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Article

Theoretical Exploration of Isomerization Pathways in H2SO4·HX (X = OH, Cl, Br) Complexes

1
Chongqing Key Laboratory of Chemical Theory and Mechanism, School of Chemistry and Chemical Engineering, Chongqing University, Chongqing 401331, China
2
Xinlian Microelectronics Co., Ltd., Chongqing 401332, China
*
Authors to whom correspondence should be addressed.
Appl. Sci. 2025, 15(14), 7642; https://doi.org/10.3390/app15147642
Submission received: 9 June 2025 / Revised: 6 July 2025 / Accepted: 7 July 2025 / Published: 8 July 2025

Abstract

Complexes formed by sulfuric acid (H2SO4) with HX (X = OH, Cl, Br) are critical in various chemical processes. In this work, we theoretically investigated the isomerization pathways of these complexes, analyzing their structures, energies, and reaction mechanisms. We identified eight, eight, and nine isomers for the H2SO4 + HX systems with X = OH, Cl, and Br, respectively, including mirror-symmetrized structures. The most stable complexes in each system are cyclic, stabilized by double hydrogen bonds forming six-membered rings. We discovered 7, 10, and 10 new transition states for the reaction involving X = OH, Cl, and Br, respectively. Isomer rearrangements primarily involve the hydrogen bond dynamics, hydrogen atom exchange, and cis–trans isomerization of H2SO4 due to wagging of its non-interacting O–H bond. Our findings underscore the dominance of hydrogen bonding in these intermolecular interactions and provide fresh insights into the nature and reactivity of these complexes.

1. Introduction

Sulfuric acid (H2SO4) is a highly reactive strong acid that plays a pivotal role in various atmospheric processes, particularly in aerosol formation [1,2,3,4,5] and cloud chemistry [6,7,8]. In the atmosphere, H2SO4 participates in heterogeneous phase reactions, contributing to the formation of aerosols and clouds, which influence climate and atmospheric chemistry [9,10]. It is involved in the nucleation of new particles, a key process in aerosol formation, and it facilitates the formation of sulfate aerosols, which serve as cloud condensation nuclei in the troposphere [11]. Moreover, H2SO4 in the presence of water vapor can form hydrates, which significantly affect its atmospheric behavior [12].
A critical aspect of atmospheric sulfuric acid chemistry is its interaction with other atmospheric species, including water vapor (H2O), hydrogen chloride (HCl), and hydrogen bromide (HBr). These interactions are not only important for aerosol formation but also for processes such as ozone depletion in the polar stratosphere [13,14]. H2SO4, HCl, HBr, and H2O are key players in the formation of particles in polar stratospheric clouds (PSCs), which are central to the catalytic destruction of ozone. These particles act as surfaces for heterogeneous reactions that generate halogen radicals, which subsequently participate in the destruction of ozone [15,16].
Numerous studies have shown that the interactions of H2SO4 with H2O, HCl, and HBr are crucial for understanding the dynamics of particle formation in the atmosphere [17,18,19]. The formation of sulfuric acid hydrates and complexes with these molecules is fundamental to atmospheric nucleation, particularly in the stratosphere, where H2SO4 and H2O molecules play a key role in particle nucleation and growth [20,21,22,23,24,25]. Experimentally, Fiacco et al. used rotational spectroscopy to identify the minimum-energy structure of the H2SO4 + H2O complex, which was found to adopt a twisted six-membered ring structure, with both H2O and H2SO4 acting as proton acceptors and donors [26]. Theoretical studies, such as those by Beichert et al., have also identified strong hydrogen bonds between H2SO4 and H2O, with HCl showing similar interactions in H2SO4-based complexes [25]. Re et al. investigated the complexes of H2SO4 (H2O)n (n = 1~5) and showed that all stable structures tend to form polycyclic forms [27]. Additionally, Miller et al. explored the vibrational frequencies of the H2SO4 + H2O system using ab initio calculations [28]. Li et al. systematically investigated the interaction of H2SO4 with HOO and H2O using density functional theory (DFT) and atoms in molecules (AIM) [23] and pointed out that H2SO4 aerosol actively participates in the absorption of HOO.
Research on the gas-phase interactions between H2SO4 and hydrogen halides has provided critical insights into the formation of pre-nucleation complexes, which are essential for understanding aerosol chemistry in the atmosphere [29,30,31]. For the H2SO4–HCl system, studies have demonstrated that the formation of hydrogen-bonded adducts plays a pivotal role in stratospheric nucleation processes [29,32,33]. For example, DFT calculations at the ωB97X-D/6-311++(2d,2p) level have revealed that these complexes exhibit binding energies closely associated with the electron density at the hydrogen bond critical points [29]. Observable shifts in the vibrational frequencies of H2SO4’s OH bonds upon complexation further underscore the directional and strong nature of the hydrogen bonds involving HCl [29]. These findings confirm that HCl acts effectively as both a hydrogen bond donor and acceptor, thereby stabilizing the complexes under atmospheric conditions.
While the H2SO4–HCl interactions are relatively well-characterized, the gas-phase reaction between H2SO4 and HBr remains less explored. Notably, despite the chemical similarities between HCl and HBr—both being hydrogen halides capable of participating in robust hydrogen bonding—the two systems exhibit important differences in reactivity and molecular properties. The relatively limited direct studies on the H2SO4–HBr system necessitate a distinct focus on it. Although direct computational analyses of H2SO4–HBr clusters are scarce, indirect evidence from related systems, such as investigations on HBr solubility in H2SO4/H2O mixtures [34], supports the plausibility of hydrogen-bonded interactions between H2SO4 and HBr. The high acidity of H2SO4 and its dual capacity to donate and accept protons remain crucial factors that drive the stabilization of these binary complexes, emphasizing the need for dedicated studies on the H2SO4–HBr system in atmospheric environments.
In the broader context of atmospheric aerosol science, the role of ternary complexes involving three of H2SO4, HCl, HNO3, HBr, and H2O has been extensively investigated [9,14,16,32,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. For example, Carslaw et al. developed a multicomponent thermodynamic model to estimate the solubility of HNO3, HCl, and HBr in stratospheric H2SO4 aerosols, contributing to our understanding of the dynamics of particle formation in the stratosphere [48]. Similarly, studies by Balcı and Nevin on the HNO3-HCl-H2O ternary system showed how the aggregation energy of clusters increases with the addition of water molecules, providing important insights into the role of water in atmospheric nucleation processes [37,38]. These works indicate that H2SO4 can form stable complexes with other molecules, with interactions dominated by hydrogen bonding, a key factor in aerosol formation.
However, despite the significant body of research on the interactions between H2SO4 and other atmospheric molecules, the isomerization mechanisms of these complexes, particularly those involving H2O, HCl, and HBr, still remain inadequately explored. Partanen et al. briefly investigated the potential energy surface of the H2SO4 + H2O system but did not provide an in-depth analysis of the structural rearrangement mechanisms [24]. In this study, we address this gap by investigating the isomerization mechanisms of the binary complexes between H2SO4 and H2O, HCl, and HBr in the gas phase. We focus on the structural rearrangement mechanisms and transition states connecting the isomers of these binary complexes. The isomerization reaction pathways identified in this work are expected to provide valuable insights into the behavior of ternary systems involving these molecules, such as H2SO4/H2O/HCl or H2SO4/H2O/HBr. Using high-level quantum chemical calculations, we identify potential isomers and transition states in the H2SO4 + HX (X = OH, Cl, Br) systems and analyze the hydrogen bonding interactions responsible for their stability and interconversion. This study aims to deepen our understanding of the intermolecular forces governing the isomerization of sulfuric acid-based aerosols and their potential role in atmospheric chemistry.

