Discontinuous transition between Zundel and Eigen for H₅O₂⁺

The hydrated-proton structure is critical for understanding the proton transport in water. However, whether the hydrated proton adopts Zundel or Eigen structure in solution has been highly debated in the past several decades. Current experimental techniques cannot directly visualize the dynamic structures in situ, while the available theoretical results on the infrared (IR) spectrum derived from current configurational models cannot fully reproduce the experimental results and thus are unable to provide their precise structures. In this work, using H₅O₂⁺ as a model, we performed first-principles calculations to demonstrate that both the structural feature and the IR frequency of proton stretching, characteristics to discern the Zundel or Eigen structures, evolve discontinuously with the change of the O-O distance. A simple formula was introduced to discriminate the Zundel, Zundel-like, and Eigen-like structures. This work arouses new perspectives to understand the proton hydration in water.