For a system of N electrons in an external scalar potential v(r) and external vector potential A(r), we prove that the wave function ψ is a functional of the gauge invariant ground state density ρ(r) and ground state physical current density j(r), and a gauge function α(R) (withIt is the presence of the gauge function α(R) that ensures the wave function functional is gauge variant. We prove this via a unitary transformation and by a proof of the bijectivity between the potentials {v(r), A(r)} and the ground state properties {ρ(r), j(r)}. Thus, the natural basic variables for the system are the gauge invariant ρ(r) and j(r). Because each choice of gauge function corresponds to the same physical system, the choice of α(R) = 0 is equally valid. As such, we construct a {ρ(r), j(r)} functional theory with the corresponding Euler equations for the density ρ(r) and physical current density j(r), together with the constraints of charge conservation and continuity of the current. With the assumption of existence of a system of noninteracting fermions with the same ρ(r) and j(r) as that of the electrons, we provide the equations describing this model system, the definitions being within the framework of Kohn-Sham theory in terms of energy functionals of {ρ(r), j(r)} and their functional derivatives. A special case of the {ρ(r), j(r)} functional theory is the magnetic-field density-functional theory of Grayce and Harris. We discuss and contrast our work with the paramagnetic current-and density-functional theory of Vignale and Rasolt in which the variables are the gauge invariant ground state density ρ(r), and vorticity ν(r) = ∇ × (j p (r)/ρ(r)), where j p (r) is the paramagnetic current density.

We generalize the quantal density functional theory (QDFT) of electrons in the presence of an external electrostatic field E (r) = −∇v(r) to include an external magnetostatic field B(r) = ∇ × A(r), where {v(r), A(r)} are the respective scalar and vector potentials. The generalized QDFT, valid for nondegenerate ground and excited states, is the mapping from the interacting system of electrons to a model of noninteracting fermions with the same density ρ(r) and physical current density j(r), and from which the total energy can be obtained. The properties {ρ(r), j(r)} constitute the basic quantum mechanical variables because, as proved previously, for a nondegenerate ground state they uniquely determine the potentials {v(r), A(r)}. The mapping to the noninteracting system is arbitrary in that the model fermions may be either in their ground or excited state. The theory is explicated by application to a ground state of the exactly solvable (2-dimensional) Hooke's atom in a magnetic field, with the mapping being to a model system also in its ground state. The majority of properties of the model are obtained in closed analytical or semi-analytical form. A comparison with the corresponding mapping from a ground state of the (3-dimensional) Hooke's atom in the absence of a magnetic field is also made.

The Gunnarsson-Lundqvist (GL) theorem of density functional theory states that there is a one-to-one relationship between the density of the lowest nondegenerate excited state of a given symmetry and the external potential. As a consequence, knowledge of this excited state density determines the external potential uniquely.[The GL theorem is the equivalent for such excited states of the Hohenberg-Kohn (HK) theorem for nondegenerate ground states.] For other excited states, there is no equivalent of the GL or HK theorem. For these states, there thus exist multiple potentials that generate the excited-state density. We show, by example, the satisfaction that the GL theorem holds and the multiplicity of potentials for excited states.

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