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

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 $t\ll U\ll Δ$ 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 $Δ$ or the $d$-electron correlation $U$ can actually be small. At times, and particularly for milder relationships between the parameters, $t\lesssim U\lesssimΔ,$ 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.

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