## 3 Results and Discussion
### The H\({}_{2}\)SO\({}_{4}\) + HCl System
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\(\cdots\)Cl hydro  
Figure 6: The normal-mode vectors corresponding to the imaginary frequencies for all transition states in the H2 SO4 + HCl system were computed at the B3 LYP-D3(B)//AVTZ level of theory, with frequencies given in cm\({}^{-1}\). gen bond has a H\(\cdots\)Cl distance of 2.298 A, while the Cl-H\(\cdots\)O has a H\(\cdots\)O distance of 1.954 A. 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 H\({}_{2}\)SO\({}_{4}\) 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(O2H3 ClH2) are 0.50 and \(-\)0.37\({}^{\circ}\), respectively, and the bond angles A(H2 ClH3) are 97.48\({}^{\circ}\) and 96.99\({}^{\circ}\). Thus, in TS6_HCl and TS7_HCl, HCl binds to H\({}_{2}\)SO\({}_{4}\) 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, H\({}_{2}\)SO\({}_{4}\) 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\(\cdots\)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 H\({}_{2}\)SO\({}_{4}\), which does not interact with the Cl atom, swings vertically, causing a cis-trans configuration transition in H\({}_{2}\)SO\({}_{4}\). Furthermore, the structural distinction between CP6_HCl and CP7_HCl is the trans-cis conformation of H\({}_{2}\)SO\({}_{4}\). 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 H\({}_{2}\)SO\({}_{4}\), which does not interact with the Cl atom, forms an approximately coplanar arrangement (D(O2 SO1H1) = 3.05 or \(-\)177.65\({}^{\circ}\)) 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 H\({}_{2}\)SO\({}_{4}\) through hydrogen bonding between the Cl atom and one O-H bond of the cis-structured H\({}_{2}\)SO\({}_{4}\).