  Since Spin Density Functional Theory was first proposed, but also recently,
examples were constructed to show that a spin-potential may share its ground
state with other spin-potentials. In fact, for collinear magnetic fields and
systems with fixed magnetization, the mapping between potentials and ground
states is invertible, provided the magnetization is not saturated and that
spin-potentials are determined within a spin-constant. We complete the proof
that the mapping is invertible also for non-collinear magnetic fields and
systems with more than one electron. We then discuss the non-collinear exchange
and correlation energy functional and suggest improvements.
