  Inspired by recent advancements in the field of single-atom magnets,
particularly those involving rare-earth (RE) elements, we present a theoretical
exploration employing DFT+$U$ calculations to investigate the magnetic
properties of selected $4f$ atoms, specifically Eu, Gd and Ho, on a monolayer
of the transition-metal dichalcogenide WSe$_2$ in the 1H-phase. This study
comparatively examines RE with diverse $4f$ orbital fillings and valence
chemistry, aiming to understand how different coverage densities atop WSe$_2$
affect the magnetocrystalline anisotropy. We observe that RE elements lacking
$5d$ occupation in the atomic limit exhibit larger magnetic anisotropy energies
at high densities, while those with outer $5d$ electrons show larger
anisotropies in dilute configurations. Additionally, even half-filled $4f$
shell atoms with small orbital magnetic moments can generate substantial energy
barriers for magnetization rotation due to prominent orbital hybridizations
with WSe$_2$. Open $4f$ shell atoms further enhance anisotropy barriers through
spin-orbit coupling effects. This aspect is crucial for the experimental
realization of stable magnetic information units.
