The accurate description of open-shell molecules, in particular of transition
metal complexes and clusters, is still an important challenge for quantum
chemistry. While density-functional theory (DFT) is widely applied in this
area, the sometimes severe limitations of its currently available approximate
realizations often preclude its application as a predictive theory. Here, we
review the foundations of DFT applied to open-shell systems, both within the
nonrelativistic and the relativistic framework. In particular, we provide an
in-depth discussion of the exact theory, with a focus on the role of the spin
density and possibilities for targeting specific spin states. It turns out that
different options exist for setting up Kohn-Sham DFT schemes for open-shell
systems, which imply different definitions of the exchange-correlation energy
functional and lead to different exact conditions on this functional. Finally,
we suggest some possible directions for future developments
