(T+d)Z//UB3LYP/6-311++G(3df,3pd)) G value  for the HSO 3 +O 2 →→HSO 5 (TS) free energy change is +0.77kcal/mol, indicating that, as for the net Reaction (2a), the value is likely to be close to zero, and even more advanced methods would be needed to reliably determine the sign of the free energy difference.Nevertheless, it is clear from the higher-level data that net barrier for the HSO 3 +O 2 →→HSO 5 (P) process is unlikely to be larger than a few kcal/mol, which supports the experimental observations (Gleason et al., 1987;Gleason and Howard, 1988) that the reaction should be reasonably fast at ambient temperatures.

Effect of hydration
Table 3 shows the formation energetics of X•H 2 O and X•(H 2 O) 2 clusters computed at various levels of theory, with X=SO 2 , OH, HSO 3 , HSO 5 (R), HSO 5 (P), SO 3 and HO 2 .For simplicity, only G3B3 and UCCSD(T)/aug-cc-pV(T+d)Z//UB3LYP/6-311++G(3df,3pd) results are given    311++G(3df,3pd) geometries computed for some of the clusters.See the supporting information for all computed Cartesian co-ordinates.) For the molecules for which previous data on hydrates are available (SO 2 , OH, HSO 3 , SO 3 and HO 2 ), our results are in fairly good agreement with previous studies.It should especially be noted that the binding energy predicted for the OH•H 2 O complex using the UCCSD(T)/augcc-pVTZ single-point energy computed at the UB3LYP/6-311++G(3df,3pd) geometry is within 0.005 kcal/mol of that computed by Du et al. (2006) using a full RCCSD(T)/augcc-pVTZ geometry optimization.This indicates that the errors induced by the DFT geometry optimization for our hydrated radical complexes are likely to be relatively minor, at least if the basis set is large enough.Similarly, it should be noted that our hydration enthalpies for HO 2 are all within the error margins of the experimental result by Kanno et al. (2006), and that our best value for the SO 3 •H 2 O binding energy is within the estimated error margins of that computed by Fliegl et al. (2006) using a combination of very advanced methods.The G3B3 and UCCSD(T)/augcc-pV(T+d)Z//UB3LYP/6-311++G(3df,3pd) values are also in reasonable agreement with each other, except for the HSO 5 (R)•H 2 O cluster, for which the coupled-cluster binding energy is almost 3 kcal/mol less than the G3B3 binding energy.(The difference in free energies is only about 1.2 kcal/mol.)The reason for this difference may be the fact that, as pointed out by Majumdar et al. (2000), the geometries predicted at the B3LYP level using double-and triple-zeta basis sets (6-31G(d,p) and TZ2P++ in their case) differ quite significantly from each other with regard to the SOH. . .OOS distance.With the smaller basis sets, the distance is predicted to be rather short (1.71 and 1.75 Å for the 6-31G(d,p) and 6-31G(d) basis sets, respectively), corresponding to an intramolecular hydrogen bond, whereas calculations with a large basis set predict the distance to be much larger (2.74 for both the TZ2P++ and 6-311++G(3df,3pd) basis sets).Presumably, the B3LYP calculations with larger basis sets correspond at least somewhat better to the high-level wavefunction-based methods employed in the G3B3 and CCSD(T) energy calculations, as indicated by the excellent agreement for the OH•H 2 O cluster mentioned above.Thus, the high HSO 5 (R)•H 2 O binding energy found at the G3B3 level is probably slightly overestimated, as the B3LYP/6-31G(d) geometry for HSO 5 (R) used in the calculation is likely to be unfavorable for the higher-level methods.
