  We show that a diatomic dot molecule made of two identical, vertically
stacked, strained InAs/GaAs self-assembled dots exhibits an asymmetry in its
single-particle and may-particle wavefunctions. The single-particle wave
function is asymmetric due to the inhomogeneous strain, while the asymmetry of
the many-particle wavefunctions is caused by the correlation induced
localization: the lowest singlet $^1\Sigma_g$ and triplet $^3\Sigma$ states
show that the two electrons are each localized on different dots within the
molecule, for the next singlet states $^1\Sigma_u$ both electrons are localized
on the same (bottom) dot for interdot separation $d>$ 8 nm. The singlet-triplet
splitting is found to be $\sim 0.1$ meV at inter-dot separation $d$=9 nm and as
large as 100 meV for $d$=4 nm, orders of magnitude larger than the few meV
found in the large (50 - 100 nm) electrostatically confined dots.
