Extending Badger's rule. I. The relationship between energy and structure in hydrogen bonds

We derive a new expression for the strength of a hydrogen bond (VHB) in terms of the elongation of the covalent bond of the donor fragment participating in the hydrogen bond (ΔrHB) and the intermolecular coordinates R (separation between the heavy atoms) and θ (deviation of the hydrogen bond from linearity). The expression includes components describing the covalent D-H bond of the hydrogen bond donor via a Morse potential, the Pauli repulsion, and electrostatic interactions between the constituent fragments using a linear expansion of their dipole moment and a quadratic expansion of their polarizability tensor. We fitted the parameters of the model using ab initio electronic structure results for six hydrogen bonded dimers, namely, NH3-NH3, H2O-H2O, HF-HF, H2O-NH3, HF-H2O, and HF-NH3, and validated its performance for extended parts of their potential energy surfaces, resulting in a mean absolute error ranging from 0.07 to 0.31 kcal/mol. The derived expression describes the energy-structure relationship in terms of a single structural parameter, namely, the elongation of the donor's covalent bond (ΔrHB), and suggests the novel relationship of 8.0 kcal/mol pm-1 (or 0.8 kcal/mol per 0.001 Å elongation). This structural parameter is easily obtained from theory and can serve as the single descriptor of the strength of individual hydrogen bonds.