Geometry optimizations of stationary points (reactants, products, and transition states) were performed at the B3LYP/6-311+G(d,p) level (Krishnan et al., 1980; McLean and Chandler, 1980; Stephens et al., 1994) with the DFT-D3 dispersion correction of Grimme et al. (2010) using Gaussian 16 package (Frisch et al., 2016). Harmonic frequency analyses were conducted at the same level. The transition states are verified to be connected with the corresponding reactants or products by means of intrinsic reaction coordinate (IRC) calculations. The solvation model based on density implicit solvent model (SMD) was applied to represent the aqueous phase. To obtain more accurate electronic energies, single-point energy calculations were subsequently performed at the DLPNO-CCSD(T) level of theory employing the aug-cc-pVTZ basis set (Guo et al., 2018; Riplinger and Neese, 2013; Riplinger et al., 2013). These calculations were carried out using the ORCA 5.0.4 quantum chemistry program package (Neese, 2011). Fukui function analysis evaluated the nucleophilic reactivity of atoms within the HOSO2- anion. Electron density difference maps were analyzed to reveal charge transfer between reactant fragments. All wave function analyses were performed by Multiwfn 3.8 (dev) code (Lu and Chen, 2012, 2023; Zhang and Lu, 2021).

Born–Oppenheimer molecular dynamics (BOMD) simulations were performed using the CP2K package (Kuhne et al., 2020). All BOMD simulations employed the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional (Perdew et al., 1996) with the DFT-D3 dispersion corrections of Grimme et al. (2010). This D3 correction has been shown to improve the description of water at the water–air interface at the GGA level (Dodia et al., 2019). A double-ζ Gaussian basis set (DZVP-MOLOPT-SR-GTH) was used for valence electrons in conjunction with Goedecker–Teter–Hutter (GTH) pseudopotentials for core electrons (Goedecker et al., 1996; VandeVondele and Hutter, 2007). The plane wave and Gaussian basis set cutoff energies were set to 300 and 40 Ry, respectively. The simulations were conducted in the canonical (NVT) ensemble using the canonical sampling through velocity rescaling (CSVR) thermostat (Bussi et al., 2007) with a time step of 1 fs. For aqueous-phase simulations, a cubic box of 1.7 nm × 1.7 nm × 1.7 nm containing 128 water molecules was employed. The air–water interface model was constructed by expanding the z-axis of this cubic box to 4.7 nm while keeping the xy-plane dimensions (1.7 nm × 1.7 nm) and water molecules (128) unchanged, which generated vacuum regions above and below the aqueous slab to establish two air–water interfaces. Free energy profiles were computed via well-tempered metadynamics (see Sect. S1 in the Supplement for details) as implemented in PLUMED (Bonomi et al., 2009). Collective variables for HMS and HMSi formation are defined in Fig. S1 in the Supplement. Convergence was assessed by monitoring Gaussian hill height evolution (Fig. S2), collective variable sampling behavior (Fig. S3), and the time evolution of free energy profiles (Fig. S4). Detailed configurations for air–ice interface simulations are provided in the Supplement (Fig. S5).

Bulk aqueous reactions represent an important pathway for HCHO–HOSO2- interactions in atmospheric chemistry. Although previous DFT studies employing implicit solvation models have provided valuable understanding of this reaction (Chen and Zhao, 2020), explicit solvation approaches offer complementary insights by capturing discrete water-reactant interactions and dynamic hydrogen bonding that may play important roles in determining reaction mechanisms and kinetics. To provide a more detailed molecular-level picture of bulk aqueous HMS formation, we employed BOMD simulations with metadynamics to investigate the HCHO–HOSO2- reaction with explicit water molecules.

