## 3 Results and Discussion
### The H\({}_{2}\)SO\({}_{4}\) + HCl System  
Figure 4 illustrates all reaction pathways for the H\({}_{2}\)SO\({}_{4}\) + HCl system, where CPi (i = 1, 2, \(\ldots\), 8) and TSi_HCl (i = 1, 2, \(\ldots\), 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 Figures 5 and 6, respectively. Additionally, Figure 5 depicts the energies of the isomers as well as the intermolecular H\(\cdots\)O bond and H\(\cdots\)Cl distances. As shown in Figure 4, the structural rearrangement mechanism for the H\({}_{2}\)SO\({}_{4}\) + 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 H\({}_{2}\)SO\({}_{4}\) 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, H\({}_{2}\)SO\({}_{4}\) undergoes a clockwise rotation, with O2-H2, connected to HCl, moving considerably to the O-H bond. The transition state is \(\sim\)0.001, which is consistent with the formation of the O-H bond. The transition state is \(\sim\)0  
align with O1-H1, which rotates to a lesser extent. During this transition, the O2-H2\(\cdots\)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 H2 SO4 near HCl aligns almost planar (D(SO2H2 OI) = 1.29\({}^{\circ}\)) with S-O1. The imaginary frequency of TS1_HCl is \(-\)147 cm\({}^{-1}\) and involves the complex motion of both O-H bonds of H2 SO4 and the HCl bond, swinging at different amplitudes. In the CP3_HCl isomer, one S=O bond and one O-H bond of H2 SO4 near HCl form an almost planar six-membered ring with HCl. In this ring, the H\(\cdots\)Cl distance is 2.317 A, while the H\(\cdots\)O distance is 1.921 A. 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\({}^{\circ}\)) with S=O3, and H2 SO4 exhibits C2\({}_{\
u}\) symmetry. The imaginary frequency of TS2_HCl is \(-\)29 cm\({}^{-1}\), corresponding to a motion where both the O-H groups of H2 SO4 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 H2 SO4 (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 H2 SO4 near HCl forms a hydrogen bond with the Cl atom, maintaining a H-Cl distance of 2.317 A, 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 H2 SO4, leading to the movement of the H3 of HCl closer to or farther away from the O3 of S=O3 (Figure 6).