4f band modulation by spin–orbit coupling in low-dimensional rare earth catalysts

) and van der Waals dispersion, we construct and relax interface models comprising single atoms, monolayers, bilayers, and quadruple layers of Eu. Layer-resolved density of states (DOS) analysis reveals that SOC lifts the degeneracy of the Eu 4f states, broadening their spectrum and shifting the principal DOS peaks closer to the Fermi level. For example, in the single-atom Eu configuration, SOC shifts the maximum 4f DOS peak from -0.759 eV to -0.063 eV while increasing the spectral width from 0.133 eV to 0.680 eV. A clear correlation emerges between the Eu-O interfacial distance and the localisation of the 4f states, with thicker Eu nano-sheets exhibiting sharper and more deeply bound electronic states. These results suggest that ultra-thin Eu configurations with SOC-broadened 4f states may enhance electron transfer to adsorbed intermediates, making them promising candidates for catalytic processes such as water splitting, hydrogen evolution, and hydrocarbon reforming.