We obtain analytic expressions for the zero temperature and finite temperature exchange interaction between two hard sphere fermions bound in a symmetric double-well potential in 1, 2, and 3 dimensions and discuss the effect of dimensionality on the model. The three-dimensional model (due to Zilm/Landesmann) and the two-dimensional model presented here are of relevance to the strongly temperature dependent exchange interaction that has been observed between hydrides bound in metal polyhydride complexes.