Fig. 1. The structures of (a) Eigen and (b) Zundel. Red: O atom; grey: H atom.

Fig. 2. Optimized H5O2+ with several selected parameters calculated through CCSD(T)-F12/cc-pVTZ.

Fig. 3. (a) Deviations of key bond lengths between those of different methods and those of CCSD(T)-F12/cc-pVTZ. (b) Proton stretch absorption band from experimental results and results of different methods.

Fig. 4. PES calculated through varying the O₁–O₂ and O₁–H⁺ distances of the optimized H5O2+ in (a) gas-phase and (b) water. Theory level: PBE0-D3/def2-TZVP.

Fig. 5. (a) Absorption band of the proton stretch of optimized H5O2+ and the O₁–H⁺ distance at various O₁–O₂ distances. (b) A 3D plot of the process of the change of O–O distance, O₁-H⁺ distance, as well as the change of the corresponding proton stretch band. Theory level: PBE0-D3/def2-TZVP. Note: 0.96 on the denominator of the equation is the O–H⁺ length in H₃O⁺ (optimized through CCSD(T)/cc-pVTZ).