  The exchange coupling of the spins of two electrons in double well potentials
in a semiconductor background is calculated within the Heitler-London (HL)
approximation. Atomic and quantum dot types of confining potentials are
considered, and a systematic analysis for the source of inaccuracies in the HL
approach is presented. For the strongly confining coulombic atomic potentials
in the H2 molecule, the most dramatic failure occurs at very large interatomic
distances, where HL predicts a triplet ground state, both in 3D and in 2D,
coming from the absence of electron-electron correlation effects in this
approach. For a 2D double well potential, failures are identified at relatively
smaller interdot distances, and may be attributed to the less confining nature
of the potential, leading to larger overlap and consequently an inadequate
representation of the two-particle states written, within HL, in terms of the
ground state orbital at each isolated well. We find that in the double dot
case, the range of validity of HL is improved (restricted) in a related 3D (1D)
model, and that results always tend to become more reliable as the interdot
distance increases. Our calculated exchange coupling is of relevance to the
exchange gate quantum computer architectures in semiconductors.
