$T>0$ ensemble-state density functional theory via Legendre transform

A logical foundation of equilibrium state density functional theory in a Kohn-Sham-type formulation is presented on the basis of Mermin’s treatment of the grand canonical state by exploiting functional Legendre transforms. It is simpler and more satisfactory compared to the usual derivation of the ground-state theory and free of most remaining open points of the latter. The existence of the functional derivative of the corresponding density functional $F\left[n\right]$ at all densities of grand canonical equilibrium states is proved even in the spin-density matrix version of the theory. It may, in particular, be relevant with respect to cases of spontaneous symmetry breaking such as noncollinear magnetism and orbital order.