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

Abstract

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.

Isomer‐Specific Vibrational Spectroscopy of Microhydrated Lithium Dichloride Anions: Spectral Fingerprint of Solvent‐Shared Ion Pairs

“…The present results are also in agreement with the previously identified correlation [20] regarding red‐shifts in the stretching frequency of the hydrogen‐bonded OH oscillators with respect to the averaged OH stretching frequency of a free water molecule. Experimentally determined red‐shifts Δν OH are plotted against the corresponding, theoretically obtained, change in the vibrationally averaged bond length Δ R 0 (with respect to R 0 of an isolated water molecule) in Figure S9.…”

“…Experimentally determined red‐shifts Δν OH are plotted against the corresponding, theoretically obtained, change in the vibrationally averaged bond length Δ R 0 (with respect to R 0 of an isolated water molecule) in Figure S9. The ratio Δν/Δ R 0 =−18.6 cm −1 /10 pm found in our current study is similar to the computed values for pure water clusters (−20.2 cm −1 /10 pm) [21] and water clusters with incorporated halide and hydronium ions (−19.1 cm −1 /10 pm) [20] …”

Theoretical Studies of IR and NMR Spectral Changes Induced by Sigma-Hole Hydrogen, Halogen, Chalcogen, Pnicogen, and Tetrel Bonds in a Model Protein Environment

“…In particular, the shifts in certain IR bands or NMR peaks are frequently interpreted as a quantitative measure of the strength of each such bond [5,6,7]. As examples [3,8,9], the red shift of the A–H stretching frequency is thought to correlate with the strength of the AH···B HB, and there is a similar type of relationship for the downfield shift of the NMR peak of the bridging proton.…”

In Situ Assessment of Intrinsic Strength of X-I⋯OA-Type Halogen Bonds in Molecular Crystals with Periodic Local Vibrational Mode Theory

“…This reveals that in essence the Badger's rule [98][99][100] still holds, namely shorter bonds have larger harmonic frequencies (or force constants) and are thus stronger. Given that the Badger's rule was generalized to polyatomic molecules based on local stretching force constant by one of the authors of this work [61] and was recently extended to O-H bonds in liquid water [129], this work demonstrates for the first time that the Badger's rule even holds for crystals. Relationship between local stretching force constant k a n and bond length r for Cl-I (green) and I-I (purple) covalent bonds, respectively.…”