Hohenberg-Kohn theorem including electron spin

The Hohenberg-Kohn theorem is generalized to the case of a finite system of N electrons in external electrostatic E(r)=-∇v(r) and magnetostatic B(r)=∇×A(r) fields in which the interaction of the latter with both the orbital and spin angular momentum is considered. For a nondegenerate ground state a bijective relationship is proved between the gauge invariant density ρ(r) and physical current density j(r) and the potentials {v(r),A(r)}. The possible many-to-one relationship between the potentials {v(r),A(r)} and the wave function is explicitly accounted for in the proof. With the knowledge that the basic variables are {ρ(r),j(r)}, and explicitly employing the bijectivity between {ρ(r),j(r)} and {v(r),A(r)}, the further extension to N-representable densities and degenerate states is achieved via a Percus-Levy-Lieb constrained-search proof. A {ρ(r),j(r)}-functional theory is developed. Finally, a Slater determinant of equidensity orbitals which reproduces a given {ρ(r),j(r)} is constructed.