2. Methods

Due to the complexity of conformation searches for systems containing more than nine atoms, the conventional workflow—relying on chemical intuition or geometries from the available literature—is impractical and insufficient for the purposes of this study. The ABCluster 3.0 software [49,50] has been successfully applied in the conformation search and global optimization of molecules and clusters [51]. Therefore, the ABCluster software was utilized to explore potential isomers of the complex systems investigated in this work. The initial guess structures of the complexes between H2SO4 and HX (X = OH, Cl, Br) were constructed based on chemical intuition. For each initial conformer, the structures were optimized using the B3LYP [52] in combination with the 6-311G(d, p) basis set. To improve the description of the weak intermolecular interactions, a dispersion correction with Becke–Johnson damping D3(BJ) [53,54] was employed. To understand how to balance accuracy and computational cost, as suggested in refs. [32,41], the B3LYP/AVTZ level of theory offers a sufficiently reasonable description of the H2SO4-involved complex. Consequently, the diffuse function broadened, which is a consistently polarized valence basis set called aug-cc-pVTZ(AVTZ) [55,56], and was employed for further geometry optimization. To probe potential reaction mechanisms among these conformers, transition states (TSs) were located at the B3LYP-D3(BJ)/AVTZ level, and an intrinsic reaction coordinate (IRC) analysis was performed to identify the correct connection between each TS and the isomers. The optimized Cartesian coordinates and harmonic vibrational frequencies of conformers, including TSs for three systems, are listed in the Supplementary Materials. All DFT calculations were carried out by the Gaussian 16 program [57].
Furthermore, for all species optimized at the B3LYP-D3(BJ)/AVTZ level, single-point energies were computed at the CCSD(T)/AVTZ level implemented in the MOLPRO 2020 program [58,59,60]. Although the explicitly correlated CCSD(T)-F12b method offers higher accuracy, the results in Table S10 indicate that the conventional CCSD(T) method yields sufficiently accurate energies, differing by only 0.03 kcal/mol from those obtained with CCSD(T)-F12b. Therefore, CCSD(T) was deemed adequate and efficient for the current systems. The vibrational zero-point energies (ZPEs) calculated at the B3LYP-D3(BJ)/AVTZ level were included in the energy comparison, unless otherwise specified.

