  We develop anisotropic pseudo-spin antiferromagnetic Heisenberg models for
monoclinically distorted double perovskites. We focus on these A$_2$BB'O$_6$
materials that have magnetic moments on the 4d or 5d transition metal B' ions,
which form a face-centered cubic lattice. In these models, we consider local
z-axis distortion of B'-O octahedra, affecting relative occupancy of $t_{2g}$
orbitals, along with geometric effects of the monoclinic distortion, and
spin-orbit coupling. The resulting pseudo-spin-1/2 models are solved in the
saddle-point limit of the Sp(N) generalization of the Heisenberg model. The
spin S in the SU(2) case generalizes as a parameter $\kappa$ controlling
quantum fluctuation in the Sp(N) case. We consider two different models that
may be appropriate for these systems. In particular, using Heisenberg exchange
parameters for La$_2$LiMoO$_6$ from a spin-dimer calculation, we conclude that
this pseudo-spin-1/2 system may order, but must be very close to a disordered
spin liquid state.
