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 epsilon(r) = -del nu(r) and magnetostatic B(r) = del x 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 rho(r) and physical current density j(r) and the potentials {nu(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 {rho(r), j(r)}, and explicitly employing the bijectivity between {rho(r), j(r)} and {nu(r), A(r)}, the further extension to N-representable densities and degenerate states is achieved via a Percus-Levy-Lieb constrained-search proof. A {rho(r), j(r)}-functional theory is developed. Finally, a Slater determinant of equidensity orbitals which reproduces a given {rho(r), j(r)} is constructed