Zero-point energies from bond orders and populations relationships

We report two analytical quantum mechanics (QM) models for approximating appropriately scaled harmonic zero-point energies (ZPEs) without Hessian calculations. Following our earlier bond energies from bond orders and populations model that takes a similar form as an extended Hückel model but uses well-conditioned orbital populations, this work demonstrates a proof of concept for approximating ZPEs, an important component in thermochemistry calculations, while eschewing unfavorably scaling algorithms involving Hessian matrices. The ZPE-BOP1 model uses Mulliken orbital populations from hybrid Kohn-Sham density functional theory calculations within an extended Hückel-type model that defines vibrational bond energy terms using two atom-pairwise parameters that are fit to reproduce ZPEs from B3LYP calculations. The more accurate ZPE-BOP2 model uses Mulliken orbital populations from Hartree-Fock calculations within a different extended Hückel-type model that includes a short-range anharmonic energy term and a coupled three-body oscillator energy term with seven atom-pairwise parameters. Both models predict ZPEs in molecules involving first row elements, but ZPE-BOP2 outperforms ZPE-BOP1 in strained and long-chain molecules and provides ZPEs more competitive with those from semi-empirical QM methods (e.g., AM1, PM6, PM7, and XTB-2) that compute ZPEs with Hessian calculations. This work shows progress and an outlook toward computational models that use well-conditioned orbital populations to efficiently predict useful physicochemical properties. It also shows opportunities for approximate QM models that would shift traditional computational bottlenecks away from costly algorithms such as Hessian calculations to others that focus on reliable orbital populations.