Frontier d-orbital center theory: A mechanism for orbital coupling and reactivity in single-atom catalysts

The traditional frontier orbital (FO) theory has achieved notable success in describing single-atom catalysts (SACs) on semiconducting supports. However, its applicability is significantly limited in metallic or semimetallic SACs due to the absence of discrete and identifiable orbitals caused by continuous states and strong orbital hybridization. Herein, we propose the Frontier d-Orbital Center (FDC) theory to analyze adsorption behavior and catalytic activity in SACs with complex electronic structures. FDC condenses the spin-polarized projected density of states (PDOS) into a scalar centroid of the most reactive d-orbital region, enabling identification of dominant orbital channels that couple with the π* orbitals of adsorbates. Crucially, this approach does not rely on spatial orbital localization and remains valid in metallic or semimetallic systems, thus addressing a key limitation of the FO model. Using the nitrate reduction reaction (NO₃RR) as a probe, we validate the FDC model across a library of 104 dual-heteroatom-doped SACs built on two-dimensional metalloporphyrin frameworks. Further analysis reveals that the d orbitals indicated by FDC govern wavefunction coupling with the π* orbitals of NO₃⁻, thereby determining adsorption strength and configuration. This work establishes FDC theory as a unified and transferable orbital-level framework for SACs, offering new insights into structure-activity relationships.