Beyond Badger’s Rule: The Origins and Generality of the Structure–Spectra Relationship of Aqueous Hydrogen Bonds

The structure of hydrogen bonded networks is intimately intertwined with their dynamics. Despite the incredibly wide range of hydrogen bond strengths encountered in water clusters, ion–water clusters, and liquid water, we demonstrate that the previously reported correlation between the change in the equilibrium bond length of the hydrogen bonded OH covalent bond and the corresponding shift in its harmonic frequency in water clusters is much more broadly applicable. Surprisingly, this correlation describes the ratios for both the equilibrium OH bond length/harmonic frequency and the vibrationally averaged bond length/anharmonic frequency in water, hydronium water, and halide water clusters. Consideration of harmonic and anaharmonic data leads to a correlation of −19 ± 1 cm–1/0.001 Å. The fundamental nature of this correlation is further confirmed through the analysis of ab initio Molecular Dynamics (AIMD) trajectories for liquid water. We demonstrate that this simple correlation for both harmonic and anharmonic systems can be modeled by the response of an OH bond to an external field. Treating the OH bond as a Morse oscillator, we develop analytic expressions, which relate the ratio of the shift in the vibrational frequency of the hydrogen-bonded OH bond to the shift in OH bond length, to parameters in the Morse potential and the ratio of the first and second derivatives of the field-dependent projection of the dipole moment of water onto the hydrogen-bonded OH bond. Based on our analysis, we develop a protocol for reconstructing the AIMD spectra of liquid water from the sampled distribution of the OH bond lengths. Our findings elucidate the origins of the relationship between the molecular structure of the fleeting hydrogen-bonded network and the ensuing dynamics, which can be probed by vibrational spectroscopy