  Using a combination of first-principles density functional theory (DFT)
calculations and exact diagonalization studies of a first-principles derived
model, we carry out a microscopic analysis of the magnetic properties of the
half-metallic double perovskite compound, Sr$_2$CrMoO$_6$, a sister compound of
the much discussed material Sr$_2$FeMoO$_6$. The electronic structure of
Sr$_2$CrMoO$_6$, though appears similar to Sr$_2$FeMoO$_6$ at first glance,
shows non trivial differences with that of Sr$_2$FeMoO$_6$ on closer
examination. In this context, our study highlights the importance of charge
transfer energy between the two transition metal sites. The change in charge
transfer energy due to shift of Cr $d$ states in Sr$_2$CrMoO$_6$ compared to Fe
$d$ in Sr$_2$FeMoO$_6$ suppresses the hybridization between Cr $t_{2g}$ and Mo
$t_{2g}$. This strongly weakens the hybridization-driven mechanism of magnetism
discussed for Sr$_2$FeMoO$_6$. Our study reveals that, nonetheless, the
magnetic transition temperature of Sr$_2$CrMoO$_6$ remains high since
additional superexchange contribution to magnetism arises with a finite
intrinsic moment developed at the Mo site. We further discuss the situation in
comparison to another related double perovskite compound, Sr$_2$CrWO$_6$. We
also examine the effect of correlation beyond DFT, using dynamical mean field
theory (DMFT).
