  We investigate the electronic and magnetic properties of the newly
synthesized double perovskites Y$_{2}$NiIrO$_{6}$ and La$_{2}$NiIrO$_{6}$,
using density functional calculations, crystal field theory, superexchange
pictures, and Monte Carlo simulations. We find that both systems are
antiferromagnetic (AFM) Mott insulators, with the high-spin Ni$^{2+}$
$t_{2g}$$^{6}e_{g}$$^{2}$ ($S=1$) and the low-spin Ir$^{4+}$ $t_{2g}$$^{5}$
($S=1/2$) configurations. We address that their lattice distortion induces
$t_{2g}$-$e_{g}$ orbital mixing and thus enables the normal Ni$^{+}$-Ir$^{5+}$
charge excitation with the electron hopping from the Ir `$t_{2g}$' to Ni
`$e_g$' orbitals, which promotes the AFM Ni$^{2+}$-Ir$^{4+}$ coupling.
Therefore, the increasing $t_{2g}$-$e_{g}$ mixing accounts for the enhanced
$T_{\rm N}$ from the less distorted La$_{2}$NiIrO$_{6}$ to the more distorted
Y$_{2}$NiIrO$_{6}$. Moreover, our test calculations find that in the otherwise
ideally cubic Y$_{2}$NiIrO$_{6}$, the Ni$^{+}$-Ir$^{5+}$ charge excitation is
forbidden, and only the abnormal Ni$^{3+}$-Ir$^{3+}$ excitation gives a weakly
ferromagnetic (FM) behavior. Furthermore, we find that owing to the crystal
field splitting, Hund exchange, and broad band formation in the highly
coordinated fcc sublattice, Ir$^{4+}$ ions are not in the $j_{\rm eff}=1/2$
state but in the $S=1/2$ state carrying a finite orbital moment by spin-orbit
coupling (SOC). This work clarifies the varying magnetism in Y$_{2}$NiIrO$_{6}$
and La$_{2}$NiIrO$_{6}$ associated with the lattice distortions.