3. Results and Discussion

3.1. The H2SO4 + H2O System

Figure 1 illustrates the reaction scheme identified for the H2SO4 + H2O system, where CPi_H2O (i = 1, 2, …, 8) represents the eight different isomers, and TSi_H2O (i = 1, 2, …, 7) labels the seven transition states. All energies reported in Figure 1 are computed at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level and are relative to that of the global complex minimum CP5_H2O. The binding energy of CP5_H2O is 10.73 kcal/mol. For a more comprehensive analysis, the optimized structures of all isomers and the normal-mode vectors corresponding to the imaginary frequencies of the transition states (TSs) are provided in Figure 2 and Figure 3, respectively. Figure 2 also shows the H···O bond distances formed within the twisted six-membered ring. The conformers on either side of the dashed line are all six-membered ring structures with identical energies but with mirror symmetry. In this work, X-H···Y represents the hydrogen bond, where X and Y denote the donor and acceptor, respectively.
As shown in Figure 1, the rearrangement mechanism among the conformers of the H2SO4 + H2O system is complicated. The entire reaction pathway involves seven transition states, all of which are reported here for the first time unless otherwise specified. CP1_H2O and CP2_H2O exhibit mirror symmetry (Figure 2) and interconvert via TS1_H2O, which has a structure similar to CP2_H2O, with an imaginary frequency of −748 cm−1, corresponding to the O-H bond stretching vibrations of H2O and H2SO4 within the six-membered ring (Figure 3). In other words, hydrogen exchange occurs within the six-membered ring, with H2O and H2SO4 acting as donors and acceptors for one another, as shown in Figure S1a. Additionally, the energies of CP1_H2O and CP2_H2O relative to CP5_H2O are 1.65 kcal/mol, and TS1_H2O is 11.00 kcal/mol higher than CP5_H2O, as depicted in Figure 1.
CP2_H2O and CP3_H2O form a pair of cis–trans isomers. The most significant structural distinction between CP2_H2O and CP3_H2O lies in the configuration of the H2SO4 moiety (Figure 2), which adopts trans and cis arrangements, respectively. The interconversion between them proceeds through TS2_H2O (Figure 1), which is 3.11 kcal/mol higher in energy relative to CP5_H2O. Partanen et al. briefly described the conversion process as a torsional rotation of the non-hydrogen-bonded hydrogen (H) in a sulfuric acid molecule [24]. The imaginary frequency of TS2_H2O is −273 cm−1, corresponding to the motion of one non-hydrogen-bonded hydrogen H1 in the sulfuric acid molecule, as shown in Figure 3.
CP3_H2O and CP4_H2O also exhibit mirror symmetry and are only 0.06 kcal/mol higher than CP5_H2O. Analogous to TS1_H2O, TS3_H2O mediates the interconversion between CP3_H2O and CP4_H2O through hydrogen exchange, as shown in Figure S1b, corresponding to an imaginary frequency of −690 cm−1 (Figure 3). Notably, TS3_H2O is 8.91 kcal/mol higher than CP5_H2O. In addition to the mirror-symmetric structure of CP4_H2O, CP3_H2O goes through TS4_H2O to CP6_H2O. The principal structural distinction between CP3_H2O and CP6_H2O lies in the orientation of the non-ring-forming H atom in H2O: in CP3_H2O, both H atoms are positioned above and below the six-membered ring plane (Figure 2). TS4_H2O is 0.95 kcal/mol relative to CP5_H2O. In TS4_H2O, H2O oscillates on either side of the five-membered ring surface (at a frequency of −159 cm−1), involving its H3 or H4 in the formation of the six-membered ring (Figure 3). This finding aligns with Partanen et al.’s conclusions that the wobbling motion of the free hydrogen of H2O facilitates the interconversion between CP3_H2O and CP6_H2O [24].
CP5_H2O, the global minimum, has a mirror-symmetric structure, CP6_H2O. A hydrogen exchange reaction occurs between them via TS5_H2O, with an energy barrier of 8.78 kcal/mol, as shown in Figure S1c. CP5_H2O was previously reported as the most stable structure, both experimentally [26] and theoretically [20,61,62]. Interestingly, Figure 2 indicates that the trans configuration of H2SO4 is more favorable for complex stabilization. In the CP5_H2O isomer, one proton of the H2SO4 forms a short, direct hydrogen bond with the oxygen atom of H2O, with a H···O distance of 1.662 Å, close to the experimental value of 1.645(5) Å obtained by Fiacco et al. using rotational spectroscopy. [26] Additionally, a weak secondary hydrogen bond between an H2O hydrogen and a nearby S=O on H2SO4 has a H···O distance of 2.118 Å, slightly larger than the experimental value of 2.05(1) Å measured by Fiacco et al. [26]. CP5_H2O and CP6_H2O undergo a hydrogen exchange reaction through TS5_H2O, with an energy barrier of 8.78 kcal/mol, as shown in Figure 1. The imaginary frequency of TS5_H2O is −683 cm−1, with a vibrational mode that is the same as that of TS1_H2O (Figure 3). Similarly, CP6_H2O transitions through TS6_H2O with an energy barrier of 2.06 kcal/mol to form CP7_H2O, involving a cis–trans isomerization of the H2SO4 molecule (Figure 1). As shown in Figure 3, an approximate plane (D(OHOS) = −2.2907°) is formed by the O5 atom of H2O, the nearby O2-H2, and the S atom in TS6_H2O, bisecting the HOH angle. The imaginary frequency of TS6_H2O corresponds to a motion in which the H2O oscillates from side to side along this plane’s axis (Figure 3). CP7_H2O and CP8_H2O are reported for the first time and are 1.10 kcal/mol higher than CP5_H2O. They can interconvert via hydrogen exchange through TS7_H2O (Figure S1d), which has an imaginary frequency of −744 cm−1 and is 10.09 kcal/mol higher than CP5_H2O (Figure 1).
Furthermore, by inspecting the newly formed H···O distances of different isomers in Figure 2, it can be observed that the H···O distances between the proton donor H2SO4 and the O atom of the acceptor in H2O are all approximately 1.660 Å. This indicates that the stability of the isomers with the same configuration of H2SO4 can be approximated by directly comparing the distance between the proton-donating H atom of H2O and the S=O group of H2SO4. The results obtained using this method align well with those obtained through energy comparisons. Notably, Partanen et al. reported a structure in which the two H atoms of H2O are bonded to the two O atoms of S=O in H2SO4 [24]. That structure is not identified in this work. However, the structure proposed by Partanen et al. is extremely unstable and nearly non-existent at atmospheric temperatures [24]. Consequently, they excluded this structure from their subsequent investigations [62]. The relative energies of CP2_H2O and CP3_H2O compared to CP5_H2O (1.65 and 0.06 kcal/mol, respectively) show small deviations from the theoretical values (1.22 and 0.09 kcal/mol) calculated at the CCSD(T)-F12b/VQZ-F12 levels by Partanen et al. [62].

