The system of two interacting electrons in one-dimensional harmonic potential or Hooke’s atom is considered, again. On one hand, it appears as a model for quantum dots in a strong confinement regime, and on the other hand, it provides us with a hard test bench for new methods with the “space splitting” arising from the one-dimensional Coulomb potential. Here, we complete the numerous previous studies of the ground state of Hooke’s atom by including the excited states and dynamics, not considered earlier. With the perturbation theory, we reach essentially exact eigenstate energies and wave functions for the strong confinement regime as novel results. We also consider external perturbation induced quantum dynamics in a simple separable case. Finally, we test our novel numerical approach based on real-time path integrals (RTPIs) in reproducing the above. The RTPI turns out to be a straightforward approach with exact account of electronic correlations for solving the eigenstates and dynamics without the conventional restrictions of electronic structure methods.
We study the properties of the Hooke's law correlation energy (
Ec), defined as the correlation energy when two electrons interact via a harmonic potential in a
D-dimensional space. More precisely, we investigate the
1S ground-state properties of two model systems: the Moshinsky atom (in which the electrons move in a quadratic potential) and the spherium model (in which they move on the surface of a sphere). A comparison with their Coulombic counterparts is made that highlights the main differences of the
Ec in both the weakly and strongly correlated limits. Moreover, we show that the Schr\"odinger equation of the spherium model is exactly solvable for two values of the dimension (
D=1and3) and that the exact wave function is based on Mathieu functions.