Singlet-triplet splitting and electron localization in elliptic quantum dots*

Experimental control of the singlet-triplet splitting in a two- electron $(2e)$ quantum dot molecule (QDM) as a function of the magnetic field is an important step in the implementation of quantum logic gates.\footnote{D.M. Zumb\"{u}hl {\it et al.\/}, cond-mat/0408276} Using symmetry breaking at the unrestricted Hartree-Fock level and subsequent symmetry restoration via projection techniques, we show that the two electrons localize and form a molecule (in the sense of Heitler- London) even when the interdot barrier {\it vanishes\/}.$^3$ This $2e$ molecule is characterized by a singlet-triplet curve similar to that of a QDM with a {\it finite\/} interdot barrier.\footnote{C. Yannouleas and U. Landman, Int. J. Quantum Chem. {\bf 90}, 699 (2002)} Most importantly, we find that a $2e$ molecule exhibiting a similar singlet-triplet curve is also formed in the case of a {\it single}, but {\it elliptic\/} QD; this is in agreement with experimental observations.$^2$