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

Abstract

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.