The structure and dynamics of excess protons in water sparked interest already two centuries ago: Grotthus hypothesized that protons move through water by rapidly hopping between water molecules, a process by which the excess proton repeatedly interchanges with water hydrogen atoms, as illustrated in Fig. 1. The shuttling process leads to a high mobility of protons in water compated with other ions, as has been confirmed in many experiments. In attempts to understand the details of the Grotthus mechanism, the structure and dynamics of excess protons in water have been investigated in numerous studies throughout the last century. Due to its high net charge, the excess proton has a pronounced spectroscopic signature. In fact, already early infrared (IR) spectroscopy experiments reported strong and broad mid-infrared signals associated with dissolved protons in acidic aqueous solutions. The so-called ‘continuum band’, located between the water bending and stretching bands, where only few organic molecules absorb, and the ‘acid bend band’, a blue-shifted water bending band, are well-known spectroscopic features of excess protons in water (see Fig. 2b). More recently, the observation of characteristic IR continuum bands in biological systems, whose function relies on mobile protons, has shifted the research focus to the role of confined water and amino acids as proton donors and acceptors in Grotthus-like proton transfer mechanisms.
The interpretation of the spectra of systems containing mobile protons started a debate about the equilibrium solvation structure of excess protons in water. On the one hand, the Eigen state was hypothesized, which corresponds to a central hydronium ion surrounded by three tightly coordinated water molecules (green, on the left in Fig. 1). On the other hand, in the Zundel state, the excess proton is shared equally by two water molecules (red, center in Fig. 1). Lately, the debate whether the dominant solvation structure of the excess proton in water corresponds to the Eigen or the Zundel state has been converging: recent studies have demonstrated geometrically asymmetric Eigen and Zundel excess proton states in water and suggested that the distinction between those states is rather small and thus mostly semantic. Several 2D-IR studies have identified significant lifetimes of Zundel-like state, where the excess proton fluctuates over large distances and encompasses asymmetric states that may equally well be characterized as Eigen-like. Obviously, such Zundel-like and Eigen-like structures form dynamic intermediates of long-distance proton transfer events in water, as illustrated in Fig. 1.
We describe the proton transfer between two water molecules, the elementary step of the Grotthus mechanism, in terms of a thermally activated stochastic barrier-crossing process and interpret the IR vibrational spectra of excess protons, which is the principle experimental observable for the rapid proton dynamics, from this perspective (see Fig. 2a). According to reaction-rate theory, two time scales characterize barrier-crossing processes: The longer time scale is the waiting time, which describes the average time the system resides in one local minimum before crossing to the other. The transfer-path time is a much faster time scale, which describes the duration of the actual barrier-crossing transfer path. A complete description of the proton dynamics is obtained when additionally, the vibrational motion in the local minima is included, which can be well described by normal-mode analysis.
We have developed a trajectory-decomposition technique to split simulated proton trajectories into contributions that correspond to these different processes and studied the excess-protons dynamics in the H5O2+ cation  as well as in bulk hydrochloric-acid solutions . For each system ab initio molecular dynamics simulations are performed using DFT on the BLYP-D3 level. Thereby, the proton dynamics as well as the electronic polarization are accounted for and IR spectra can be computed from the polarization fluctuations via linear response theory that quantitatively compare to experimental data.
For bulk hydrochloric-acid solutions (data shown in Fig. 2b-d) we find that the excess-proton spectral signatures are proportional to the total infrared difference spectra, obtained by subtracting the neat-water absorption spectrum from hydrochloric-acid solution spectra. Furthermore, since the difference spectra scale linearly with HCl concentration, it follows that correlated dynamic effects between excess protons are negligible (see Fig. 2d). Therefore, the total difference spectra report rather accurately on single excess-proton dynamics and vice versa. By the aforementioned trajectory-decomposition technique, the spectral signatures of each contribution can be related to distinct features in the total difference spectra. The waiting time (TW) is the slowest time scale with around 200-300 fs and its signature is found deep in the THz regime. It corresponds to the interconversion time between Eigen and Zundel-like configurations. Our experimental THz spectra of hydrochloric acid solution likely show a signal in this regime, that however is partially hidden by a band of the rattling chloride counter ions. The transfer-path (TP) contribution is related to a pronounced feature at 1000-1200 cm-1, that in the literature is associated with Zundel-like configurations. The contribution related to vibrations in the meta-stable states and amenable to normal-mode (NM) analysis makes up for the IR continuum between 2000-3000 cm-1, which in the literature is associated with Eigen-like configurations.
The protonated gas-phase cluster H5O2+, illustrated in Fig. 2a , presents a tunable model system. By constraining the two oxygens to a fixed distance, the barrier height for the proton transfer between the two water molecules can be adjusted, and thereby the stochastic time scales of the transfer process are affected according to the Arrhenius exponential scaling. For this system, long ab initio trajectories are obtained for various oxygen-oxygen distances. It is thus possible to validate predictions for the spectral contributions based on stochastic barrier-crossing theory. We find that the spectral line shape of the waiting-time contribution corresponds to a Debye function. On the other hand, the line shape of the transfer-path contribution corresponds to the product of a Debye and a Lorentzian shape and thus is much sharper. By hydronium/deuterium exchange of the excess proton the transfer-path contribution is red-shifted, which reflects that the actual proton-transfer path is not affected much by friction. The spectral waiting-time contribution on the other hand does not shift upon hydronium/deuterium exchange, from which we conclude that the barrier-crossing waiting time is dominated by friction effects.
An animation is provided online https://fu-berlin.eu.vbrickrev.com/sharevideo/df2d94a4-6e7f-499a-a256-17d73b6124e4.
 Florian N Brünig, Manuel Rammler, Ellen M Adams, Martina Havenith, and Roland R Netz. "Spectral signatures of excess-proton waiting and transfer-path dynamics in aqueous hydrochloric acid solutions". Nat Commun 13(1):4210, (2022).
 Florian N Brünig, Paul Hillmann, Won Kyu Kim, Jan O Daldrop, and Roland R Netz. Proton-transfer spectroscopy beyond the normal-mode scenario. J Chem Phys 157, 174116 (2022).