Unveiling the intermediate hydrated proton in water through vibrational analysis on the 1750 cm^-1 signature

Hydration of proton is the key to understand the acid-base chemistry and biochemical processes, for which the Zundel and Eigen cations have been recognized as the foundation. However, their dominance remains contentious due to the challenge of attributing the infrared signature at  ~1750 cm^-1, stemming from the theoretical dilemma of balancing structural diversity and solvent fluctuations. Herein, we circumvent this obstacle by devising an integrated approach for computing frequency-specific vibrational vectors via inverse Fourier transform of the vibrational density of states. When applied to aqueous acid, it unveils an additional "Intermediate" configuration, linked to the aforementioned spectral signature, which exhibits a higher population (44%) and longer lifetime (51 fs), compared to Zundel-like (28%, 25 fs) and Eigen-like (28%, 36 fs), benefitting from the local electric field induced by surrounding solvent molecules.