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

TL;DR

This paper investigates how the Lieb energy functional in density functional theory depends on particle and spin numbers, clarifying the existence of derivatives and their relation to total energy derivatives.

Contribution

It generalizes Lieb functionals for non-standard norms and establishes the relationship between their derivatives and total energy derivatives in spin-polarized DFT.

Findings

Derivatives of Lieb functionals with respect to N and N_s are linked to total energy derivatives.

Nonuniqueness of magnetic fields affects the existence of derivatives in spin-polarized DFT.

Generalized Lieb functionals can be defined for norms different from N and N_s.

Abstract

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.