Given that HOSO2- contains multiple potential nucleophilic sites (sulfur, non-hydroxyl oxygens, and hydroxyl oxygen), we first assessed their relative reactivities through preliminary quantum chemical calculations (Fig. S6). These analyses confirmed that the hydroxyl oxygen O(H) in HOSO2- exhibits significantly lower nucleophilic reactivity compared to the sulfur atom and non-hydroxyl oxygen atoms. Furthermore, reactions involving the O(H) did not yield viable products. Based on these findings, reactions involving the O(H) were excluded from subsequent metadynamics simulations. As illustrated in Fig. 1, two distinct reaction pathways dominate the bulk-phase HCHO + HOSO2- reaction. In Path I (Fig. 1b), the sulfur atom of the HOSO2- attacks the carbon atom of formaldehyde, forming a seven-membered ring transition state with the participation of a water molecule. The calculated free energy barrier for the formation of HMS is 7.7 kcal mol⁻¹. Path II (Fig. 1d) proceeds via nucleophilic attack of a non-hydroxyl oxygen atom of HOSO2- on the carbonyl carbon atom of HCHO. This pathway proceeds through a ring transition state mediated by water molecules (Fig. 1d). The free energy barrier for HMSi formation via this route is 7.6 kcal mol⁻¹. These values are in close agreement with barriers derived from high-level quantum chemical calculations (Fig. S7), validating the reliability of the simulation results. The comparable barriers for HMS and HMSi formation in bulk water (7.7 vs. 7.6 kcal mol-1) indicate that both isomers form competitively in bulk water. Despite these moderate free energy barriers, bulk phase HMS formation alone is insufficient to reconcile the atmospheric sulfur budget gap. Atmospheric modeling studies incorporating bulk-phase HMS chemistry can account for only approximately one-third of the discrepancy between modeled and observed sulfate concentrations during severe haze episodes (Song et al., 2019). This shortfall may be partly attributed to the high surface-to-volume ratios of atmospheric aerosols and cloud droplets, where interfacial processes may play a more significant role than bulk-phase reactions.

Given the limitations of bulk-phase chemistry identified above, understanding HCHO-HOSO2- reactivity at air–water interfaces becomes critical. Recent investigations reveal significant accumulation of both HCHO and HOSO2- at the air-water interface (Martins-Costa et al., 2012; Yang et al., 2019). This interfacial enrichment suggests potentially significant atmospheric reaction mechanisms previously overlooked. Herein, we employed BOMD simulations to investigate these interfacial heterogeneous reactions at the air-water interface.

Similar to bulk-phase reactions, interfacial processes proceed via two distinct pathways. Remarkably, HMS formation via Path I proceeds rapidly at the air–water interface. Starting from reactant complex configurations optimized at the B3LYP-D3/6-311+G(d,p) level, unbiased BOMD trajectories reveal that the reaction completes within ∼ 3 ps (Figs. 2a, S8), in contrast to the bulk-phase reaction. During the reaction, the distance between the C atom of HCHO and the S atom of HOSO2- gradually decreases to approximately 1.95 Å (Fig. 2d, e), with a corresponding Mayer bond order of 0.57, indicating that an S–C bonding interaction has been established at this intermediate stage. The resulting anionic intermediate structurally resembles HMS, differing only in proton location, which remains on the O atom of the HOSO2-. Electron density difference analysis demonstrates decreased electron density surrounding the S atom with concurrent increase at the formaldehyde O atom during intermediate formation (Fig. 2f). Mulliken population analysis quantifies this charge transfer: the charge on S atom in HOSO2- increases from approximately 0.6 to 0.81 e, while the charge on O atom in HCHO decreases from approximately -0.3 to -0.57 e (Fig. S9). These analyses can provide clear evidence for electron transfer from HOSO2- to HCHO in the initial nucleophilic addition step. Subsequently, at approximately 2.42 ps, a transition state structure forms wherein the O2 atom of proximal H₂O approaches the H2 atom of the reaction intermediate, while the H1–O2 bond in H₂O lengthens. Interfacial water molecules facilitate the final step of the reaction, completing HMS formation while shortening the S–C bond from approximately 1.95 to 1.81 Å (Fig. 2g). Unlike the bulk-phase reaction, where S–C bond formation and proton transfer occur synchronously (Fig. 1b), the interfacial HMS formation proceeds via a stepwise mechanism: the S–C bond forms initially, followed by water molecule-mediated proton transfer from HOSO2-.

To obtain quantitative free energy profiles and systematically explore the interfacial reaction pathways, we performed computationally expensive metadynamics-biased BOMD simulations. Metadynamics simulations reveal an exceptionally low free energy barrier of 0.6 kcal mol⁻¹ for interfacial HMS formation (Fig. 2b), corroborating the fast reaction observed in unbiased trajectories. We validated our BOMD findings using high-level quantum chemical calculations using a ten-water-molecule cluster model (W10) at the DLPNO-CCSD(T)/aug-cc-pVTZ//B3LYP-D3/6-311+G(d,p) level. These high-level calculations further validated the near-barrierless HMS formation, providing complementary evidence to our metadynamics simulations (Fig. S10).

