Théorie de la fonctionnelle de la densité avec spin. VIII. Équation d'Euler-Lagrange pour f(r)

In the previous paper we expressed the Euler-Lagrange variational equation for the non-relativistic energy of spin multiplicities of a multifermionic system, implicit in the charge distribution ρ(r), and showed how to solve it iteratively — and derive optimal values of observables, including the spin distribution σ(r) — through using the scalar function f(r) of the local-scaling transformation of the orbit. In the present paper we derive a similar equation, more explicit but also more complex, for the total energy expressed as a functional of f(r), and compare the problems raised by the different approaches.