3.2. The H2SO4 + HCl System

Figure 4 illustrates all reaction pathways for the H2SO4 + HCl system, where CPi (i = 1, 2, …, 8) and TSi_HCl (i = 1, 2, …, 11) denote the various isomers and transition states, respectively. All energies reported in Figure 4 are presented relative to the most stable isomer CP4_HCl, with a binding energy of 5.46 kcal/mol. The optimized geometries of all isomers and the normal-mode vectors of the imaginary frequencies of the TSs are displayed in Figure 5 and Figure 6, respectively. Additionally, Figure 5 depicts the energies of the isomers as well as the intermolecular H···O bond and H···Cl distances.
As shown in Figure 4, the structural rearrangement mechanism for the H2SO4 + HCl system is complex, involving multiple isomers and transition states. CP1_HCl is 2.57 kcal/mol higher than CP4_HCl, with a hydrogen bond formed by the O-H bond of H2SO4 and the Cl atom. CP6_HCl is the mirror-symmetric geometry of CP1_HCl, with the same energy. No transition state directly connecting CP1_HCl and CP6_HCl has been found so far. Additionally, TS1_HCl, connecting CP1_HCl and CP3_HCl, has an energy of 5.78 kcal/mol. In the transition from CP1_HCl to CP3_HCl, H2SO4 undergoes a clockwise rotation, with O2–H2, connected to HCl, moving considerably to align with O1–H1, which rotates to a lesser extent. During this transition, the O2–H2···Cl hydrogen bond is maintained, while H3 of HCl shifts closer to O4 of S=O4, forming a six-membered ring. In TS1_HCl, the O2–H2 bond of H2SO4 near HCl aligns almost planar (D(SO2H2O1) = 1.29°) with S–O1. The imaginary frequency of TS1_HCl is −147 cm−1 and involves the complex motion of both O-H bonds of H2SO4 and the HCl bond, swinging at different amplitudes. In the CP3_HCl isomer, one S=O bond and one O–H bond of H2SO4 near HCl form an almost planar six-membered ring with HCl. In this ring, the H···Cl distance is 2.317 Å, while the H···O distance is 1.921 Å. The isomers CP2_HCl and CP3_HCl have mirror symmetry with a six-membered ring structure and an energy of 1.25 kcal/mol relative to CP4_HCl, as shown in Figure 5. The interconversion process between CP3_HCl and CP2_HCl occurs via TS2_HCl, which is 1.70 kcal/mol higher in energy than CP4_HCl (Figure 4). In TS2_HCl, HCl aligns approximately in a plane (D(ClH3O3S) = 0.00°) with S=O3, and H2SO4 exhibits C2v symmetry. The imaginary frequency of TS2_HCl is −29 cm−1, corresponding to a motion where both the O–H groups of H2SO4 rotate in the same direction, bringing one O–H closer to the Cl atom, resulting in hydrogen bond formation within the six-membered ring (Figure 6).
A cis–trans interconversion is proceeded through TS3_HCl with an energy barrier of 2.92 kcal/mol between CP3_HCl and CP4_HCl, as shown in Figure 4. In TS3_HCl, the imaginary frequency of −177 cm−1 corresponds to the oscillation of the O2–H2 bond of H2SO4 (Figure 6). This O-H bond participates in ring formation, allowing it to move closer to or further from HCl. Another reaction path starting from CP3_HCl is also a cis–trans interconversion from CP3_HCl to CP7_HCl, as evidenced in Figure 5. In CP7_HCl, an O–H bond of H2SO4 near HCl forms a hydrogen bond with the Cl atom, maintaining a H–Cl distance of 2.317 Å, identical to that in CP3_HCl. The interconversion proceeds via TS4_HCl, with an energy of 3.63 kcal/mol relative to CP4_HCl, as shown in Figure 4. The imaginary frequency of TS4_HCl is −91 cm−1, which corresponds to HCl oscillating above H2SO4, leading to the movement of the H3 of HCl closer to or farther away from the O3 of S=O3 (Figure 6).