Regardless of the method used, it can be seen from Table 3 that SO 2 and OH are only weakly bound to water, while SO 3 and HO 2 are moderately and HSO 3 and HSO 5 strongly hydrated.Especially the reactant complex HSO 5 (R) is very strongly bound to water, even after the probable overbinding in the G3B3 energies is accounted for.Hydration by one water molecule stabilizes the reactant form of HSO 5 by around 3-5 kcal/mol compared to the product form, while both HSO 5 dihydrates are more strongly bound than any of the other dihydrate clusters.However, the difference in hydration energies between the dihydrates HSO 5 (P)•(H 2 O) 2 and HSO 5 (R)•(H 2 O) 2 is only around 1-2 kcal/mol, less than half of the difference between the monohydrates.The reason for this is apparent from Fig. 2i): the HSO 5 (P)•(H 2 O) 2 minimum-energy structure actually corresponds to an ion pair cluster SO (Other, local minima without proton transfer were also found at the G3B3 level, but they were all less stable than the ion pair structure by several kcal/mol.See the supporting information for their Cartesian co-ordinates.)As the SO − 5 radical ion is known to be an effective oxidant in aqueous solutions (Wayne, 2000;Das, 2001) this prediction, if true, would have significant implications for the reactivity and nucleating potential of hydrated HSO 5 .For example, two SO − 5 •H 3 O + •H 2 O clusters could be expected to react rapidly to yield S 2 O 2− 8 •(H 3 O + ) 2 •(H 2 O) 2 +O 2 , and the resulting H 2 S 2 O 8 cluster can then be expected to nucleate more efficiently than pure sulfuric acid (as discussed by Salonen et al., 2009).Furthermore, SO − 5 is known to be a strong oxidant for sulfite in aqueous solutions (Das, 2001).By analogy, SO − 5 ) y (y=2, 3, . . . ) can efficiently oxidize further SO 2 picked up from the gas phase.This process leads to growth of the initial HSO 5 hydrates.
However, the B3LYP/6-31G(d) method used for geometry optimizations within the G3B3 combination approach is rather modest, and the predicted ion pair structure could be an artifact.To check if this is the case, we performed further geometry optimizations on the HSO 5 (P)•(H 2 O) 2 cluster at the UB3LYP/6-311++G(3df,3pd), UMP2/6-311++G(2d,2p) and RI-MP2/aug-cc-pV(T+d)Z levels.The UB3LYP/6-311++G(3df,3pd) optimization also led to an ion pair structure, while the UMP2/6-311++G(2d,2p) and RI-MP2/augcc-pV(T+d)Z optimizations led to structures where the hydrogen atom was equally shared between the S-O-O and OH 2 groups.We then computed the G3B3 single-point electronic energies at these geometries, and found them to be 0.55 kcal/mol higher and 0.32 lower than that computed at the UB3LYP/6-31G(d) minimum geometry, for the UMP2/6- Dashed lines correspond to hydrogen bonds.Color coding: yellow=sulfur, red=oxygen, white=hydrogen.311++G(2d,2p) and RI-MP2/aug-cc-pV(T+d)Z methods, respectively.(It should be noted that the G3B3 method is parametrized to the B3LYP/6-31G(d) geometries, so the accuracy of these single-point energy calculations may not be as good as that of the original G3B3 method.However, they should still be good indications of the higher-level potential energy surface.)Given the influence of higher-level correlation on the reaction energetics of the HSO 5 system (see e.g.Tables 2-3), neither the B3LYP nor the MP2 structures are likely to be totally trustworthy.Ideally, geometry optimizations would need to be performed at e.g. the UCCSD(T)/aug-cc-pV(T+d)Z level to determine how many water molecules are needed to induce proton transfer from HSO 5 to H 2 O. Unfortunately, this is beyond the capabilities of present computer hardware.Still, the fact that both B3LYP and MP2 methods predict at least a partial proton transfer for the HSO 5 (P)•(H 2 O) 2 cluster is a strong indication that the minimum-energy geometry for HSO 5 (P)•(H 2 O) 2 is likely to have at least some degree of an ion-pair character.
Using the data in Table 3, the influence of hydration on the net Reaction (2a) and the dissociation Reaction (2f) can now be estimated.G3B3 and UCCSD(T)/augcc-pV(T+d)Z//UB3LYP/6-311++G(3df,3pd) reaction energy and free energy values for Reactions (2a), (2d)+(2e) and (2f) Table 4. Electronic energy change ( E elec ) and Gibb's free energy change ( G) for various reaction pathways corresponding to Reactions (2a), (2c)+(2d) and ( 2f), but including the effects of hydration.All values in kcal/mol."CCSD(T)" corresponds to UCCSD(T)/aug-cc-pV(T+d)Z single-point energies and UB3LYP/6-311++G(3df,3pd) geometries and frequencies.A scaling factor of 0.967 has been used in the UB3LYP/6-311++G(3df,3pd) frequency calculations.Gibbs free energies correspond to 298 K and 1 atm reference pressure.with one or more of the participating species hydrated are given in Table 4. Reaction pathways including the dissociation of water from the clusters were not included in the table as they are all predicted to be strongly endothermal.

