Two interacting electrons confined in a 3D parabolic cylindrically symmetric potential, in presence of axial magnetic field: A finite element approach

In this work, we explore the performance and accuracy of finite element numerical simulations for the relevant and well-known problem of two interacting electrons confined in a parabolic cylindrically symmetric potential, and under the influence of axial magnetic field. By using a full configuration interaction method as reference, we compare two different ways to obtain the eigenvalues of the system by finite element simulations, one of which approximately separates the Coulomb interaction and averages the longitudinal part. We find that in the regime of low aspect ratios, the results from the approximate scheme with approximation are quite optimal (<0.05% of difference respect to the reference), but once the dots turn more and more thicker, the results become just acceptable (>0.5% of difference), due to underestimation of the Coulomb interaction. On the other hand, for the non-approximated finite element approach the results are consistently reliable along the different field and aspect ratio regimes (<0.02% of difference respect to the reference). This allows us to present a novel, efficient and highly accurate method for obtaining electronic structure of interacting particles in 0D nanostructures.