## 4 Conclusions  
In this study, we investigate the intermolecular interactions within the H\({}_{2}\)SO\({}_{4}\) + 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 B3 LYP-D3(BJ)/AVTZ level, with single-point energy calculations refined at the CCSD(T)/AVTZ level. We first identified many potential isomers of the H\({}_{2}\)SO\({}_{4}\) + HX (X = OH, Cl, Br) system. For the H\({}_{2}\)SO\({}_{4}\) + H\({}_{2}\)O system, eight isomers with six-membered ring structures were found, in which H\({}_{2}\)O and H\({}_{2}\)SO\({}_{4}\) 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_H\({}_{2}\)O and CP8_H\({}_{2}\)O are newly reported here. The H\({}_{2}\)SO\({}_{4}\) + HCl system also has eight isomers, organized as four pairs of equivalent structures. For the H\({}_{2}\)SO\({}_{4}\) + 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 H\({}_{2}\)SO\({}_{4}\) + HCl system, differentiated by the position of HX within the complex and cis-trans configuration of H\({}_{2}\)SO\({}_{4}\). In both the H\({}_{2}\)SO\({}_{4}\) + HX (X = Cl, Br) systems, HX interacts with H\({}_{2}\)SO\({}_{4}\) in two configurations: H\({}_{2}\)SO\({}_{4}\) donates a proton to the X atom on HX, and the protons on HX either form hydrogen bonds with proton-free O atoms on H\({}_{2}\)SO\({}_{4}\) or do not participate in intermolecular interactions. Finally, in the H\({}_{2}\)SO\({}_{4}\) + HX (X = OH, Cl, Br) system, the minimum-energy structures are consistently characterized by ring configurations with H\({}_{2}\)SO\({}_{4}\) 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 H\({}_{2}\)SO\({}_{4}\) + 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 H\({}_{2}\)SO\({}_{4}\) + HX (X = OH, Cl, Br) system. For the H\({}_{2}\)SO\({}_{4}\) + H\({}_{2}\)O 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 H\({}_{2}\)SO\({}_{4}\) or through H\({}_{2}\)O forming complexes with H\({}_{2}\)SO\({}_{4}\) in various orientations. The cis-trans change involves an O-H swing in H\({}_{2}\)SO\({}_{4}\), not participating in ring formation, while complex formation involves H\({}_{2}\)O vibrating symmetrically within a hydrogen-bonded state, resulting in a shorter hydrogen bond. For the H\({}_{2}\)SO\({}_{4}\) + 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.