## 3 Results and Discussion
### The H\({}_{2}\)SO\({}_{4}\) + HBr System  
An overview of all reaction pathways for the H\({}_{2}\)SO\({}_{4}\) + HBr system is depicted in Figure 7, where CPi_HBr (i = 1, 2, \(\ldots\), 9) and TSi_HBr (i = 1, 2, \(\ldots\), 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 H2 SO4 + HBr system, Figure 8 shows the optimized geometries of all isomers and labels the H\(\cdots\)O bond and H\(\cdots\)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  
Figure 8: Optimized geometries of isomers in the H2 SO4 + HBr system. The relative energies (in kcal/mol) were calculated at the CCSD(T)/AVTZ//B3 LYP-D3(B)//AVTZ level of theory. The intermolecular distances in Å, H\(\cdots\)O and H\(\cdots\)Br, are also indicated. Figure 7: Schematic reaction path for the H2 SO4 + 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//B3 LYP-D3(B)//AVTZ level. 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, H2 SO4 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 H2 SO4, 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\(\cdots\)O and H\(\cdots\)Br distances are 2.046 and 2.451 A, 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\(\cdots\)Br distance in CP5_HBr is 2.446 A. For TS5_HBr, the dihedral angle (D(SO2 BrH3) = 88.48\({}^{\circ}\)) formed by H3-Br with the neighboring S-O2 is nearly 90\({}^{\circ}\). 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 H2 SO4, while the latter alters how HBr binds to H2 SO4 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.