## 3 Results and Discussion
### The H\({}_{2}\)So\({}_{4}\) + H\({}_{2}\)O System
TS4_H\({}_{2}\)O, H\({}_{2}\)O oscillates on either side of the five-membered ring surface (at a frequency of \(-159\) cm\({}^{-1}\)), involving its H3 or H4 in the formation of the six-membered ring (Figure 3). This finding aligns with [PERSON] et al. 's conclusions that the wobbling motion of the free hydrogen of H\({}_{2}\)O facilitates the interconversion between CP3_H\({}_{2}\)O and CP6_H\({}_{2}\)O [24]. CP5_H\({}_{2}\)O, the global minimum, has a mirror-symmetric structure, CP6_H\({}_{2}\)O. A hydrogen exchange reaction occurs between them via TS5_H\({}_{2}\)O, with an energy barrier of 8.78 kcal/mol, as shown in Figure S1c. CP5_H\({}_{2}\)O was previously reported as the most stable structure, both experimentally [26] and theoretically [20; 61; 62]. Interestingly, Figure 2 indicates that the trans configuration of H\({}_{2}\)SO\({}_{4}\) is more favorable for complex stabilization. In the CP5_H\({}_{2}\)O isomer, one proton of the H\({}_{2}\)SO\({}_{4}\) forms a short, direct hydrogen bond with the oxygen atom of H\({}_{2}\)O, with a H\(\cdots\)O distance of 1.662 A, close to the experimental value of 1.645(5) A obtained by [PERSON] et al. using rotational spectroscopy. [26] Additionally, a weak secondary hydrogen bond between an H\({}_{2}\)O hydrogen and a nearby S=O on H\({}_{2}\)SO\({}_{4}\) has a H\(\cdots\)O distance of 2.118 A, slightly larger than the experimental value of 2.05(1) A measured by [PERSON] et al. [26]. CP5_H\({}_{2}\)O and CP6_H\({}_{2}\)O undergo a hydrogen exchange reaction through TS5_H\({}_{2}\)O, with an energy barrier of 8.78 kcal/mol, as shown in Figure 1. The imaginary frequency of TS5_H\({}_{2}\)O is \(-683\) cm\({}^{-1}\), with a vibrational mode that is the same as that of TS1_H\({}_{2}\)O (Figure 3). Similarly, CP6_H\({}_{2}\)O transitions through TS6_H\({}_{2}\)O with an energy barrier of 2.06 kcal/mol to form CP7_H\({}_{2}\)O, involving a cis-trans isomerization of the H\({}_{2}\)SO\({}_{4}\) molecule (Figure 1). As shown in Figure 3, an approximate plane (D(OHOS) = \(-2.290\)\(\%\)) is formed by the O5 atom of H\({}_{2}\)O, the nearby O2-H2, and the S atom in TS6_H\({}_{2}\)O, bisecting the HOH angle. The imaginary frequency of TS6_H\({}_{2}\)O corresponds to a motion in which the H\({}_{2}\)O oscillates from side to side along this plane's axis (Figure 3). CP7_H\({}_{2}\)O and CP8_H\({}_{2}\)O are reported for the first time and are 1.10 kcal/mol higher than CP5_H\({}_{2}\)O. They can interconvert via hydrogen exchange through TS7_H\({}_{2}\)O (Figure S1d), which has an imaginary frequency of \(-744\) cm\({}^{-1}\) and is 10.09 kcal/mol higher than CP5_H\({}_{2}\)O (Figure 1). Furthermore, by inspecting the newly formed H\(\cdots\)O distances of different isomers in Figure 2, it can be observed that the H\(\cdots\)O distances between the proton donor H\({}_{2}\)SO\({}_{4}\) and the O atom of the acceptor in H\({}_{2}\)O are all approximately 1.660 A. This indicates that the stability of the isomers with the same configuration of H\({}_{2}\)SO\({}_{4}\) can be approximated by directly comparing the distance between the proton-donating H atom of H\({}_{2}\)O and the S=O group of H\({}_{2}\)SO\({}_{4}\). The results obtained using this method align well with those obtained through energy comparisons. Notably, [PERSON] et al. reported a structure in which the two H atoms of H\({}_{2}\)O are bonded to the two O atoms of S=O in H\({}_{2}\)SO\({}_{4}\)[24]. That structure is not identified in this work. However, the structure proposed by [PERSON] et al. is extremely unstable and nearly non-existent at atmospheric temperatures [24]. Consequently, they excluded this structure from their subsequent investigations [62]. The relative energies of CP2_H\({}_{2}\)O and CP3_H\({}_{2}\)O compared to CP5_H\({}_{2}\)O (1.65 and 0.06 kcal/mol, respectively) show small deviations from the theoretical values (1.22 and 0.09 kcal/mol) calculated at the CCSD(T)-F12b/VQZ-F12 levels by [PERSON] et al. [62].