A Geometric-Electronic Principle for Curvature-Driven Single-Atom Catalysis

Substrate curvature offers a powerful handle for catalytic tuning, yet a general quantitative framework for arbitrary anisotropic morphologies remains lacking. Here, by combining differential geometry with a symmetry-constrained Taylor expansion, we derive a geometric descriptor, φ, from the full curvature tensor that compresses complex local curvature into a predictive variable for catalytic properties. Using Fe–N 4 –C single-atom catalysts as a model for CO 2 RR, systematic calculations across 602 nonequivalent curved sites reveal that the curvature-dependent activity variation is governed by the out-of-plane d -orbital centroid, 〈 z 〉. This electronic descriptor captures the asymmetric polarization of d orbitals along the surface normal and naturally distinguishes convex from concave active sites. We trace this electronic polarization to its mechanical origin: the intrinsic stiffness contrast between pentagonal and hexagonal rings converts geometric strain into opposite vertical displacements of the metal center that drive the orbital response. This establishes a quantitative causal pathway from geometry ( φ ) through mechanics and electronic structure (〈 z 〉) to catalytic activity. This framework elevates local curvature from a qualitative structural feature to a quantitative and tunable design axis for catalyst engineering.