Reaction
A comparison of Table 4 with Tables 1 and 2 demonstrate that hydration does indeed stabilize HSO 5 via both mechanisms 1 and 2 presented in the Introduction.E.g. the HSO 5 (R)→HSO 5 (P) conversion is predicted to have a G of 3.39 kcal/mol at the UCCSD(T)/aug-cc-pV(T+d)Z//UB3LYP/6-311++G(3df,3pd) level, increasing to 6.81 kcal/mol after hydration by one water molecule.(Corresponding values for the G3B3 method are 0.08, 4.67 and 1.36 kcal/mol for zero, one and two water molecules, respectively, the latter low value being due to the predicted ion pair formation.)Similarly, the free energy values for the most favorable routes for the net Reaction (2a) at the same level are 1.60 kcal/mol and 2.20 kcal/mol after hydration by one and two water molecules, respectively, compared to −0.26 kcal/mol for the unhydrated reaction.(Corresponding G3B3 values are −1.76,−1.08 and 0.51 kcal/mol, for zero, one and two waters, respectively).While the difference is relatively small -about 2 kcal/mol -the effect on the equilibrium constant for the reaction is significant, and hydration is likely to strongly increase the lifetime of HSO 5 in ambient conditions.If the prediction of an ion pair structure for HSO 5 (P)•(H 2 O) 2 is correct, then the monohydrate will be stabilized mainly by mechanism 2 (thermodynamic stabilization with respect to the HSO 5 (R)→HSO 5 (P) conversion) while the dihydrate (and, presumably, higher hydrates) will be stabilized mainly by mechanism 1 (stabilization with respect to dissociation into SO 3 +HO 2 ).The dominant stabilization mechanism would then dependent on the relative humidity (RH).If, on the other hand, the prediction of an ion pair structure is incorrect, then mechanism 2 is likely to dominate both for the mono-and dihydrates.It should be noted that the net reactions HSO 5 ,1,2) are in any case all strongly endothermal, independent of the prediction of proton transfer for HSO 5 (P)•(H 2 O) 2 .Also, possible errors in the configurational sampling of HSO 5 hydrates (i.e. the minimum-energy geometries described here may not be the global minima) would only serve to further increase the stability of these structures with respect to hydrated SO 3 and HO 2 .

Kinetic modeling of flow-tube experiments
The data in Tables 3 and 4 unequivocally show that if HSO 5 clusters are hydrated, they will be significantly stabilized both with respect to the HSO 5 (R)→HSO 5 (P) conversion and with respect to dissociation into SO 3 +HO 2 .The key question that then remains is the mechanism by which HSO 5 can be hydrated in the first place.If only a fraction of unhydrated HSO 3 +O 2 collisions lead to the formation of HSO 5 , then HSO 5 +H 2 O collisions are not likely to be an effective source of hydrated HSO 5 , especially given the presumably short lifetime of unhydrated HSO 5 .The major source of water molecules is thus likely to be the hydration of the preceding reactants in the SO 2 oxidation chain.O 2 is unlikely to be hydrated due to the lack of a dipole moment, and as shown in Table 3, SO 2 is only weakly bound to water.OH is somewhat more strongly hydrated, but using the UCCSD(T)/augcc-pV(T+d)Z//UB3LYP/6-311++G(3df,3pd) free energy of hydration at 298 K together with a H 2 O concentration of 0.015488 atm (corresponding to RH 50% at 298 K) in the law of mass action leads to the conclusion that less than 0.1% of OH molecules will be bound to water at equilibrium.Thus, the only remaining possibility is the hydration of HSO 3 .The free energy changes in Table 3 are large enough that a significant fraction (on the order of 10-20%) of HSO 3 molecules are bound to water at equilibrium in ambient conditions.
