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

The nature of the explicit dependence on the particle number N and on the spin number N_s of the Lieb definition for the energy density functional is examined both in spin-free and in spin-polarized density functional theory. First, it is pointed out that for ground states, the nonuniqueness of the external magnetic field B(r) corresponding to a given pair of density n(r) and spin density s(r) in spin-polarized density functional theory implies the nonexistence 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)'s with norms not equal to N and N_s of the functionals' subscripts, it is then shown that the Lieb functionals' derivatives with respect to N and N_s are equal to the derivatives, with respect to N and N_s, of the total energies E[N,v] and E[N,N_s,v,B] minus the external-field energy components, respectively.