## 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

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Taut, M.

Machon, P.

Eschrig, H.

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We have shown by using the exact solutions for the two-electron system in a
parabolic confinement and a homogeneous magnetic field [ M.Taut, J Phys.A{\bf
27}, 1045 (1994) ] that both exact densities (charge- and the paramagnetic
current density) can be non-interacting $\cal V$-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 $M_L$ is even) both
densities can be NIVR if the vorticity of the exact solution vanishes. For
$M_L=0$ this is trivially guaranteed because the paramagnetic current density
vanishes. The vorticity based on the exact solutions for the higher $|M_L|$
does not vanish, in particular for small r. In the limit $r \to 0$ this can
even be shown analytically. For triplets ($M_L$ 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 $|M_L|= 1$ can be NIVR with KS states having angular
momenta $m_1=0$ and $|m_2|=1$. 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.

Comment: 18 pages, 4 figures

Comment: 18 pages, 4 figures

##### Keywords

Condensed Matter - Mesoscale and Nanoscale Physics