  We analyze here a model for single-electron charging in semiconductor quantum
dots that includes the standard Anderson on-site repulsion (U) as well as the
spin-exchange ($J_d$) that is inherently present among the electrons occupying
the various quantum levels of the dot. We show explicitly that for
ferromagnetic coupling ($J_d>0$), an s-d exchange for an S=1 Kondo problem is
recovered. In contrast, for the antiferromagnetic case, $J_d<0$, we find that
the Kondo effect is present only if there are an odd number of electrons on the
dot. In addition, we find that spin-exchange produces a second period in the
conductance that is consistent with experimental measurements.