As presented in Figure 5, CP4_HCl and CP5_HCl are global minimum-energy structures that are structurally equivalent, with two intermolecular hydrogen bonds formed similar to those in CP2_HCl (or CP3_HCl). In this arrangement, the O–H⸳⸳⸳Cl hydrogen bond has a H···Cl distance of 2.298 Å, while the Cl–H···O has a H···O distance of 1.954 Å. The interconversion barrier between CP4_HCl and CP5_HCl is notably high, reaching 10.07 kcal/mol for TS5a with an imaginary frequency of −610 cm−1. As shown in Figure 6, this imaginary frequency corresponds to the stretching vibration of both the H3-Cl bond in HCl and the O2–H2 bond in H2SO4 within the six-membered ring, leading to the exchange of H atoms. The intrinsic reaction coordinate pathway of TS5_HCl is shown in Figure S2a. Specifically, two transition states, TS6_HCl and TS7_HCl, are found connecting CP5_HCl and CP6_HCl, as shown in Figure 4. For TS6_HCl and TS7_HCl, the dihedral angles D(O2H3ClH2) are 0.50 and −0.37°, respectively, and the bond angles A(H2ClH3) are 97.48° and 96.99°. Thus, in TS6_HCl and TS7_HCl, HCl binds to H2SO4 in the opposite direction. In addition, the imaginary frequencies, −86 and −79 cm−1, of TS6_HCl and TS7_HCl correspond with HCl oscillating (Figure 6) such that HCl swings clockwise and counterclockwise, respectively, bringing the H3 atom of HCl close to or away from the adjacent O3, thereby forming or disrupting the six-membered ring. The energy barriers from CP5_HCl to CP6_HCl via TS6_HCl and TS7_HCl are 2.70 and 2.65 kcal/mol, respectively. For CP6_HCl, H2SO4 binds to HCl in a trans configuration, similar to that of CP5_HCl. In this case, only the Cl atom is involved in the formation of a hydrogen bond, specifically via O–H···Cl. The energy of CP6_HCl is 2.57 kcal/mol higher than that of CP5_HCl. Another pathway starting from CP5_HCl is going to CP7_HCl, involving TS8_HCl with an energy barrier of 3.63 kcal/mol (Figure 4). The imaginary frequency of TS8a, −55 cm−1, corresponds with a wobbling motion of HCl that brings the H3 atoms closer to or farther from the adjacent O4 (Figure 6). Simultaneously, the O2–H2 bond in H2SO4, which does not interact with the Cl atom, swings vertically, causing a cis–trans configuration transition in H2SO4. Furthermore, the structural distinction between CP6_HCl and CP7_HCl is the trans–cis conformation of H2SO4. The interconversion between CP6_HCl and CP7_HCl proceeds through two reaction pathways involving TS9_HCl and TS10_HCl, as shown in Figure 4. As observed in Figure 6, TS9_HCl and TS10_HCl are structurally similar. Notably, the O1–H1 bond in H2SO4, which does not interact with the Cl atom, forms an approximately coplanar arrangement (D(O2SO1H1) = 3.05 or −177.65°) with S-O2. The key difference is that O1–H1 adopts opposite orientations in the plane. It is noteworthy that the imaginary frequencies of TS9_HCl and TS10_HCl are −310 and −94 cm−1, respectively, corresponding with the up-and-down swinging of O1-H1 within the plane. The energies of TS9_HCl and TS10_HCl are 5.84 and 3.63 kcal/mol higher than that of CP5_HCl, respectively, suggesting that TS10_HCl is more favorable.
For CP7_HCl and CP8_HCl, HCl binds to H2SO4 through hydrogen bonding between the Cl atom and one O-H bond of the cis-structured H2SO4. Their energies are 3.45 kcal/mol higher than CP5_HCl, as shown in Figure 5, and the H···Cl distance for this intermolecular hydrogen bond is 2.317 Å. CP7_HCl and CP8_HCl are mirror-symmetric and interconvert through TS11_HCl (an imaginary frequency of only −31 cm−1), which is 15.49 kcal/mol higher in energy than CP5_HCl, as depicted in Figure 4. In TS11_HCl, HCl is perpendicular to the plane defined by the two O–H bonds of H2SO4 (D(O1H1H2O2) = 0.00°; D(O1H1ClH3) = 91.18°), with the Cl atom equidistant from the two O–H bonds. Similar to TS2_HCl, TS11_HCl connects CP7_HCl and CP8_HCl through the O–H swinging motion, where two O-H bonds compete to approach the Cl atom to form a stronger hydrogen bond, as shown in Figure S2b.

