Nonuniqueness of magnetic fields and energy derivatives in spin-polarized density functional theory

The effect of the recently uncovered nonuniqueness of the external magnetic field B(r) corresponding to a given pair of density n(r) and spin density n(s)(r) on the derivative of the energy functional of spin-polarized density functional theory, and its implications for the definition of chemical reactivity descriptors, is examined. For ground states, the nonuniqueness of B(r) implies the nondifferentiability of the energy functional E(v,B)[n,n(s)] with respect to n(s)(r). It is shown, on the other hand, that this nonuniqueness allows the existence of the one-sided derivatives of E(v,B)[n,n(s)] with respect to n(s)(r). Although the N-electron ground state can always be obtained from the minimization of E(v,B)[n,n(s)] without any constraint on the spin number N(s)=integraln(s)(r)dr, the Lagrange multiplier mu(s) associated with the fixation of N(s) does not vanish even for ground states. Mu(s) is identified as the left- or right-side derivative of the total energy with respect to N(s), which justifies the interpretation of mu(s) as a (spin) chemical potential. This is relevant not only for the spin-polarized generalization of conceptual density functional theory, the spin chemical potential being one of the elementary reactivity descriptors, but also for the extension of the thermodynamical analogy of density functional theory for the spin-polarized case. For higher-order reactivity indices, B(r)'s nonuniqueness has similar implications as for mu(s), leading to a split of the indices with respect to N(s) into one-sided reactivity descriptors.