Correlation energy of anisotropic quantum dots

We study the $D$-dimensional high-density correlation energy $\text{Ec}$ 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 $\text{Ec}$ 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 $\text{Ec}$ over the entire range of $\alpha$.