  We have implemented a systematic LSDA and LSDA+U study of the double
perovskites $A_{2}$FeReO$_{6}$ (A=Ba,Sr,Ca) and Sr$_{2}$$M$MoO$_{6}$
(M=Cr,Mn,Fe,Co) for understanding of their intriguing electronic and magnetic
properties. The results suggest a ferrimagnetic (FiM) and half-metallic (HM)
state of $A_{2}$FeReO$_{6}$ (A=Ba,Sr) due to a pdd-$\pi$ coupling between the
down-spin Re$^{5+}$/Fe$^{3+}$ $t_{2g}$ orbitals via the intermediate O
$2p_{\pi}$ ones, also a very similar FiM and HM state of Sr$_{2}$FeMoO$_{6}$.
In contrast, a decreasing Fe $t_{2g}$ component at Fermi level ($E_{F}$) in the
distorted Ca$_{2}$FeReO$_{6}$ partly accounts for its nonmetallic behavior,
while a finite $pdd$-$\sigma$ coupling between the down-spin
Re$^{5+}$/Fe$^{3+}$ $e_{g}$ orbitals being present at $E_{F}$ serves to
stabilize its FiM state. For Sr$_{2}$CrMoO$_{6}$ compared with
Sr$_{2}$FeMoO$_{6}$, the coupling between the down-spin Mo$^{5+}$/Cr$^{3+}$
$t_{2g}$ orbitals decreases as a noticeable shift up of the Cr$^{3+}$ 3d
levels, which is likely responsible for the decreasing $T_{C}$ value and weak
conductivity. Moreover, the calculated level distributions indicate a
Mn$^{2+}$(Co$^{2+}$)/Mo$^{6+}$ ionic state in Sr$_{2}$MnMoO$_{6}$
(Sr$_{2}$CoMoO$_{6}$), in terms of which their antiferromagnetic insulating
ground state can be interpreted. While orbital population analyses show that
owing to strong intrinsic pd covalence effects, Sr$_{2}M$MoO$_{6}$
(M=Cr,Mn,Fe,Co) have nearly the same valence state combinations, as accounts
for the similar M-independent spectral features observed in them.
