We have calculated the chemical trend of magnetic exchange parameters ($J_{dd}$, $N alpha$, and $N beta$) of Zn-based II-VI semiconductors ZnA (A=O, S, Se, and Te) doped with Co or Mn. We show that a proper treatment of electron correlations by the LSDA+$U$ method leads to good agreement between experimental and theoretical values of the nearest-neighbor exchange coupling $J_{dd}$ between localized 3$d$ spins in contrast to the LSDA method. The exchange couplings between localized spins and doped electrons in the conduction band $N alpha$ are in good agreement with experiment as well. But the values for $N beta$ (coupling to doped holes in the valence band) indicate a cross-over from weak coupling (for A=Te and Se) to strong coupling (for A=O) and a localized hole state in ZnO:Mn. That hole localization explains the apparent discrepancy between photoemission and magneto-optical data for ZnO:Mn.
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