## 3 Results and Discussion  
### The H\({}_{2}\)So\({}_{4}\) + H\({}_{2}\)O System  
Figure 1 illustrates the reaction scheme identified for the H\({}_{2}\)SO\({}_{4}\) + H\({}_{2}\)O system, where CPi_H\({}_{2}\)O (i = 1, 2, \(\ldots\), 8) represents the eight different isomers, and TSi_H\({}_{2}\)O (i = 1, 2,  
\(\ldots\), 7) labels the seven transition states. All energies reported in Figure 1 are computed at the CCSD(T)/AVTZ//B3 LYP-D3(BJ)/AVTZ level and are relative to that of the global complex minimum CP5_H\({}_{2}\)O. The binding energy of CP5_H\({}_{2}\)O is 10.73 kcal/mol. For a more comprehensive analysis, the optimized structures of all isomers and the normal-mode vectors corresponding to the imaginary frequencies of the transition states (TSs) are provided in Figures 2 and 3, respectively. Figure 2 also shows the H\(\cdots\)O bond distances formed within the twisted six-membered ring. The conformers on either side of the dashed line are all six-membered ring structures with identical energies but with  
As shown in Figure 1, the rearrangement mechanism among the conformers of the H\({}_{2}\)SO\({}_{4}\) + H\({}_{2}\)O system is complicated. The entire reaction pathway involves seven transition states, all of which are reported here for the first time unless otherwise specified. CP1_H\({}_{2}\)O and CP2_H\({}_{2}\)O exhibit mirror symmetry (Figure 2) and interconvert via TS1_H\({}_{2}\)O, which has a structure similar to CP2_H\({}_{2}\)O, with an imaginary frequency of \(-\)748 cm\({}^{-1}\), corresponding to the O-H bond stretching vibrations of H\({}_{2}\)O and H\({}_{2}\)SO\({}_{4}\) within the six-membered ring (Figure 3). In other words, hydrogen exchange occurs within the six-membered ring, with H\({}_{2}\)O and H\({}_{2}\)SO\({}_{4}\) acting as donors and acceptors for one another, as shown in Figure S1a. Additionally, the energies of CP1_H\({}_{2}\)O and CP2_H\({}_{2}\)O relative to CP5_H\({}_{2}\)O are 1.65 kcal/mol, and TS1_H\({}_{2}\)O is 11.00 kcal/mol higher than CP5_H\({}_{2}\)O, as depicted in Figure 1.  
CP2_H\({}_{2}\)O and CP3_H\({}_{2}\)O form a pair of cis-trans isomers. The most significant structural distinction between CP2_H\({}_{2}\)O and CP3_H\({}_{2}\)O lies in the configuration of the H\({}_{2}\)SO\({}_{4}\) moiety (Figure 2), which adopts trans and cis arrangements, respectively. The interconversion between them proceeds through TS2_H\({}_{2}\)O (Figure 1), which is 3.11 kcal/mol higher in energy relative to CP5_H\({}_{2}\)O. [PERSON] et al. briefly described the conversion process as a torsional rotation of the non-hydrogen-bonded hydrogen (H) in a sulfuric acid molecule [24]. The imaginary frequency of TS2_H\({}_{2}\)O is \(-\)273 cm\({}^{-1}\), corresponding to the motion of one non-hydrogen-bonded hydrogen H1 in the sulfuric acid molecule, as shown in Figure 3.  
CP3_H\({}_{2}\)O and CP4_H\({}_{2}\)O also exhibit mirror symmetry and are only 0.06 kcal/mol higher than CP5_H\({}_{2}\)O. Analogous to TS1_H\({}_{2}\)O, TS3_H\({}_{2}\)O mediates the interconversion between CP3_H\({}_{2}\)O and CP4_H\({}_{2}\)O through hydrogen exchange, as shown in Figure S1b, corresponding to an imaginary frequency of \(-\)690 cm\({}^{-1}\) (Figure 3). Notably, TS3_H\({}_{2}\)O is 8.91 kcal/mol higher than CP5_H\({}_{2}\)O. In addition to the mirror-symmetric structure of CP4_H\({}_{2}\)O, CP3_H\({}_{2}\)O goes through TS4_H\({}_{2}\)O to CP6_H\({}_{2}\)O. The principal structural distinction between CP3_H\({}_{2}\)O and CP6_H\({}_{2}\)O lies in the orientation of the non-ring-forming H atom in H\({}_{2}\)O: in CP3_H\({}_{2}\)O, both H atoms are positioned above and below the six-membered ring plane (Figure 2). TS4_H\({}_{2}\)O is 0.95 kcal/mol relative to CP5_H\({}_{2}\)O. In  
Figure 3: The illustration of the normal-mode vectors of the imaginary frequency for all transition states in the H\({}_{2}\)SO\({}_{4}\) + H\({}_{2}\)O system. They were determined at the B3 LYP-D3(BJ)/AVTZ level, with the imaginary frequency (in cm\({}^{-1}\)) shown in parentheses.