We investigate multi-band Hubbard models for the three iron 3$d$-$t_{2g}$ bands and the two iron 3$d$-$e_g$ bands in ${rm La O Fe As}$ by means of the Gutzwiller variational theory. Our analysis of the paramagnetic ground state shows that neither Hartree--Fock mean-field theories nor effective spin models describe these systems adequately. In contrast to Hartree--Fock-type approaches, the Gutzwiller theory predicts that antiferromagnetic order requires substantial values of the local Hunds-rule exchange interaction. For the three-band model, the antiferromagnetic moment fits experimental data for a broad range of interaction parameters. However, for the more appropriate five-band model, the iron $e_g$ electrons polarize the $t_{2g}$ electrons and they substantially contribute to the ordered moment.
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