Rui Wang and co-authors have used computational methods to study the formation and clustering of H2S2O7 - known (depending on the source) as either disulfuric acid, pyrosulfuric acid, or oleum. The technical methods used in the study are broadly appropriate, and the context (atmospheric new-particle formation involving different sulfur compounds) is certainly relevant and broadly interesting. The study is thus without doubt publishable. However, I have some critical notes about the interpretation of the results, and their atmospheric implications (which I believe to the overstated, at least in the context of Earth’s lower atmosphere). 

Major issues:

1)As shown by Torrent-Sucarrat (JACS 2012; cited in the present study), the SO3 + H2SO4 reaction in the presence of water can also lead directly to H2SO4 + H2SO4 (instead of H2S2O7). Given that H2SO4 is pretty much always hydrated, water is - as the authors themselves argue here - essentially always present in the reaction system, at least in the lower troposphere. Thus, an explicit consideration of the competition between the two channels would be warranted - however this seems to be missing in the study. The authors should try to estimate what percentage of SO3 + H2SO4 collisions, in different hydration environments, we can expect to yield (at least transiently, see below) H2S2O7, as compared to H2SO4 + H2SO4? (Note that this question should be asked on top of the question that they DO address, i.e. “what fraction of SO3 will collide with H2SO4 as opposed to H2O, or H2O*X, where X is any other catalyst for the SO3 hydration reaction. As per the authors own calculation in their Table S6, already this percentage is very small - despite their neglect of many other known candidates for X.)

2)While interesting, I’m not sure the BOMD simulations are saying much about the relevance of H2S2O7 for actual new-particle formation: H2S2O7 formed in water droplets will presumably stay there, and never evaporate to participate in NPF. (Overall, “air-water interfaces” have little to do with actual NPF, as the interfaces are by definition found in particles that *have already formed*: many of the claims of “NPF-relevance” made in the study are thus by definition incorrect). While there may be some relevance of the studied process to particle growth, even H2SO4 has an essentially zero evaporation rate from any particles larger than a few nanometers - so it may makes little difference to the growth of larger aerosol whether the sulfur is taken up as H2SO4 or H2S2O7 (also see issue 4 for a further caveat). 

3)The authors spend much time discussing the results they obtain for the “enhancement factor” R (equation 5). As cautioned in Elm et al. (https://www.sciencedirect.com/science/article/abs/pii/S0021850220301099), the excessive use of such abstract “enhancement factors” is questionable and risky. In this particular case, I don’t believe the results are actually technically badly wrong - for example including the effect of H2SO4 depletion (caused by a fraction of the SO3 forming DSA rather than H2SO4) would probably not change the qualitative results, as the clustering ability of DSA is much greater than that of H2SO4. (For completeness sake, I would nevertheless recommend this is done). However, many of the presented “results” are in reality rather trivial consequence of how the simulation is set up, and the parameters defined. For example, the R values are quite obviously “greater than or equal to 1”, as the J values with added DSA cannot (in the way the authors run ACDC) be lower than the J values without the added DSA. Similarly, the various correlations between R and different parameters are not particularly informative or novel. I recommend the authors first of all account for all relevant effects (including sulfur depletion - ie run the code with a constant SO3 source rather than constant [H2SO4]), and also condense the discussion on “enhancement factors”.

4)The most problematic part of the overall claim for atmospheric relevance is the neglect of H2S2O7 decomposition by hydration (i.e. the H2S2O7 + H2O => H2SO4 + H2SO4 reaction), which is very well known (e.g. from industrial sulfur chemistry) to be rapid and spontaneous. (Indeed, H2S2O7 is one of the strongest dehydrating agents in the known universe - its hydration reaction is so strong and favourable that it can even extract water molecules from sugar.) The BOMD simulations indicate that H2S2O7 is stable for 10 picoseconds - but this is nowhere near enough time for a H2S2O7 to, for example, collide with a H2SO4 (timescale: seconds) in the gas phase (and thus participate in NPF). Recently, another group showed that preliminary results on the role of sulfamic acid in new-particle formation are invalidated by rapid hydrolysis (https://pubs.acs.org/doi/full/10.1021/acs.jpca.3c04982) - I anticipate something very similar may end up being the case for di/pyrosulfuric acid in the Earth’s lower atmosphere. Having said that, the presented results may very well have relevance for stratospheric chemistry, as well as for cloud chemistry on Venus (where there is much less water, and much more H2SO4). I recommend the atmospheric implications and relevance discussion be reformulated to target the appropriate atmospheres /or regions of them. Or at the very least, the possible (even likely) rapid hydrolysis of H2S2O7 should be mentioned as a major caveat of the results (and as a strongly recommended subject for follow-up studies!) 