3.3. The H2SO4 + HBr System

An overview of all reaction pathways for the H2SO4 + HBr system is depicted in Figure 7, where CPi_HBr (i = 1, 2, …, 9) and TSi_HBr (i = 1, 2, …, 10) represent the various isomers and transition states, respectively. All energies reported in Figure 7 are relative to that of the isomer CP3_HBr, and the binding energy of CP3_HBr is 5.46 kcal/mol. For the H2SO4 + HBr system, Figure 8 shows the optimized geometries of all isomers and labels the H⸳⸳⸳O bond and H⸳⸳⸳Br distances as well as their energies. Figure 9 displays the normal mode of the imaginary frequency vectors for these TSi_HBr transition states.
As seen in Figure 7, two transition states, TS1_HBr and TS2_HBr, connect CP1_HBr and CP3_HBr. In TS1_HBr, H3–Br forms an approximately five-atom plane with a nearby O2–H2 and S atom. The imaginary frequency of −107 cm−1 for TS1_HBr corresponds to the back-and-forth swinging of the H3-Br bond in this plane, moving closer to the O3 or O4 atom (Figure 9). For TS2_HBr, the imaginary frequency is −174 cm−1, corresponding to an O2–H2 swing that brings the H2 atom close to the Br atom, forming a six-membered ring (S–O2–H2–Br–H3–O4), namely CP1_HBr. Alternatively, when H2 is far away from the Br atom, H2SO4 rotates in the direction of the O2–H2 swing, bringing H1 closer to the Br atom, forming a six-membered ring structure, namely CP3_HBr. The energy barriers of TS1_HBr and TS2_HBr are 2.76 and 2.23 kcal/mol, respectively. In addition, CP2_HBr converts to CP3_HBr through TS3_HBr, which is 3.23 kcal/mol higher than CP3_HBr. TS3_HBr has an imaginary frequency of −306 cm−1, corresponding to the swinging motion of O1-H1 on H2SO4, which is not involved in the ring formation. This causes the two O-H bonds to swing either in the same direction or in the opposite direction. CP3_HBr and CP4_HBr are mirror-symmetric, and the H···O and H···Br distances are 2.046 and 2.451 Å, respectively. CP3_HBr and CP4_HBr can interconvert via TS4_HBr with a high energy barrier of 9.50 kcal/mol. The imaginary frequency of TS4_HBr is −343 cm−1, corresponding to the O2–H3 and H3–Br stretching vibrations, which brings H2 and H3 closer to Br and O4, respectively, facilitating hydrogen atom exchange, as shown in Figure S3.
CP4_HBr and CP5_HBr are connected by TS5_HBr, which is 2.16 kcal/mol higher than CP4_HBr, and the relative energy difference between CP5_HBr and CP4_HBr is 1.81 kcal/mol, as depicted in Figure 7. The H···Br distance in CP5_HBr is 2.446 Å. For TS5_HBr, the dihedral angle (D(SO2BrH3) = 88.48°) formed by H3-Br with the neighboring S–O2 is nearly 90°. As seen from Figure 9, the imaginary frequency of −87 cm−1 for TS5_HBr corresponds to the swinging motion of H3–Br, which brings the H3 atom closer to O3 to form a six-membered ring (B2) or farther away from O3 (B5). Furthermore, CP4_HBr transitions to CP6_HBr via TS6_HBr, which has an imaginary frequency of −79 cm−1, corresponding to vibrational modes involving the swinging of O1–H1 and H3–Br. The former causes a cis–trans conformational change in H2SO4, while the latter alters how HBr binds to H2SO4 by moving H3 closer to or further away from O3. The energies of TS6_HBr and CP6_HBr are 3.14 and 2.73 kcal/mol higher than CP4_HBr, respectively. Also, CP5_HBr transitions to CP6_HBr via two different transition states, TS7_HBr and TS8_HBr. The H···Br distance in CP6_HBr is 2.486 Å. The imaginary frequencies of TS7_HBr and TS8_HBr are −89 cm−1 and −310 cm−1, respectively. As shown in Figure 9, the conformational and normal-mode vectors of the imaginary frequency of TS7_HBr and TS8_HBr are very similar to those of TS10_HCl and TS9_HCl for H2SO4 + HCl in Figure 6. These correspond to the cis–trans conformational change of H2SO4, resulting from a clockwise–counterclockwise rotation of O1–H1, which does not interact with HBr. Compared to CP4_HBr, the energies of TS7_HBr and TS8_HBr are higher by 2.88 and 5.03 kcal/mol, respectively. Interestingly, the transition from CP4_HBr to CP6_HBr encompasses multiple reaction pathways, either in one step or after two consecutive rearrangements.
From CP6_HBr to CP8_HBr, the transition state TS9_HBr is involved. Its imaginary frequency is −100 cm−1, and the corresponding vibration mode is similar to that of TS3_HCl (Figure 6). In this mode, the H3–Br swing moves the H3 atoms closer to O3 or further away from O3, facilitating ring-forming (B3) or ring-breaking (B8) (Figure 9). TS9_HBr is 3.16 kcal/mol higher than CP4_HBr. In addition, TS10_HBr connects CP8_HBr and CP9_HBr. In TS10_HBr, the O2–H2 near Br forms an approximate plane (D(O1SO2H2) = 2.51°) with S-O1. TS10_HBr has an imaginary frequency of −159 cm−1, and H3–Br, O1–H1, and O2–H2 all undergo swinging motions of varying amplitudes. TS10_HBr is 5.30 kcal/mol higher than CP4_HBr. It should be noted that CP7_HBr is not shown in Figure 7 as no reaction pathway involving CP7_HBr was found.
As shown in Figure 8, nine isomers are reported in the H2SO4 + HBr system. The configurations of CP1_HBr to CP8_HBr isomers are essentially the same as those of CP1_HCl to CP8_HCl in the H2SO4 + HCl system. The hydrogen bonding distances (H···O or H···Br distances) formed in these isomers are all larger than those in the H2SO4 + HCl system, as expected, because Br is less electronegative than Cl. The structure of CP1_HBr is similar to that of CP8_HBr, with similar H···O and H···Br distances of 2.051 and 2.451 Å for the former and 2.013 and 2.457 Å for the latter. CP1_HBr differs from CP8_HBr mainly in the configuration of H2SO4. CP1_HBr is 1.42 kcal/mol higher than CP3_HBr.

