To investigate the MSA-assisted CH₂OO hydrolysis in the gas phase, geometric configurations and vibrational frequencies of all relevant configurations, including reactants, pre-reactive complexes, transition states (TSs), post-reactive complexes, and products, were calculated at the M06-2X/6-311++G(2df,2pd) level (Mardirossian and Head-Gordon, 2016; Pereira et al., 2017) using Gaussian 09. Intrinsic reaction coordinate analyses, performed at the same computational level, verified the correspondence of each TS to its associated pre- and post-reactive complexes. Subsequently, the ORCA 4.2.0 package (Neese, 2012) was performed to compute the single point energies using the CCSD(T)-F12/cc-pVDZ-F12 method.

To identity the global minimum energy configurations of (MSA)_x(MA)_y(HMHP)_z clusters, the ABCluster program (Zhang and Dolg, 2015) was utilized to systematically generate initial configurations for various clusters. Subsequently, these structures were further optimized using different levels of theoretical methods. Specifically, the ABCluster program generated n × 1000 (1 < n ≤ 4) initial structures for each cluster system. Then, the PM6 semi-empirical method (Partanen et al., 2016) was used for preliminary optimization of these structures. As the M06-2X functional performs well for noncovalent binding and structural predictions of thermochemical and atmospheric aggregates, 100 of the lowest-energy configurations were chosen from n × 1000 (1 < n ≤ 3) structures and re-optimized using M06-2X/6-31G(d,p). Next, the 10 most stable configurations were re-optimized at the M06-2X/6-311++G(2df,2pd) level, and the configuration with the lowest free energy was identified. Ultimately, single-point energies were evaluated at the DLPNO-CCSD(T)-F12/cc-pVDZ-F12-CABS level (Tsona Tchinda et al., 2022) in ORCA, using geometries optimized for the stable clusters at the M06-2X/6-311++G(2df,2pd) level.

The rate coefficients of MSA-mediated CH₂OO hydrolysis were investigated through a two-step approach. Initially, high-pressure-limit (HPL) rate coefficients were obtained using the VRC-VTST (Zhang et al., 2023, 2024) approach implemented in Polyrate 2017-C (Meana-Pañeda et al., 2024). Subsequently, MESMER (Master Equation Solver for Multi-Energy Well Reactions) (Glowacki et al., 2012) was employed to calculate the rate coefficients of MSA-catalyzed CH₂OO hydrolysis across the temperature range of 280.0–320.0 K. The rate coefficients for the barrierless transition from separated reactants to pre-reactive complexes were estimated using the Inverse Laplace Transform (ILT) (Kumar et al., 2021) method. Concurrently, RRKM theory was employed to calculate the rate coefficients describing the conversion of the pre-reactive complex into the post-reactive species through the transition state. Further descriptions of the ILT methods and RRKM theory are provided in Sects. S1 and S2 of the Supplement, respectively.

BOMD calculations were carried out in CP2K program (Hutter et al., 2014), with exchange and correlation interactions described by the BLYP functional (Becke, 1988). Dispersion effects were incorporated through Grimme's dispersion (Grimme et al., 2010) (BLYP-D3) method. The Goedecker-Teter-Hutter (GTH) (Goedecker et al., 1996) pseudopotential was adopted for the core region, whereas the valence electrons were represented through a Gaussian DZVP basis (Phillips et al., 2005) in conjunction with an auxiliary plane-wave set. The calculations employed a plane-wave energy cutoff of 280 Ry together with a 40 Ry cutoff for the Gaussian basis set. In the gas phase, a supercell with a side length of 15 Å was employed to minimize periodic boundary effects, and the integration was performed with a 0.5 fs time step. A water droplet containing 191 molecules, serving as the interfacial model, was subjected to BOMD pre-optimization for roughly 5.0 ps at 300 K. CH₂OO and MSA were then positioned at the air-water interface, followed by 10 ps of molecular dynamics simulation. A supercell length of 35 Å was adopted to prevent interactions between periodic images of the droplet, while the dynamics were advanced with a timestep of 1.0 fs. The gas-phase and interfacial simulations were conducted under NVT conditions at ∼ 300 K, with the temperature maintained via a Nosé-Hoover thermostat.

