  Dimers are the smallest chemical objects that show magnetic anisotropy. We
focus on 3$d$ and 4$d$ transition metal dimers that have magnetic ground states
in most cases. Some of these magnetic dimers have a considerable barrier
against re-orientation of their magnetization, the so-called magnetic
anisotropy energy, MAE. The height of this barrier is important for
technological applications, as it determines, e.g., the stability of
information stored in magnetic memory devices. It can be estimated by means of
relativistic density functional calculations. Our approach is based on a
full-potential local-orbital method (FPLO) in a four-component Dirac-Kohn-Sham
implementation. Orbital polarization corrections to the local density
approximation are employed. They are discussed in the broader context of
orbital dependent density functionals. Ground state properties (spin
multiplicity, bond length, harmonic vibrational frequency, spin- and orbital
magnetic moment, and MAE) of the 3$d$ and 4$d$ transition metal dimers are
evaluated and compared with available experimental and theoretical data. We
find exceptionally high values of MAE, close to 0.2 eV, for four particular
dimers: Fe$_2$, Co$_2$, Ni$_2$, and Rh$_2$.
