Density and physical current density functional theory

Abstract 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 ) (with R = r 1 ,…, r N ):ψ=ψ [ρ, j ,α]. It 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. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010