  In this work, the zero-temperature limit of the thermodynamic spin-density
functional theory is investigated. The coarse-grained approach to the
equilibrium density operator is used to describe the equilibrium state. The
characteristic functions of a macrostate are introduced and their
zero-temperature limits are investigated. A detailed discussion of the
spin-grand-canonical ensemble in the entropy and energy representations is
performed. The maps between the state function variables at 0K limit are
rigorously studied for both representations. In the spin-canonical ensemble,
the energy surface and the discontinuity pattern are investigated. Finally,
based on the maps between the state function variables at 0K limit, the
Hohenberg-Kohn theorem for the systems with non-integer electron and spin
numbers at zero-temperature limit is formulated.
