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We discuss the well-known three-center cation-anion-cation model forsuperexchange in insulating transition-metal compounds using limitingexpansions for the Anderson-Hubbard model. We find that due to the threeinterfering energy scales in the model, a limiting expression for thesuperexchange $J$ for the idealized Mott-Hubbard (M-H) case $t\ll U\ll \Delta$cannot be formally defined. We further show that no single expansion variablecan describe any type of limiting behaviour for superexchange. The well-known$t^4$ expression for M-H insulators, obtained from path-dependent seriesexpansions, is not unique to these systems as it can also be obtained with manyother 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, $t\lesssimU\lesssim\Delta,$ the reverse form of the series expansions can yield betteragreement with the exact results. This implies that the fitting of experimentaldata to the simple expressions derived from path-dependent series expansionscan lead to qualitatively incorrect relationships between the parameters,fictitiously within the M-H regime.

Non-Convergent Perturbation Theory and Misleading Inferences about Parameter Relationships: the Case of Superexchange