ACDC (McGrath et al., 2012) was utilized to investigate the cluster formation rates, steady-state cluster concentrations, and growth mechanisms within the extensive MSA-MA-HMHP system. Thermodynamic data, calculated at the DLPNO-CCSD(T)-F12/cc-pVDZ-F12-CABS//M06-2X/6-311++G(2df,2pd) level of theory, were employed as input parameters for the ACDC simulations. The temporal progression of cluster concentrations was numerically resolved by solving the birth-death equations, employing the ode15s solver within the MATLAB-R2013a program. The birth-death equations are given below,

1dcidt=12∑j<iβj,(i-j)CjC(i-j)+∑jγ(i+j)→iCi+j-∑jβi,jCiCj-12∑j<iγi→jCi+Qi-Si In this formulation, ci denotes the concentration of cluster i,βi,j represents the collision coefficient of clusters i with j and then γ(i+j)→i represents the evaporation coefficient of clusters evaporating from i+j to i and j clusters. represents the potential source for cluster i, while Si denotes the sink term for cluster i. External losses of cluster i were represented using a fixed condensation sink coefficient of 0.02 s⁻¹ (Qiao et al., 2024; Zhang et al., 2022). Besides, the clusters (MSA)₄ ⋅ (MA)₃ and (MSA)₄ ⋅ (MA)₄ were selected as boundary clusters in the MSA-MA-HMHP system, as boundary clusters in ACDC are required to be sufficiently stable to allow continued growth.

To evaluate the catalytic role of MSA in the CH₂OO hydrolysis reaction, the potential energy surface was investigated both in the presence and absence of MSA and H₂O. As depicted in Fig. 1a and b, the potential energy surfaces with and without water closely match previously reported data (Wang et al., 2021a, b), suggesting that the CCSD(T)-F12/cc-pVDZ-F12//M06-2X/6-311++G(2df,2pd) method is suitable for assessing MSA's catalytic effect. The MSA-catalyzed CH₂OO hydrolysis reaction (Fig. 1c) follows a continuous bimolecular process, involving CH₂OO ⋯ H₂O + MSA and MSA ⋯ H₂O + CH₂OO. The stabilization energy of MSA ⋯ H₂O is 9.8 kcal mol⁻¹ higher than that of CH₂OO ⋯ H₂O. Considering the atmosphere concentrations of MSA (Li et al., 2024b), H₂O (Anglada et al., 2013), and CH₂OO (Khan et al., 2018), the concentration of MSA ⋯ H₂O is two orders of magnitude greater than that of CH₂OO ⋯ H₂O, as shown in Table S3, predicting that the MSA ⋯ H₂O + CH₂OO route is the dominant pathway for the MSA-catalyzed reaction. Starting from MSA ⋯ H₂O + CH₂OO, the reaction proceeds through the IM_MSA intermediate, which has stabilization energy 6.5 and 0.7 kcal mol⁻¹ higher than IM and IM_WM, respectively. The increase in stabilization energy is primarily due to the addition of MSA, which reduces ring strain and favours C-O bond formation. Atoms in Molecules (AIM) topological analysis results also show that in IM_MSA (ρ = 3.74 × 10⁻²), the strength of the density of all electrons (C-O) is 1.64 times stronger than in IM_WM (ρ = 2.27 × 10⁻²), further stabilizing the intermediate. The reaction then proceeds through the barrierless transition state TS_MSA, forming a nine-membered ring complex HOCH₂OOH ⋯ MSA (labeled as IMF_MSA). This process has a 1.4 kcal mol⁻¹ lower energy barrier compared to the H₂O-catalyzed reaction, due to MSA's enhanced ability to facilitate proton transfer. Additionally, IMF_MSA is more stable by 7.6 kcal mol⁻¹ compared to IMF_WM.

