  An intriguing idea of spin-orbit Mott insulator has been proposed to explain
magnetic insulating behavior in various iridates. This scenario relies on the
strength of the spin-orbit coupling being comparable to electronic
correlations, and it is not a priori obvious whether this picture is valid for
all iridates. In particular, Sr3Ir2O7 exhibits a small charge gap and magnetic
moment compared to Sr2IrO4, questioning the validity of such hypothesis. To
understand the microscopic mechanism for magnetism in Sr3Ir2O7, we construct a
tight binding model taking into account the full t2g- orbitals, the staggered
rotation of the local octahedra, and the bilayer structure. The bands near the
Fermi level are mainly characterized by the total angular momentum Jeff = 1/2,
except below the {\Gamma} point, supporting a reasonably strong spin-orbit
coupling picture. A first order transition to a collinear antiferromagnet via
multi-orbital Hubbard interactions is found within the mean field
approximation. The magnetic moment jump at the transition is consistently
smaller than Sr2IrO4, originated from the underlying band structure of a barely
band insulator. Given the small charge gap and moment observed in Sr3Ir2O7, the
system is close to a magnetic transition. A comparison to a spin-model is
presented and connection to the Mott insulator is also discussed.
