Exact density functionals for two-electron systems in an external magnetic field

In principle, the extension of density functional theory (DFT) to Coulombic systems in a nonvanishing magnetic field is via current DFT (CDFT). Though CDFT is long established formally, relatively little is known with respect to any generally applicable, reliable approximate E(XC) and A(XC) functionals analogous with the workhorse approximate functionals (local density approximation and generalized gradient approximation) of ordinary DFT. Progress can be aided by having benchmark studies on a solvable correlated system. At zero field, the best-known finite system for such purposes is Hooke's atom. Recently we extended the exact ground state solutions for this two-electron system to certain combinations of nonzero external magnetic fields and confinement strengths. From those exact solutions, as well as high-accuracy numerical results for other field and confinement combinations, we construct the correlated electron density and paramagnetic current density, the exact Kohn-Sham orbitals, and the exact DFT and CDFT exchange-correlation energies and potentials. We compare with results from several widely used approximate functionals, all of which exhibit major qualitative failures, whether in CDFT or in naive application of ordinary DFT. We also illustrate how the CDFT vorticity variable nu is a computationally difficult quantity which may not be appropriate in practice to describe the external B field effects on E(XC) and A(XC).