  We have theoretically designed three families of the half-metallic (HM)
antiferromagnets (AFM), namely, La$A$VOsO$_6$, La$A$MoTcO$_6$ and
La$A$MoReO$_6$ ($A$ = Ca, Sr, Ba), based on a systematic {\it ab initio} study
of the ordered double perovskites La$ABB'$O$_6$ with the possible $B$ and $B'$
pairs from all the 3$d$, 4$d$ and 5$d$ transtion metal elements being
considered. Electronic structure calculations based on first-principles
density-functional theory with generalized gradient approximation (GGA) for
more than sixty double perovskites LaCa$BB'$O$_6$ have been performed using the
all-electron full-potential linearized augmented-plane-wave method. The found
HM-AFM state in these materials survives the full {\it ab initio} lattice
constant and atomic position optimizations which were carried out using
frozen-core full potential projector augmented wave method. It is found that
the HM-AFM properties predicted previously in some of the double perovskites
would disappear after the full structural optimizations. The AFM is attributed
to both the superexchange mechanism and the generalized double exchange
mechanism via the $B$ ($t_{2g}$) - O (2$p_{\pi}$) - $B'$ ($t_{2g}$) coupling
and the latter is also believed to be the origin of the HM. Finally, in our
search for the HM-AFMs, we find La$A$CrTcO$_6$ and La$A$CrReO$_6$ to be AFM
insulators of an unconventional type in the sense that the two
antiferromagnetic coupled ions consist of two different elements and that the
two spin-resolved densities of states are no longer the same.
