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

Abstract Out-of-plane strain associated with nonzero curvature enhances the reactivity of M–N–C single-atom catalytic sites embedded in graphene toward electrochemical oxygen reduction as well as CO2 reduction reactions. 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. These results establish out-of-plane curvature as a geometric descriptor that simultaneously modulates adsorbate binding thermodynamics and proton vibrational overlap, offering mechanistic insight into out-of-plane-strain-induced reactivity modulation at catalytic sites.