We study the $D$-dimensional high-density correlation energy $E_c$ of the singlet ground state of two electrons confined by a harmonic potential with Coulombic repulsion. We allow the harmonic potential to be anisotropic and examine the behavior of $E_c$ as a function of the anisotropy ${\alpha }^{-1}$. In particular, we are interested in the limit where the anisotropy goes to infinity ($\alpha \to 0$) and the electrons are restricted to a lower-dimensional space. We show that tuning the value of $\alpha$ from 0 to 1 allows a smooth dimensional interpolation and we demonstrate that the usual model, in which a quantum dot is treated as a two-dimensional system, is inappropriate. Finally, we provide a simple function which reproduces the behavior of $E_c$ over the entire range of $\alpha$.

• Received 1 August 2011

DOI:https://doi.org/10.1103/PhysRevA.84.032513