4. Conclusions

In this study, we investigate the intermolecular interactions within the H2SO4 + HX (X = OH, Cl, Br) systems by exploring the isomerization mechanisms of the corresponding complexes. Reaction pathways were identified based on optimized geometries at the B3LYP-D3(BJ)/AVTZ level, with single-point energy calculations refined at the CCSD(T)/AVTZ level.
We first identified many potential isomers of the H2SO4 + HX (X = OH, Cl, Br) system. For the H2SO4 + H2O system, eight isomers with six-membered ring structures were found, in which H2O and H2SO4 form hydrogen bonds and serve as both proton donors and acceptors. Among these, some isomers are mirror-symmetric, sharing identical energies and frequencies, and can thus be grouped into four pairs of equivalent structures. Notably, isomers CP7_H2O and CP8_H2O are newly reported here. The H2SO4 + HCl system also has eight isomers, organized as four pairs of equivalent structures. For the H2SO4 + HBr system, nine isomers were identified, four of which form pairs of equivalent structures. All isomers except for CP1_HBr show configurations similar to those in the H2SO4 + HCl system, differentiated by the position of HX within the complex and cis–trans configuration of H2SO4. In both the H2SO4 + HX (X = Cl, Br) systems, HX interacts with H2SO4 in two configurations: H2SO4 donates a proton to the X atom on HX, and the protons on HX either form hydrogen bonds with proton-free O atoms on H2SO4 or do not participate in intermolecular interactions. Finally, in the H2SO4 + HX (X = OH, Cl, Br) system, the minimum-energy structures are consistently characterized by ring configurations with H2SO4 in the trans configuration.
Based on optimized stationary points, we search for connections between them. To the best of our knowledge, this is the first theoretical investigation into the isomerization reaction pathways of the H2SO4 + HX (X = OH, Cl, Br) system. The transition states identified in this study have not been previously reported, and our findings on structural rearrangement mechanisms contribute to the understanding of intermolecular interactions in the H2SO4 + HX (X = OH, Cl, Br) system. For the H2SO4 + H2O system, each pair of equivalent structures undergoes interconversion via a hydrogen exchange reaction, with energy barriers ranging from 8 to 10 kcal/mol. This process involves transition states with imaginary frequencies exceeding −680 cm−1, indicating that these hydrogen exchange reactions are not readily feasible, and thus, equivalent structures do not coexist. Transitions between different isomers occur either through a cis–trans conformational change in H2SO4 or through H2O forming complexes with H2SO4 in various orientations. The cis–trans change involves an O–H swing in H2SO4, not participating in ring formation, while complex formation involves H2O vibrating symmetrically within a hydrogen-bonded state, resulting in a shorter hydrogen bond. For the H2SO4 + HCl system, the hydrogen exchange reaction between CP4_HCl and CP5_HCl has a high energy barrier of 10.07 kcal/mol, making it unlikely to occur. The interconversion between CP7_HCl and CP8_HCl has an energy barrier exceeding 12 kcal/mol and includes the transition state TS11_HCl in a symmetric hydrogen-bonded configuration. Similarly, the transition between CP2_HCl and CP3_HCl involves competition between symmetric hydrogen bonds (TS2_HCl), but with a much lower reaction energy barrier of only 0.46 kcal/mol. This significant difference is attributed to the higher structural stability of TS2_HCl, which contains three hydrogen bonds atop the ring structure. Additionally, the least stable isomer, CP8_HCl, can interconvert with the other three isomers. While CP1_HCl and CP6_HCl can interconvert through multistep reactions, a direct transition between them is difficult due to the high energy barriers associated with CP4_HCl and CP5_HCl interconversion. In the H2SO4 + HBr system, CP3_HBr and CP4_HBr represent the global minimum-energy structures, with interconversion between them (a hydrogen exchange reaction) exhibiting an energy barrier of 9.50 kcal/mol. Similarly, the interconversion between CP5_HBr and CP9_HBr involves multistep reactions with two reaction channels. For the remaining two pairs of equivalent structures, no connecting transition states were identified. Notably, for isomerization reactions involving only the cis–trans conformational change of H2SO4, two transition states are observed, such as the conversion from CP5_HBr to CP6_HBr in the H2SO4 + HCl system. The imaginary frequencies of these transition states correspond to vibrational modes of O–H swinging motions unrelated to intermolecular interactions in H2SO4, but with O–H pointing in opposite directions. Based on imaginary frequency values and reaction energy barriers of these transition states, the counterclockwise rotation of O–H is more favorable.
Overall, isomerization in the H2SO4 + HX (X = OH, Cl, Br) system involves three main processes: (I) hydrogen exchange reactions within the ring structure with O–H stretching vibrations; (II) cis–trans conformational changes in H2SO4, primarily due to O–H swinging not involved in intermolecular interactions; and (III) ring structure formation and dissociation, involving HX swinging and altering H2SO4–HX binding. The most active groups in these rearrangements are O–H and oxygen atoms without protons, playing crucial roles in forming intermolecular hydrogen bonds.
Through reliable theoretical calculations, we have illuminated the pathways and mechanisms of isomerization in the H2SO4 + HX (X = OH, Cl, Br) binary complex systems. We hope that our findings will offer insights into the H2SO4/H2O/HX (X = Cl, Br) ternary system and the proton exchange reaction taking place in the gas–liquid interface between HCl, HBr, and supercooled sulfuric acid [63].

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/app15147642/s1, Figure S1: Intrinsic reaction coordinate pathway of TS1_H2O, TS3_H2O, TS5_H2O, and TS7_H2O. Figure S2: Intrinsic reaction coordinate pathway of TS5_HCl and TS11_HCl. Figure S3: Intrinsic reaction coordinate pathway of TS4_HBr. Table S1–S6: Optimized structural parameters for isomers and transition states of the H2SO4 + HX (X = OH, Cl, Br) systems at the B3LYP-D3(BJ)/AVTZ level. Table S7–S9: Optimized harmonic vibrational frequencies of isomers and transition states of the H2SO4 + HX (X = OH, Cl, Br) systems at the B3LYP-D3(BJ)/AVTZ level. Table S10: Comparison of energies for CP1_H2O, CP2_H2O, CP3_H2O, CP4_H2O, and CP5_H2O using CCSD(T)/AVTZ and CCSD(T)-F12b/AVTZ methods, including ZPE corrections at B3LYP-D3(BJ)/AVTZ level.

Author Contributions

Q.Z.: formal analysis (lead), investigation (lead), visualization (lead), writing—original draft (lead), writing—review and editing (equal). K.S.: formal analysis (equal), visualization (equal), writing—original draft (lead), writing—review and editing (lead). J.L.: conceptualization (lead), methodology (lead), project administration (lead), supervision (equal), resources (lead), funding acquisition (lead), writing—review and editing (lead). All authors have read and agreed to the published version of the manuscript.

Funding

This work was financially supported by the National Natural Science Foundation of China (Grant No. 22473019).

