Extending Badger's rule. II. The relationship between energy and vibrational spectra in hydrogen bonds

We describe the energetic-spectral relationship between the energy of a hydrogen bond (EHB) and the redshift in the vibrational frequency of the covalent hydrogen bond donor (ΔωHB). The relationship was derived by convoluting expressions relating EHB and ΔωHB to the covalent bond distance of the hydrogen bond donor (ΔrHB), while ensuring balanced treatment of the exchange repulsion. We relied on reduced parts of the potential energy surfaces (PESs) of six hydrogen bonded dimers, namely, NH3-NH3, H2O-H2O, HF-HF, H2O-NH3, HF-H2O, and HF-NH3 derived from ab initio electronic structure calculations to fit the parameters of the model and validated its performance for extended parts of the PESs that include non-linear hydrogen bonds. The developed model suggests a novel relationship of a strength of 4.5 kcal/mol per 100 cm-1 redshift in the covalent donor (D-H) vibrational frequency, while non-linear effects become important for redshifts >200 cm-1. The single descriptor (ΔωHB), which is measurable either experimentally via gas-phase spectroscopy or theoretically via electronic structure calculations, was able to predict the experimental or calculated hydrogen bond energies of typical hydrogen bonded dimers using the developed relationship.