5)Another issue to consider is the timescale associated with participation in NPF of compounds with mixing ratios well below a part per quadrillion. The gas-kinetic bimolecular collision rate for small molecules and their clusters is around 1E-10…1E-9 cm3 per molecule and second. If the DSA concentration is 1 molecule per cm3, then on average a H2SO4 molecule, or a H2SO4-containing cluster, will thus collide with DSA molecules about once per 1E9 seconds or so (as the pseudo-unimolecular collision rate is k_coll times the concentration, and the lifetime with respect to collisions is the inverse of this) . This is more than 30 YEARS. Even for a DSA concentration of 10 per cm3, the timescale of a given molecule or cluster colliding with a DSA molecule is more than 3 YEARS.  Or for 100 per cm3, more than 3 months. It is quite clear from this that the pseudo-steady-state assumed by ACDC simulations will simply never have time to form, when some of the participating molecules have such low concentrations. Or in other words, the basic assumptions required for modelling clustering with ACDC do not apply in these cases (or, to put it yer another way, an ACDC - type code needs to be run in a very different way, explicitly accounting for these timescales). 

Technical issues

1)The kinetic approach used here seems quite elaborate, given that the authors are not actually treating (or at least not discussing) any sort of pressure dependence, non-thermalisation, etc. How different are the rates compared to what one would obtain using a simple transition state theory framework (plus assuming kinetic gas theory forward rates for the initial complex formation)? I’m not criticising the use of elaborate methods as such, I’m just trying to assess how much difference they make, compared to a much simpler approach. 

2)The method references for M06-2X, CCSD(T)-F12, and the ORCA program are not correct - the first is completely wrong, while the latter refer to studies which have also used these approaches. Please refer to the actual publications introducting the methods/codes instead.  

General Comments

Using computational methods Wang and co-workers study the reaction between H₂SO₄ and SO₃ leading to the formation of H₂S₂O₇. The gas-phase formation mechanism is studied using well-established methodologies, both with and without a water molecule present. The reaction is also studied at the air-water interface using Born-Oppenheimer molecular dynamics simulations. Finally, the authors study the potential of the formed H₂S₂O₇ product in “enhancing” new particle formation involving sulfuric acid and ammonia.

Overall, the applied quantum chemical methods are up to the current standard and the study is broadly atmospherically interesting, but I believe many of the conclusions are erroneously drawn and not supported by the data. Remember negative results are equally as important as positive results. So try to frame the results in a more transparent fashion. In addition, there is heavy referencing to the SI, which makes the paper difficult to follow in some places and the reader is left wondering if the claims are actually correct. I believe the paper might be worth publishing, but some critical changes must made.

Specific Comments:

Overall: When referring to the SI, please add the numbers to the text as well and elaborate on what the reader is supposed to look at in the SI. In several places it is very difficult to comprehend how the authors draw the conclusions.

Line 48: ”As a typical inorganic acid, SA can act as an important role in the new particle formation …”

What is meant by “typical here? Please rephrase this sentence.

Line 82: “It has been shown that the products of SO3 with some important atmospheric species have been identified in promoting NPF process. ”

Such reaction would lead to the consumption of an SO₃ molecule potentially at the expense of forming less sulfuric acid. This competition should be further discussed in the manuscript.

Line 104: I am missing some justification to why the M06-2X functional has been used and why the 6-311++G(2df,2pd) basis set was chosen. In addition, the M06-2X reference is incorrect.

Line 108: The ORCA reference is incorrect.

Line 110-116: I am in doubt whether the applied configurational sampling of the clusters is sufficient to identify the lowest free energy cluster structures. Only calculating 1000 local minima from the ABCluster search sounds a bit low on the low side. How certain are the authors that they have located the global minimum? As the CHARMM forcefield cannot handle bond breaking a more diverse pool of clusters is needed. This is usually done by performing ABCluster runs with ionic monomers as well (see Kubečka et al., https://doi.org/10.1021/acs.jpca.9b03853). Only selecting the lowest 100 cluster configurations based on PM6 could lead to the global minimum cluster being missed (see Kurfman et al., https://doi.org/10.1021/acs.jpca.1c00872).

