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- independent and in spin- polarized density functional theory. It is pointed out that the nonuniqueness of the external magnetic field B(r) corresponding to a given pair of ground-state density n(r) and spin density s(r) in spin-polarized density functional theory implies the nonexistence of the total derivative of the SDFT Lieb functional F-N(,Ns)L [n, s] with respect to N-s. By giving a suitable extension of F-N(L)[n] and F-N(L),(Ns) [n,s] for N not equal integral n(r) dr and N-s not equal integral s(r) dr, it is then shown that their derivatives with respect to N and N s are equal to the derivatives, with respect to N and Ns, of the total energies E[N, upsilon] and E[N, N-s, upsilon, B] minus the external- field energy components, respectively.