Correlation energy of anisotropic quantum dots

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 α^-1. In particular, we are interested in the limit where the anisotropy goes to infinity (α→0) 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 E_c over the entire range of α.