Analytical Solutions for States of the 3D Hooke's Atom in an External B Field

Hooke's atom is a 2-electron model with the nuclear-electron attraction replaced by an isotropic harmonic potential. Closed-form solutions for certain eigenstates are known in 3D at B=0; B $\ne$ 0 solutions are known only for the 2D quantum dot. Both solutions are products of center-of-mass oscillators and relative motion factors. Because the uniform-B confining potential is quadratic in the cartesian coordinates normal to B, we can find related analytical solutions for certain eignvalues of the 3D Hooke's atom at B $\ne$ 0. They are more complicated because of the imposed axial symmetry. We sketch the somewhat tedious solution techniques, then compare the analytical and numerical solutions for a large range of field strengths. We have used these results to obtain exact Kohn-Sham orbitals for current density functional theory (CDFT) [``Exact Current DFT Study of Hooke's Atom in Magnetic Fields,'' W. Zhu and S.B. Trickey to be published]