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

Potential in spin-density-functional theory of noncollinear magnetism determined by the many-electron ground state