Magnetic Field Effects upon Exchange-Correlation in the Hooke's Atom

Extending Density Functional Theory (DFT) to coulombic systems in a non-vanishing magnetic field in a computationally feasible way is highly desirable. Even though the current DFT (CDFT) formalism is long-established, there still are no generally applicable, reliable $E_{xc}, {\mathbf A}_{xc}$ functionals analogous with the LDA. Progress can be made by comparison study on a solvable correlated system. Hooke's atom is well-known in ordinary DFT because its Schr\"{o}dinger equation can be solved exactly for some coupling strengths and numerically with high accuracy for the rest. Hence exact Kohn-Sham quantities are readily available. Using our extensions (exact and numerical) to non-zero B-field, we examined the effects on exchange- correlation holes and energies and considered possible ways to include the essential ones in $E_{xc}, {\mathbf A}_{xc} $. In our tests, the CDFT vorticity variable, $\nu$, turns out to be a computationally difficult quantity which may not be appropriate in practice to describe external B field effects on $E_{xc}, {\mathbf A}_{xc}$.