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) (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.

Hohenberg-Kohn theorem including electron spin

“…Note that the equations for this {ρ(r), j(r)} functional theory reduce to those where the interaction of the magnetic field with the electron spin is ignored [1,2]. The latter set of equations in turn reduce to HK DFT in the absence of a magnetic field.…”

“…(2), we have proved [1,2] that the basic variables are the gauge invariant nondegenerate ground state density ρ(r) and the physical current density j(r). We arrived at this conclusion by proving for the nondegenerate ground state a bijective relationship between {ρ(r), j(r)} and the external potentials be many-to-one [13] and even infinite-to-one [14].…”

“…It is known [13] that {ρ(r), j p (r)} cannot uniquely determine the potentials {v(r), A(r)}. Thus, the arguments for {ρ(r), j p (r)} being the basic variables are based on solely a Map D type proof presupposing [1,2,16] the existence of a Map C. But as noted above, there is no Map C in this case.…”

“…A basic variable is defined [1,2] as a gauge invariant property of the system that uniquely determines the Hamiltonian H, and thereby via the solution Ψ of the Schrödinger equationĤΨ = EΨ, all the properties of the system. This path from the basic variable to the wave function Ψ emanates from the Hohenberg-Kohn (HK) theorem [3].…”

“…In our recent work [1,2], we have considered the case of the presence of both an external electrostatic field E(r) = −∇v(r) and magnetostatic field B(r) = ∇ × A(r), with A(r) the vector potential. The Hamiltonian, when the interaction of the magnetic field is only with the orbital angular momentum is then…”