Proton disorder in ice Ih and inhomogeneous broadening in two-dimensional infrared spectroscopy

It is well-known that in ice Ih the oxygen atoms form a regular hexagonal lattice while the positions of the hydrogen atoms are disordered, called proton disorder in the literature. Various OH (OD) stretch vibrational spectroscopies (e.g., IR, Raman, two-dimensional IR (2DIR), and hole burning) have been used to probe this proton disorder in the past several decades. However, the presence and the magnitude of the inhomogeneous broadening due to this proton disorder in the vibrational spectroscopy is still controversial. In this work, we calculate 2DIR spectroscopy for HOD in D2O ice Ih at 80 K with a mixed quantum/classical approach, and make comparison to a recent 2DIR experiment on the same system. Fair agreement is achieved between theory and experiment, although the calculated 2DIR line shape shows inhomogeneous broadening that was not observed in the experiment. However, the theory reproduces the linear IR for the same system fairly well, and the inhomogeneous broadening from the calculation is consistent with the extrapolation of the experimental IR line-widths in the literature. The effect of this proton disorder on the 2DIR line shape is explored in detail. We also calculate the vibrational three-pulse photon echo peak shift signal, which shows signatures of both low-frequency dynamics and inhomogeneous broadening.