  Access to magnetic excitation spectra of single atoms deposited on surfaces
is nowadays possible by means of low-temperature inelastic scanning tunneling
spectroscopy. We present a first-principles method for the calculation of
inelastic tunneling spectra utilizing the Korringa-Kohn-Rostoker Green function
method combined with time-dependent density functional theory and many-body
perturbation theory. The key quantity is the electron self-energy describing
the coupling of the electrons to the spin excitation within the adsorbate. By
investigating Cr, Mn, Fe and Co adatoms on a Cu(111) substrate, we
spin-characterize the spectra and demonstrate that their shapes are altered by
the magnetization of the adatoms, of the tip and the orbital decay into vacuum.
Our method also predicts spectral features more complex than the steps obtained
by simpler models for the adsorbate (e.g., localized spin models).