The following reaction scheme was used for a more detailed description of the processes going on in the course of SO 2 oxidation by OH using as the OH source the photolysis of O 3 , as applied in the IFT-LFT experiments (Berndt et al., 2005(Berndt et al., , 2006)): Simulations have been carried out using the following rate constants.(Note that the degree of hydration of HSO 3 is independent of rate constants k M1 . . .k M3 ).For the reaction of all HSO 3 •(H 2 O) x species with O 2 , the rate constant for the unhydrated molecule k M4 =k M7 =k M10 =4.3×10 −13 cm 3 molecule −1 s −1 (Gleason et al., 1987;Gleason and Howard, 1988) was used.
HSO 5 •H 2 O, HSO 5 •(H 2 O) 2 or higher hydrates can be treated as potential nucleation precursors reacting, for instance, in a self-reaction, or oxidizing further SO 2 after picking it up from the gas phase.For example, if the prediction of an ion pair structure of HSO 5 (P)•(H 2 O) 2 is correct, then the self-reaction of this species is likely to proceed with little or no barrier like the self-reaction of sulfamic acid (Lovejoy and Hanson, 1996): Using an order-of-magnitude estimate of k M11 =1×10 −10 cm 3 molecule −1 s −1 for the rate constant of the HSO 5 •(H 2 O) 2 ion pair self-reaction, and assuming that only this step is responsible for critical cluster formation, the rate at which the reaction products are produced via path (M11) is slightly larger than the formation rate of detected particles measured for commonly used conditions in the laboratory experiments (Berndt et al., 2005(Berndt et al., , 2006(Berndt et al., , 2008)).The corresponding steady-state concentrations for HSO 5 •(H 2 O) 2 (assuming that self-reaction is the main sink) are on the order of a few 10 5 cm −3 .Further addition of HSO 5 •(H 2 O) 2 or HSO 5 •H 2 O, and/or other SO 2 oxidation to the produced clusters could govern the growth process.
This rough estimate is affected by a few assumptions, some of which require further research.It shows, however, that the formation of hydrates HSO 5 •(H 2 O) x (x=1, 2, . . . ) can help to explain the observed nucleation in the atmosphere, as well as in the laboratory, starting from OH+SO 2 .
We have investigated the effect of hydration on the HSO 3 +O 2 → SO 3 +HO 2 reaction, and especially on the stability of HSO 5 intermediate radicals, using advanced quantum chemical methods.Without hydration, the standard enthalpy and free energy of reaction are found to be very close to zero, in accordance with very recent high-level results.After hydration, the reaction becomes moderately endothermal.Hydration is found to stabilize the reactant form of HSO 5 (corresponding to a HSO 3 •O 2 complex) compared to the product form (corresponding to a SO 3 •HO 2 complex), and significantly increase the energy barrier for dissociation of HSO 5 into SO 3 +HO 2 .The dihydrated product complex was predicted to have at least a partial ion pair structure at several different levels of theory.Kinetic modeling indicates that HSO 3 hydration is likely to produce significant amounts of hydrated HSO 5 in atmospheric and laboratory conditions.The reactions of these hydrated complexes may help explain recent laboratory observations of unexpectedly efficient nucleation from SO 2 oxidation initiated by OH radicals.

Table 1 .
Electronic energy change ( E elec ), enthalphy change ( H) and Gibbs free energy change ( G) for the reaction HSO 3 +O 2 → SO 3 +HO 2 , computed at various levels of theory.All values in kcal/mol.Enthalpies and Gibbs free energies correspond to 298 K and 1 atm reference pressure.A scaling factor of 0.967 has been used in the UB3LYP/6-311++G(3df,3pd) frequency calculations.

Table 3 .
Electronic energy change ( E elec ), enthalphy change ( H) and Gibb's free energy change ( G) for hydration reactions of various molecules: X+n(H 2 O) → X•(H 2 O) n .A scaling factor of 0.967 has been used in the UB3LYP/6-311++G(3df,3pd) frequency calculations.All values in kcal/mol.Enthalpies and Gibbs free energies correspond to 298 K and 1 atm reference pressure.HSO 5 (R), HSO 5 (TS) and HSO 5 (P) correspond to the reactant, transition state and product complexes, respectively.(Seethetext and Fig.1for details.)When multiple literature values exist for some parameter, only the highest-level ones are given.