## 4 Conclusions
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 H\({}_{2}\)SO\({}_{4}\) + 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 H\({}_{2}\)SO\({}_{4}\), two transition states are observed, such as the conversion from CP5_HBr to CP6_HBr in the H\({}_{2}\)SO\({}_{4}\) + HCl system. The imaginary frequencies of these transition states correspond to vibrational modes of O-H swinging motions unrelated to intermolecular interactions in H\({}_{2}\)SO\({}_{4}\), 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 H\({}_{2}\)SO\({}_{4}\) + 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 H\({}_{2}\)SO\({}_{4}\), primarily due to O-H swinging not involved in intermolecular interactions; and (III) ring structure formation and dissociation, involving HX swinging and altering H\({}_{2}\)SO\({}_{4}\)-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 H\({}_{2}\)SO\({}_{4}\) + HX (X = OH, Cl, Br) binary complex systems. We hope that our findings will offer insights into the H\({}_{2}\)SO\({}_{4}\)/H\({}_{2}\)O/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](https://www.mdpi.com/article/10.3390/app15147642/s1)]([https://www.mdpi.com/article/10.3390/app15147642/s1](https://www.mdpi.com/article/10.3390/app15147642/s1)), Figure S1: Intrinsic reaction coordinate pathway of TS1_H2O, TS3_H2O, TSS_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 H2 SO4 + HX (X = OH, Cl, Br) systems at the B3 LYP-D3(BJ)/AVTZ level. Table S7-S9: Optimized harmonic vibrational frequencies of isomers and transition states of the H2 SO4 + HX (X = OH, Cl, Br) systems at the B3 LYP-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 B3 LYP-D3(BJ)/AVTZ level. 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. This work was financially supported by the National Natural Science Foundation of China (Grant No.