To further examine the impact of MSA on the CH₂OO hydrolysis reaction, we calculated the effective rate constants for the CH₂OO hydrolysis assisted by H₂O ([H₂O] = 10¹⁵–10¹⁷ molec. cm⁻³), NH₃ ([NH₃] = 10⁷–10¹¹ molec. cm⁻³), H₂SO₄ ([H₂SO₄] = 10⁴–10⁸ molec. cm⁻³) and MSA ([MSA] = 10⁴–10⁸ molec. cm⁻³) in Table S5. When the MSA concentration ranges from 10⁶ to 10⁸ molec. cm⁻³, its catalytic effect is substantially stronger than that of NH₃ ([NH₃] = 10⁷–10¹¹ molec. cm⁻³), with kMSA′ being approximately 2–6 orders of magnitude over the temperature range of 280.0–320.0 K. Similarly, for MSA concentrations between 10⁵ and 10⁸ molec. cm⁻³, MSA exhibits a significantly higher catalytic activity than SA ([SA] = 10⁴–10⁷ molec. cm⁻³). In this case, kMSA′ exceeds kSA′ by about 1–3 orders of magnitude. Taken together, these results demonstrate that MSA is a more effective catalyst than both NH₃ and SA under atmospherically relevant conditions. Nevertheless, even under extreme conditions, with MSA at its upper-limit concentration ([MSA] = 10⁸ molec. cm⁻³) and H₂O at its lower-limit concentration ([H₂O] = 10¹⁶ molec. cm⁻³), kMSA′ is approximately five orders of magnitude smaller than kWM′, indicating that the catalytic efficiency of MSA remains lower than that of H₂O.

The mechanism of MSA-catalyzed CH₂OO hydrolysis at the air-water interface remained unclear. To elucidate this process, BOMD simulations were employed to investigate possible reaction pathways at the aqueous interface. Based on analogies with CH₂OO reactions involving other atmospheric species (Ding et al., 2024; Li et al., 2023; Cheng et al., 2025), three potential routes were considered: (i) MSA interacting with adsorbed CH₂OO at the air-water interface, (ii) CH₂OO interacting with adsorbed MSA at the air-water interface, or (iii) the MSA-CH₂OO complex reacting at the air-water interface. Because both CH₂OO and MSA exhibit high reactivity toward interfacial water, their lifetimes at the droplet interface remain very brief, making pathway (i) and (ii) far less likely than pathway (iii). Therefore, only the reaction of the MSA ⋯ CH₂OO complex at the aqueous interface was considered, owing to its high stability in the gas phase.

As displayed in Figs. 2 and S1 and Movie S1 in the Supplement, the MSA ⋯ CH₂OO complex initially interacts with a water molecule at the aqueous interface. At 0.76 ps, this water molecule forms both a van der Waals interaction with the carbon atom in CH₂OO (d(C-O₃) = 4.00 Å) and a hydrogen bond with the oxygen atom in MSA (d(O₄-H₂) = 3.27 Å), thereby generating a nine-membered ring structure (CH₂OO ⋯ MSA ⋯ H₂O). As the reaction progresses, a hydration transition-state-like ring configuration forms at 1.06 ps, with the C-O₃, O₄-H₂ and O₂-H₁ bonds shorten to 1.75, 1.27 and 0.97 Å, respectively, while the O₃-H₂ and O₁-H₁ bonds elongate to 1.16 and 1.85 Å, respectively. By 1.16 ps, the bond lengths of C-O₃, O₄-H₂ and O₂-H₁ further shorten to 1.43, 1.05 and 0.96 Å, while the O₃-H₂ and O₁-H₁ bonds further increase to 1.61 and 1.93 Å, respectively, indicating the formation of the hydrogen-bonded CH₂OOHOH ⋯ MSA complex. During this process, the interfacial water molecule acts as a reactant, with MSA serving as a proton transfer bridge. Notably, due to the higher abundance of interfacial water molecules compared to MSA, the formation of HMHP at the interface primarily proceeds through the direct hydration of CH₂OO. However, the source of HMHP from MSA-catalyzed CH₂OO hydrolysis at the interface occurs at a significantly faster rate than its corresponding gas-phase formation, with a computed ratio (r1) of 1.01 × 10² at 298.0 K (Table 1), which is detailed computational results are provided in Sect. S3. Traditionally, the loss of CH₂OO in the troposphere has been primarily attributed to its hydrolysis. Therefore, it is crucial to examine the rate ratio r2 (Eq. S4) between interfacial MSA-catalyzed CH₂OO hydrolysis and the corresponding gas-phase process mediated by water. At 298.0 K, r2 is 13.4 (Table 1), indicating that the formation of HMHP from interfacial MSA-mediated CH₂OO hydrolysis is much closer to that catalyzed by water in the gas phase. These results indicate that interfacial MSA-catalyzed CH₂OO hydrolysis is a significant source of HMHP formation in MSA-polluted areas under relatively humid conditions. Consequently, when evaluating the comprehensive sources of HMHP in MSA-rich regions, it is essential to consider the formation involving MSA-catalyzed hydrolysis of CH₂OO at the air-water interface.

