When Are Dopant <i>d</i>-States Free-Atom-Like? Periodic Trends and Confinement Effects in Single-Atom Alloys

The free-atom-like character of dopant <i>d</i>-states is a defining feature of single-atom alloys (SAAs), contributing to their exceptional selectivity and activity in heterogeneous catalysis. Based on reliable density functional theory (DFT) calculations for the full series of 4<i>d</i> transition metal (TM) dopants embedded in various hosts, we provide a unified perspective on when dopants exhibit this characteristic and how it can be tuned. Only late TM dopants exhibit the narrow, free-atom-like <i>d</i>-bands typically associated with SAAs, whereas early TM dopants display significantly broader <i>d</i>-bands that approach the width of host metal <i>d</i>-bands. This variation is not driven by increased dopant-host <i>d</i>-state mixing, which remains minimal across the series. Instead, we attribute the observed periodic trend to differences in the spatial extent of the localized dopant states and their overlap with surrounding host atoms, as well as to avoided hybridization associated with <i>d</i>-state filling. We further corroborate that dopant confinement, quantified by the number and proximity of surrounding host atoms, is as a key factor: more confined dopants exhibit broader <i>d</i>-bands, whereas less confined dopants feature narrower <i>d</i>-bands. Reduced dopant confinement also stabilizes high-spin configurations, enhancing spin polarization for certain 4<i>d</i> elements. Together, these findings offer fundamental insights into the origins of the unique electronic structure of SAAs. Building on these findings, we establish design principles for tuning dopant <i>d</i>-band shape and spin and illustrate how such modifications impact catalytic selectivity. The developed guidelines are also encapsulated in a machine learning model that predicts <i>d</i>-band widths, facilitating the rational design of SAAs.