  Single adatoms offer an exceptional playground for studying magnetism and its
associated dynamics at the atomic scale. Here we review recent results on
single adatoms deposited on metallic substrates, based on time-dependent
density functional theory. First we analyze quantum zero-point
spin-fluctuations (ZPSF) as calculated from the fluctuation-dissipation
theorem, and show how they affect the magnetic stability by modifying the
magnetic anisotropy energy. We also assess the impact of ZPSF in the limit of
small hybridization to the substrate characteristic of semi-insulating
substrates, connecting to recent experimental investigations where magnetic
stability of a single adatom was achieved for the first time. Secondly, we
inspect further the dynamics of single adatoms by considering the longitudinal
and transverse spin-relaxation processes, whose time-scales are analyzed and
related to the underlying electronic structure of both the adatom and the
substrate. Thirdly, we analyze spin-fluctuation modes of paramagnetic adatoms,
i.e., adatoms where the Stoner criterion for magnetism is almost fulfilled.
Interestingly, such modes can develop well-defined peaks in the meV range,
their main characteristics being determined by two fundamental electronic
properties, namely the Stoner parameter and the density of states at the Fermi
level. Furthermore, simulated inelastic scanning tunneling spectroscopy curves
reveal that these spin-fluctuation modes can be triggered by tunneling
electrons, opening up potential applications also for paramagnetic adatoms.
Lastly, an overview of the outstanding issues and future directions is given.
