Correlation energy of anisotropic quantum dots

We study the $D$-dimensional high-density correlation energy $\Ec$ 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 $\Ec$ as a function of the anisotropy $α^{-1}$. In particular, we are interested in the limit where the anisotropy goes to infinity ($α\to0$) and the electrons are restricted to a lower-dimensional space. We show that tuning the value of $α$ 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 $\Ec$ over the entire range of $α$.

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