Line 116: Here it is stated that the free energies are calculated at the M06-2X/6-311++G(2df,2pd) level of theory. However, Table S8 indicates that DLPNO-CCSD(T) single point energy calculations were carried out on top of the clusters.

Line 147: How was the 191 water cluster obtained? Has this cluster been equilibrated before the SO₃ and H₂SO₄ was added? Or after? Some more details about how the system was setup is needed. Is a 1 fs timestep adequate to capture the desired dynamics? I.e. can it actually capture the hydrogen bond stretching vibration?

Line 156: A am not entirely convinced that the 3x3 system “box” size is large enough to ensure meaningful cluster dynamics of the systems. For instance, the work by Besel et al. (https://doi.org/10.1021/acs.jpca.0c03984) showed how the sulfuric acid-ammonia system is impacted by the studied box size. Please elaborate on this aspect.

Also is the sulfur concentration constrained in the simulations? A single DSA molecule would consume 2 sulfuric acids. 1 SA and 1 SO₃ that could form SA. Hence, the simulations might actually “push” additional sulfur into the system.

Line 162:  I do not believe the factors of 1/2 should be in this equation.

Line 168-169: Please explicitly mention the boundary conditions and concentration ranges in the text here instead of referring to the SI.

Section 3.1: I am missing some comments on why the titled reaction is of interest and how much the competitive pathway of SO₃+H₂O matters. Would SO₃ not react with water instead of H₂SO₄? What are the branching ratios between these reaction pathways?

Line 181: “Therefore, it can be said that the direct reaction between SO3 and SA is more favorable over H2O-catalyzed hydrolysis of SO3 energetically and kinetically. ”

I believe this conclusion should be based on the “reaction rates” and not the “reaction rate constants”.

Section 3.2: The first two sentences are contradicting each other. Is the mechanism lacking or does it have high reactivity? I am also missing some information about how the system was setup:

• Would the studied compounds (SO₃, H₂SO₄ and H₂S₂O₇) actually be at the interface or would they be solvated in the water cluster?
• Is the reaction an artefact of not equilibrating the system before setting up the reaction?
• How many trajectories were carried out? Are adequate statistics ensured or can this be considered a “rare event”.
• What was the starting geometries? At the transition state?
• Was the SO₃+H₂O reaction observed in any of the trajectories? The reaction without SA should also be tested.

Section 3.3: There is heavy referencing to the SI. Please also add the relevant data to the text. For instance, at line 303, how can the H₂S₂O₇ formation reaction matter if SO₃ + (H2O)₂ is the major sink?

Line 307-308: The “stability analysis” should be added to the manuscript.

Line 312: The application of the enhancement factor yields an incorrect picture of the importance of H₂S₂O₇ for cluster formation. Sulfuric acid and ammonia form very weakly bound electrically neutral clusters. Usually, ions are required to facilitate the process. Hence, large enhancement factors (R) are an artefact of dividing with a very small number. Please mention the absolute formation rates to ensure that the cluster formation rate is not zero.

Line 316: An R value of 1.0 will mean that there is no enhancement. Hence, I do not believe that this can be stated. In addition, please add the numbers and explain how this conclusion of DSA being a “better enhancer” is drawn.

Line 325: Please mention the absolute rates here to let the reader know if this enhancement of many orders of magnitude is actually meaningful.

Line 336-339: “Hence, it can be forecasted that the participation of DSA in SA-A-based NPF can likely enhance the number concentration of atmospheric particulates significantly in the polluted atmospheric boundary layer (278.15 K) areas with relatively high [DSA] and [A]. ”

I do not believe this claim is adequately supported by the data. Please report the absolute values to support the conclusion.

Line 367-368: “Furthermore, the adsorption capacity of the S2O72-, H3O+ and SA- to gasous precursors in the atmosphere was further investigated.”

How was this evaluated? From Table 2 it looks like only the binding free energies were calculated. I guess the addition free energy of a given species should represent adsorption?

Line 373-374: I do not believe you can use a charged 2-3 molecular cluster in the gas-phase to draw conclusions about the “acceleration of particle growth”

Line 380-382: “It was demonstrated that S2O72- has the highest potential to stabilize SA-A clusters and promote SA-A nucleation in these clusters due to its acidity and structural factors such as more intermolecular hydrogen bond binding sites”

I do not understand how this conclusion is drawn. What is the acidity of each of the compounds?

Line 384: An ion at a particle interface does not influence NPF.