The role of HMHP on MSA-MA-driven ternary nucleation was assessed. Initial assessments focused on the potential interaction sites of HMHP with MSA and MA, followed by an analysis of the conformation and stability of the (MSA)_x(MA)_y(HMHP)_z (0 ≤ y ≤ x + z ≤ 3) clusters. The MSA-MA nucleation mechanism involving HMHP was then examined, along with the effects of temperature and precursor concentrations on the MSA-MA-HMHP system. Finally, the atmospheric implication of HMHP for MSA-MA nucleation were calculated for urban industrial areas.

Building on the above findings, low T coupled with low [MSA] and [MA] appear to favor an enhanced role of HMHP in MSA-MA nucleation. To assess the atmospheric significance of these variations, we quantified the contributions of MSA-MA cluster growth pathways, with and without HMHP involvement (Fig. 6), under favorable conditions of temperature (T = 258.15 K) and precursor concentrations ([MSA] = 1.00 × 10⁴ molec. cm⁻³, [MA] = 2.50 × 10⁷ molec. cm⁻³). Indeed, substantial variability in atmospheric HMHP concentrations has been observed across diverse environments worldwide. For example, levels range from 2.50 × 10⁹ to 6.25 × 10⁹ molec. cm⁻³ in Central Portugal, Pabstthum, and Beijing. Higher concentrations, between 1.15 × 10¹⁰ to 3.00 × 10¹⁰ molec. cm⁻³, have been observed in Guang Zhou and Niwot Ridge, while the Southeastern United States exhibit the highest concentrations, reaching up to 1.25 × 10¹¹ molec. cm⁻³. As shown in Fig. 6, in low [HMHP] regions such as Pabstthum and Beijing, HMHP-involved pathways account for only 11 % and 12 % of total NPF, respectively. In contrast, in environments characterized by high HMHP concentrations, such as the southeastern United States (1.25 × 10¹¹ molec. cm⁻³) and Niwot Ridge (3.00 × 10¹⁰ molec. cm⁻³), HMHP-involving nucleation pathways become dominant. Under these conditions, HMHP acts both as a “catalyst”, facilitating the formation of MSA-MA clusters, and as an “participant” in the assembly of critical clusters (Figs. S5 and S6). These two roles contribute up to 59 % and 42 %, respectively, to the overall nucleation process. These results highlight that HMHP exerts a markedly stronger influence on MSA-MA nucleation at elevated concentrations, particularly in urban industrial regions, where its contribution to NPF can be substantial.

Previous studies have revealed that SA-MA and SA-A nucleation mechanisms are widely regarded as key contributors to new particle formation in urban industrial regions (Yin et al., 2021; Liu et al., 2021). To underscore the importance of MSA-MA-HMHP nucleation in urban industrial regions, the cluster formation rates (J) of the MSA-MA-HMHP system have been compared with those of the SA-MA and SA-A systems (Qiao et al., 2024) (Fig. S7). The results show that, over the temperature range of 238.15–298.15 K, the J of MSA-MA-HMHP system is 1–5 orders of magnitude higher than that of SA-MA system at equivalent precursor concentrations ([SA] = 1.00 × 10⁶ molec. cm⁻³ and [MA] = 1.00 × 10⁸ molec. cm⁻³). Similarly, under the conditions of [SA] = 1.00 × 10⁶ molec. cm⁻³ and [A] = 1.00 × 10¹⁰ molec. cm⁻³, the J of MSA-MA-HMHP systems slightly exceeds that of SA-A system by approximately 5–6 orders of magnitude. These comparisons suggest that HMHP plays a key role in enhancing MSA-MA nucleation, particularly in urban industrial environments.