Violation of non-interacting $\cal V$-representability of the exact solutions of the Schr\"odinger equation for a two-electron quantum dot in a homogeneous magnetic field

We have shown by using the exact solutions for the two-electron system in a parabolic confinement and a homogeneous magnetic field [1, 2] that both exact densities (chargeand the paramagnetic current density) can be non-interacting V-representable (NIVR) only in a few special cases, or equivalently, that an exact Kohn-Sham (KS) system does not always exist. All those states at non-zero B can be NIVR, which are continuously connected to the singlet or triplet ground states at B = 0. In more detail, for singlets (total orbital angular momentum ML is even) both densities can be NIVR if the vorticity γ(r) = ∇× ( jp(r)/n(r) ) of the exact solution vanishes. For ML = 0 this is trivially guaranteed because the paramagnetic current density vanishes. The vorticity based on the exact solutions for the higher |ML| does not vanish, in particular for small r. In the limit r→ 0 this can even be shown analytically. For triplets (ML is odd) and if we assume circular symmetry for the KS system (the same symmetry as the real system) then only the exact states with |ML| = 1 can be NIVR with KS states having angular momenta m1 = 0 and |m2| = 1. Without specification of the symmetry of the KS system the condition for NIVR is that the small-r-exponents of the KS states are 0 and 1. PACS numbers: 31.15.EDensity-functional theory 31.15.ec Hohenberg-Kohn theorem and formal mathematical properties ... 73.21.La Quantum dots