Out-of-Plane Strain Modulates Frontier-Orbital Splitting and Vibrational Wave Function Overlap at M–N₄ Single-Atom Catalytic Sites

Here, by combining density functional theory calculations, quantized proton vibrational analysis, and machine-learning feature selection, we show that curvature-induced deformation drives coupled electronic-structure and vibrational responses at M–N₄ single-atom catalytic sites.

Systematic comparison across signed curvature values for multiple metal–adsorbate combinations reveals two coupled consequences of deformation: (i) splitting of metal-centered frontier orbitals and local symmetry breaking of the M–N₄ coordination pocket, and (ii) modulation of the reactant–product proton vibrational wave function overlap, with implications for proton-transfer and proton-coupled electron-transfer kinetic descriptors.