Institutional Review Board Statement

Not applicable.

Data Availability Statement

Detailed computational data, including Gaussian input and output files for the structural optimization of isomers and transition states, as well as for the intrinsic reaction coordinate (IRC) calculations of all transition states, are available in the Zenodo repository at https://doi.org/10.5281/zenodo.15622742, accessed on 9 June 2025.

Conflicts of Interest

Author Qi Zhang was employed by the company Xinlian Microelectronics Co. Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Figure 1. Schematic reaction path for the H2SO4 + H2O system. The electronic energies with ZPE corrections (in kcal/mol) are shown relative to the most stable isomer CP5_H2O at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level.
Figure 1. Schematic reaction path for the H2SO4 + H2O system. The electronic energies with ZPE corrections (in kcal/mol) are shown relative to the most stable isomer CP5_H2O at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level.
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Figure 2. The structural representations for eight optimized H2SO4 + H2O complexes are presented. The relative energies (in kcal/mol) were calculated at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level of theory. The H···O distances (in Å) within the ring structure are also indicated (note: the two isomers on either side of the dashed line are mirror images of each other).
Figure 2. The structural representations for eight optimized H2SO4 + H2O complexes are presented. The relative energies (in kcal/mol) were calculated at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level of theory. The H···O distances (in Å) within the ring structure are also indicated (note: the two isomers on either side of the dashed line are mirror images of each other).
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Figure 3. The illustration of the normal-mode vectors of the imaginary frequency for all transition states in the H2SO4 + H2O system. They were determined at the B3LYP-D3(BJ)/AVTZ level, with the imaginary frequency (in cm−1) shown in parentheses.
Figure 3. The illustration of the normal-mode vectors of the imaginary frequency for all transition states in the H2SO4 + H2O system. They were determined at the B3LYP-D3(BJ)/AVTZ level, with the imaginary frequency (in cm−1) shown in parentheses.
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Figure 4. Schematic reaction path for the H2SO4 + HCl system. The energies in kcal/mol are the electronic energies with ZPE corrections, relative to those of the isomer CP4_HCl at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level.
Figure 4. Schematic reaction path for the H2SO4 + HCl system. The energies in kcal/mol are the electronic energies with ZPE corrections, relative to those of the isomer CP4_HCl at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level.
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Figure 5. The optimized complex isomers in the H2SO4 + HCl system. The relative energies (in kcal/mol) were calculated at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level. The intermolecular distances, H···O and H···Cl, are indicated.
Figure 5. The optimized complex isomers in the H2SO4 + HCl system. The relative energies (in kcal/mol) were calculated at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level. The intermolecular distances, H···O and H···Cl, are indicated.
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Figure 6. The normal-mode vectors corresponding to the imaginary frequencies for all transition states in the H2SO4 + HCl system were computed at the B3LYP-D3(BJ)/AVTZ level of theory, with frequencies given in cm−1.
Figure 6. The normal-mode vectors corresponding to the imaginary frequencies for all transition states in the H2SO4 + HCl system were computed at the B3LYP-D3(BJ)/AVTZ level of theory, with frequencies given in cm−1.
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Figure 7. Schematic reaction path for the H2SO4 + HBr system. The energies in kcal/mol are the electronic energies with ZPE corrections, relative to those of the CP3_HBr isomer at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level.
Figure 7. Schematic reaction path for the H2SO4 + HBr system. The energies in kcal/mol are the electronic energies with ZPE corrections, relative to those of the CP3_HBr isomer at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level.
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Figure 8. Optimized geometries of isomers in the H2SO4 + HBr system. The relative energies (in kcal/mol) were calculated at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level of theory. The intermolecular distances in Å, H···O and H···Br, are also indicated.
Figure 8. Optimized geometries of isomers in the H2SO4 + HBr system. The relative energies (in kcal/mol) were calculated at the CCSD(T)/AVTZ//B3LYP-D3(BJ)/AVTZ level of theory. The intermolecular distances in Å, H···O and H···Br, are also indicated.
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Figure 9. The normal-mode vectors corresponding to the imaginary frequencies for all transition states in the H2SO4 + HBr system were calculated at the B3LYP-D3(BJ)/AVTZ level, with given frequencies in cm−1.
Figure 9. The normal-mode vectors corresponding to the imaginary frequencies for all transition states in the H2SO4 + HBr system were calculated at the B3LYP-D3(BJ)/AVTZ level, with given frequencies in cm−1.
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Zhang, Q.; Song, K.; Li, J. Theoretical Exploration of Isomerization Pathways in H2SO4·HX (X = OH, Cl, Br) Complexes. Appl. Sci. 2025, 15, 7642. https://doi.org/10.3390/app15147642

AMA Style

Zhang Q, Song K, Li J. Theoretical Exploration of Isomerization Pathways in H2SO4·HX (X = OH, Cl, Br) Complexes. Applied Sciences. 2025; 15(14):7642. https://doi.org/10.3390/app15147642

Chicago/Turabian Style

Zhang, Qi, Kaisheng Song, and Jun Li. 2025. "Theoretical Exploration of Isomerization Pathways in H2SO4·HX (X = OH, Cl, Br) Complexes" Applied Sciences 15, no. 14: 7642. https://doi.org/10.3390/app15147642

APA Style

Zhang, Q., Song, K., & Li, J. (2025). Theoretical Exploration of Isomerization Pathways in H2SO4·HX (X = OH, Cl, Br) Complexes. Applied Sciences, 15(14), 7642. https://doi.org/10.3390/app15147642

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