Fig. 5: The distributions of the local electric field computed with conditional ensemble average.

a The local electric field distributions along collective coordinate q_l, for Zundel-like (0–0.13 V/Å), Intermediate (0.13–0.28 V/Å) and Eigen-like (0.28–0.40 V/Å) configurations, respectively. b The local electric field distributions along collective coordinate q_s, for Zundel-like, Intermediate and Eigen-like configurations, respectively. The color bar of the scatter points signifies the structural probability densities (PD), and it is worth noting that the color bar is uniquely defined for each sub-graph. The local electric field surrounding the excess proton is evaluated using Coulomb’s law within the first solvation shell of the \({{{{\rm{H}}}}}_{5}{{{{\rm{O}}}}}_{2}^{+}\) moiety, as long-range contributions have minor impact on the internal electric field³⁷. The atomic charges utilized in this estimation are derived from the SPC/E water model, which are set as  −0.8476e for oxygen and  +0.4238e for hydrogen⁵⁰. This model has been demonstrated to provide a valid and efficient approximation, with no significant impact on the robustness of the conclusions in our previous work³⁷. Specifically, E₁ is defined as the projection of the local electric field onto the O₁-H^* bond, where a positive value of E₁ would stabilize the current configuration by hindering the proton’s transfer towards the opposite atom O₂. The introduction of an additional “intermediate” configuration can be necessitated by inspecting the plot of local electric field distribution presented as Supplementary Fig. 13, classifying configurations merely into Zundel-like and Eigen-like cations will result in non-negligible dependence of the field intensity on q_s and overlapped electric field between categories of configurations.