  We investigate the dynamics of Fe adatoms and dimers deposited on the Cu(111)
metallic surface in the presence of spin-orbit coupling, within time-dependent
density functional theory. The \textit{ab initio} results provide
material-dependent parameters that can be used in semiclassical approaches,
which are used for insightful interpretations of the excitation modes. By
manipulating the surroundings of the magnetic elements, we show that elliptical
precessional motion may be induced through the modification of the magnetic
anisotropy energy. We also demonstrate how different kinds of spin precession
are realized, considering the symmetry of the magnetic anisotropy energy, the
ferro- or antiferromagnetic nature of the exchange coupling between the
impurities, and the strength of the magnetic damping. In particular, the normal
modes of a dimer depend on the initial magnetic configuration, changing
drastically by going from a ferromagnetic metastable state to the
antiferromagnetic ground state. By taking into account the effect of the
damping into their resonant frequencies, we reveal that an important
contribution arises for strongly biaxial systems and specially for the
antiferromagnetic dimers with large exchange couplings. Counter intuitively,
our results indicate that the magnetic damping influences the quantum
fluctuations by decreasing the zero-point energy of the system.
