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The nature of the explicit dependence on the particle number N and on thespin number N_s of the Lieb definition for the energy density functional isexamined both in spin-free and in spin-polarized density functional theory.First, it is pointed out that for ground states, the nonuniqueness of theexternal magnetic field B(r) corresponding to a given pair of density n(r) andspin density s(r) in spin-polarized density functional theory implies thenonexistence of the derivative of the SDFT Lieb functional with respect to N_s.Giving a suitable generalization of the Lieb functionals for n(r)'s and s(r)'swith norms not equal to N and N_s of the functionals' subscripts, it is thenshown that the Lieb functionals' derivatives with respect to N and N_s areequal to the derivatives, with respect to N and N_s, of the total energiesE[N,v] and E[N,N_s,v,B] minus the external-field energy components,respectively.

Derivative of the Lieb definition for the energy functional of density functional theory with respect to the particle number and the spin number