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The on-site Coulomb repulsion $U$ is the key ingredient for describing themagnetic properties of Mott insulators, leading to a popular believe that manylimitations of the density-functional theory based methods can be cured byartificially incorporating such on-site interactions for localized electrons inthe model form. The layered antiferromagnet NaMnCl$_3$ reveals quite adifferent story: while the Coulomb $U$ on the Mn sites controls the strength ofantiferromagnetic superexchange interactions, an equally important parameter isthe Stoner coupling ${\cal I}_{\rm Cl}$ on the ligand sites. The latter isresponsible for large ferromagnetic contributions to the interatomic exchangeinteractions, which in NaMnCl$_3$ nearly cancel the effect of the superexchangeinteractions. Although such behavior is anticipated from the phenomenologicalGoodenough-Kamanori-Anderson rules, the quantitative description of theligand-related contributions remains disputable. Considering NaMnCl$_3$ as anexample, we discuss how they can be generally taken into account in the linearresponse theory to regain the dependence of the exchange interactions on ${\calI}_{\rm Cl}$. The problem is complicated by the fact that, for the nearlyfilled Cl $3p$ shell, the parameters ${\cal I}_{\rm Cl}$ are sensitive to themodel assumptions.

The origin of ferromagnetic interactions in NaMnCl$_3$: how the response theory reconciles with Goodenough-Kanamori-Anderson rules