Our results differ from Li et al. (2025)'s recent report of a ∼ 7.6 kcal mol⁻¹ free energy barrier for this process. This discrepancy stems from their use of a single distance collective variable d(C⋯S) in metadynamics simulations, which overlooked the critical water-mediated cyclic proton transfer mechanism. In contrast to the rapid formation of HMS, no fast formation of HMSi was observed across multiple simulation trajectories. Metadynamics simulations revealed a higher free energy barrier of 6.1 kcal mol⁻¹ for HMSi formation at the interface (Fig. 2c), with the reaction mechanism paralleling that observed in the bulk phase. Although this barrier represents a 1.5 kcal mol⁻¹ reduction from the bulk-phase value (7.6 kcal mol⁻¹), it remains substantially higher than the negligible barrier for HMS formation.

To elucidate the potential contributor to the enhanced interfacial reactivity, we analyzed the solvation environments of HCHO and HOSO2- at the interface versus bulk water (Fig. S11). The partial solvation characteristic of the air-water interface results in reduced water coordination around both reactants relative to their bulk-phase counterparts, potentially facilitating reactant approach. Furthermore, the HOMO–LUMO gap of the reactive complex is smaller at the interface than in bulk water, suggesting that the partially solvated environment may contribute to the reaction acceleration (Fig. S12) (Ishiyama et al., 2022; Kusaka et al., 2021). It is important to note that the interfacial acceleration mechanism is multifaceted. Beyond partial solvation effects, other factors including the interfacial electric field and synergistic enthalpy-entropy effects play crucial roles in promoting HMS formation at the air-water interface (Li et al., 2025). These computational findings demonstrate a clear mechanistic preference for HMS formation over HMSi at the air-water interface, with a markedly lower free energy barrier (ΔG=0.6 kcal mol⁻¹ for HMS versus 6.1 kcal mol⁻¹ for HMSi). The exceptionally low barrier indicates that HMS formation proceeds very rapidly, especially in the polluted regions with abundant HCHO and SO₂ in the atmosphere, which can provide an important theoretical foundation for more accurate simulation of atmospheric sulfur budgets.

The strongly acidic nature of many atmospheric environments, as discussed earlier, leads to protonation of HCHO, forming HCHOH⁺. Experimental studies (Jayne et al., 1996) have demonstrated that when pH drops below 2, formaldehyde undergoes efficient uptake by acidic droplets, leading to its protonation and formation of HCHOH⁺. However, the detailed mechanistic consequences of this protonation, especially as they manifest in heterogeneous environments, remain poorly understood. To capture the complex reactivity landscape of HCHOH⁺ with HOSO2- under strongly acidic conditions, we conducted 25 independent BOMD simulations (detailed configurations in Fig. S13). Figure 3a displays the temporal evolution of product formation across all simulations, with each horizontal line representing an individual BOMD trajectory. The colored markers indicate specific chemical transformations. Hydroxymethanesulfonic acid (HMSA) and hydroxymethyl hydrogen sulfite (HMHSi) formed within 2 ps in most trajectories. The subsequent deprotonation processes led to the formation of HMS and HMSi within approximately 10 ps. The mechanistic diversity enabled by formaldehyde protonation under strongly acidic conditions represents a fundamental departure from reactions involving neutral formaldehyde molecules. Figure 3b illustrates the three distinct mechanistic pathways identified through our simulations. Path A proceeds via nucleophilic attack by sulfur atom of HOSO2- at the carbonyl carbon of HCHOH⁺, leading to HMSA formation before transforming to HMS. Path B involves nucleophilic attack by either O1 or O2 of HOSO2- on the carbonyl carbon, forming HMSi.

Our trajectory analysis shows no significant preference between O1 and O2 attack, suggesting that both oxygen atoms experience similar activation in the interfacial acidic environment. The subsequent deprotonation of HMHSi to HMSi proceeds through water-mediated proton transfer, with interfacial water molecules playing a crucial catalytic role. Perhaps most remarkably, Path C demonstrates that the typically unreactive hydroxyl oxygen O(H) of HOSO2- becomes a viable nucleophile under acidic conditions.

