Correlation energy of anisotropic quantum dots

Abstract

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 ${\ensuremath{\alpha}}^{\ensuremath{-}1}$. In particular, we are interested in the limit where the anisotropy goes to infinity ($\ensuremath{\alpha}\ensuremath{\rightarrow}0$) and the electrons are restricted to a lower-dimensional space. We show that tuning the value of $\ensuremath{\alpha}$ 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 $\ensuremath{\alpha}$.