  The search for room-temperature magnetocaloric materials for refrigeration
has led to investigations of double perovskites. In particular, a puzzle has
appeared in the La$_2$MnNiO$_6$, La$_2$MnCoO$_6$ and La$_2$MnFeO$_6$ family of
compounds. They share the same crystal structure, but while La$_2$MnNiO$_6$ and
La$_2$MnCoO$_6$ are ferromagnets below room temperature, La$_2$MnFeO$_6$,
contrary to simple expectations, is a ferrimagnet. To solve this puzzle, we use
density-functional theory calculations to investigate the electronic structure
and magnetic exchange interactions of the ordered double perovskites. Our study
reveals the critical role played by local electron-electron interaction in the
Fe-$d$ orbital to promote the Fe$^{3+}$ valence state with half-filled
$d$-shell over Fe$^{2+}$ and to establish a ferrimagnetic ground state for
La$_2$MnFeO$_6$. The importance of Hund's coupling and Jahn-Teller distortion
on the Mn$^{4+}$ ion is also pointed out. Exchange constants are extracted by
comparing different magnetically ordered states. Mean-field and classical
Monte-Carlo calculations on the resulting model give trends in $T_C$ that are
in agreement with experiments on this family of materials.