Although the O(H) site of HOSO2- is generally weakly nucleophilic (Fig. S6), protonation fundamentally alters the electronic structure of HCHO by introducing a new δ-hole on the oxygen atom while substantially intensifying the δ-hole at the carbonyl carbon (Fig. S14). These multiple electron-deficient sites work synergistically to attract even weak nucleophiles such as O(H), thereby enabling direct HMSi formation. This reactivity highlights the unique ability of highly acidic interfaces to modulate electronic structure, thereby activating HCHOH⁺ toward unconventional nucleophilic pathways. Statistical analysis of the product distribution reveals a notable preference for HMSi formation under acidic conditions: among the 25 trajectories, 6 proceeded through Path A to form HMS, while 19 yielded HMSi via either Path B or Path C, establishing an HMS : HMSi ratio of approximately 1 : 3. This product distribution contrasts sharply with the preferential HMS formation observed at neutral pH interfaces, highlighting the profound influence of acidity on reaction selectivity. The pH-dependent reaction selectivity becomes even more pronounced under extreme acidic conditions. Under extremely acidic conditions (pH ≈ 0), the speciation of sulfurous acids at interfaces undergoes significant shifts. As observed by Buttersack et al. (2024) HSO3- becomes the predominant species at the air-water interface under extremely acidic conditions. With the sulfur atom protonated and therefore unavailable for nucleophilic attack, S–C bond formation leading to HMS becomes inhibited, yielding only HMSi (Fig. S15). These results establish HMSi as a potentially significant contributor to atmospheric sulfur budgets in strongly acidic environments. From an analytical perspective, HMS and HMSi differ in charge distribution, molecular geometry, and bonding arrangements at the sulfur center, which may in principle enable their separation by targeted ion chromatography and their differentiation by high-resolution tandem mass spectrometry. Both techniques have been successfully applied to HMS detection (Wei et al., 2020; Campbell et al., 2022), although dedicated method development and experimental verification will be required to confirm the resolution of these two isomers in ambient samples.

Although air–water interfaces have been widely recognized as critical sites for heterogeneous reactions, the cold conditions prevalent in the upper troposphere and polar regions give rise to an additional interfacial environment in the form of atmospheric ice. These ice surfaces provide a distinct medium for heterogeneous chemistry beyond conventional water vapor and aqueous surfaces, particularly relevant given that HCHO and SO₂ are ubiquitous throughout the atmosphere, including regions where temperatures favor ice formation (Anderson et al., 2017; Bai et al., 2023; Höpfner et al., 2015; Joppe et al., 2024; Sumner and Shepson, 1999). Despite the atmospheric abundance of ice crystals and the widespread distribution of these reactive species, the heterogeneous chemistry occurring at ice surfaces remains poorly characterized compared to reactions in other atmospheric environments. Considering the potential significance of ice-surface heterogeneous reactions for atmospheric sulfur cycling, we investigated the reaction between HCHO and HOSO2- at the air-ice interface to elucidate the associated reaction pathway. Given that the freezing process can lead to ion redistribution, we employed classical molecular dynamics (MD) simulations to track HOSO2- spatial distribution during ice formation (see Sect. S3 for details). The results showed that HOSO2- ions preferentially accumulate at the ice-air interface (Fig. S16), which is likely attributed to their exclusion from the growing ice crystal during crystallization (Ning et al., 2024; Tsironi et al., 2020). Next, we explored the reaction between HCHO and HOSO2- at the air-ice interface, with a quasi-liquid layer (QLL) on the ice crystal (Nagata et al., 2019) at T=243 K as shown in Fig. 4a. This temperature represents a typical polar temperature (Zhang et al., 2024a).

Energy decomposition analysis (Lu and Chen, 2023) (Fig. 4b; see Sect. S2 for methodology) revealed that the interaction between HCHO and HOSO2- fragments is driven by electrostatic (ΔEels) and orbital interaction (ΔEorb) components, with modest contributions from dispersion forces (ΔEdisp). Electronic structure analysis (Fig. S17) confirmed HOSO2- as the active electron donor throughout the reaction pathway. These results show that strong electrostatic attraction and orbital interactions jointly facilitate the approach of the two reactants and promote S–C bond formation on the ice surface. Subsequent water-mediated proton transfer yielded HMS. In addition, we investigated HMS formation at temperatures characteristic of the upper troposphere and lower stratosphere. The reaction remains feasible at 223 K, which represents a typical temperature of the high-altitude free troposphere (Li et al., 2024), and HMS formation proceeded rapidly in unbiased BOMD trajectories, completing within ∼ 3 ps through similar mechanistic pathways (Fig. S18). In contrast, HMSi formation was not observed in unbiased trajectories. Metadynamics simulations revealed free energy barriers of 5.4 kcal mol⁻¹ at 243 K and 5.7 kcal mol⁻¹ at 223 K for HMSi formation (Figs. S19 and S20). Additionally, the protonated formaldehyde pathway remained viable on ice surfaces (Figs. S21 and S22). Our simulations revealed three distinct reaction pathways on ice surfaces (Fig. S21): nucleophilic attack at the sulfur atom (Path A, forming HMS), at oxygen atoms O1/O2 (Path B, forming HMSi), and at the hydroxyl group O(H) (Path C, also forming HMSi). Furthermore, when HSO3- serves as the sulfur species, the reaction exclusively yields HMSi (Fig. S22).

