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.