  We discuss the well-known three-center cation-anion-cation model for
superexchange in insulating transition-metal compounds using limiting
expansions for the Anderson-Hubbard model. We find that due to the three
interfering energy scales in the model, a limiting expression for the
superexchange $J$ for the idealized Mott-Hubbard (M-H) case $t\ll U\ll \Delta$
cannot be formally defined. We further show that no single expansion variable
can describe any type of limiting behaviour for superexchange. The well-known
$t^4$ expression for M-H insulators, obtained from path-dependent series
expansions, is not unique to these systems as it can also be obtained with many
other different expansions, in which either the $d-p$ energy difference
$\Delta$ or the $d$-electron correlation $U$ can actually be small. At times,
and particularly for milder relationships between the parameters, $t\lesssim
U\lesssim\Delta,$ the reverse form of the series expansions can yield better
agreement with the exact results. This implies that the fitting of experimental
data to the simple expressions derived from path-dependent series expansions
can lead to qualitatively incorrect relationships between the parameters,
fictitiously within the M-H regime.