Our mechanistic findings provide a potential explanation for field observations reporting enhanced HMS concentrations during cold atmospheric periods (Song et al., 2019; Campbell et al., 2024). While HMS formation involves multiple complex atmospheric processes, our results demonstrate that ice-mediated heterogeneous chemistry may significantly enhance this process through some key contributing factors. First, the substantially increased Henry law constants for both HCHO and SO₂ at reduced temperatures facilitate enhanced partitioning of these precursors to condensed phases (Campbell et al., 2022). Second, efficient HMS formation on ice surfaces establishes a previously unrecognized heterogeneous pathway operating across cold atmospheric environments from polar regions to the upper troposphere. This ice-mediated chemistry represents an important yet underappreciated contributor to global organosulfur compound formation and atmospheric sulfur cycling. It should be noted that real atmospheric aerosol and cloud-droplet interfaces are chemically more complex than the simplified system examined here, and they may contain dissolved oxidants, organic species, transition metal ions, and other S(IV) components that could influence interfacial reactivity by competing for reactive sites or modifying the local environment. In future work, we intend to confirm the impacts of other atmospheric components.

To further analyze the atmospheric implications of these products, the dynamic interfacial behavior of HMS and HMSi was characterized through quantitative analysis of hydrogen-bonded water molecules at the air-water interface. As shown in Fig. 5, the reaction products are stabilized at the interface through hydrogen bonds (HBs) with surrounding water molecules. Specifically, HMS and HMSi exhibit the highest probability of forming hydrogen bonds with 4–7 interfacial water molecules, exceeding 80 % probability. These products remain bound to the interface through multiple hydrogen-bonding interactions with water molecules, which consequently impedes their transfer into the gas phase. This interfacial retention reflects their strong hydrophilicity, as evidenced by the extensive hydration distribution of HMS and HMSi under different relative humidities (RH = 25 %–100 %) and temperatures (T = 240–298 K; Fig. S23), which further suggests their potential to enhance aerosol hygroscopic growth.

To further elucidate the interfacial behavior of HMS and HMSi, we investigated the positional correlations between their active sites (Oa, Ob, Oc, and Od) and the air–water interface. Figure 5c depicts the distances between these active sites and the Gibbs dividing surface (GDS) to determine their positions at the interface, while Fig. 5d quantifies the probability distributions of each oxygen atom (Oa, Ob, Oc, and Od) across the air-water interface. For HMS, the Ob, Oc, and Od atoms predominantly reside within the interfacial region (yellow shaded area), while the hydroxyl oxygen, designated as Oa, exhibits maximum probability at positive distances from the GDS plane. This indicates preferential orientation of the hydroxyl group toward the gas phase. For HMSi, while the distribution pattern is comparable, the probability distribution curve for Oa shows a significant reduction in extension toward the gas phase compared to HMS. Notably, the spatial distributions of HMS and HMSi at the air-ice interface exhibit similar orientational preferences to those observed at the air-water interface (Fig. S24). This differential orientation of the hydroxyl oxygen may have atmospheric implications, as the greater gas-phase exposure of the hydroxyl group in HMS compared to HMSi suggests a potential enhancement of interfacial oxidation due to orientation preferences, which could facilitate its interaction with atmospheric oxidants such as hydroxyl radicals (OH) and oxygen (O₂). These molecular-level insights into the interfacial behavior of HMS and HMSi provide mechanistic understanding of their atmospheric processing and environmental fate, enabling a more accurate assessment of the contributions of organosulfur compounds to aerosol formation.