  Na$_2$IrO$_3$, a honeycomb 5$d^5$ oxide, has been recently identified as a
potential realization of the Kitaev spin lattice. The basic feature of this
spin model is that for each of the three metal-metal links emerging out of a
metal site, the Kitaev interaction connects only spin components perpendicular
to the plaquette defined by the magnetic ions and two bridging ligands. The
fact that reciprocally orthogonal spin components are coupled along the three
different links leads to strong frustration effects and nontrivial physics.
While the experiments indicate zigzag antiferromagnetic order in Na$_2$IrO$_3$,
the signs and relative strengths of the Kitaev and Heisenberg interactions are
still under debate. Herein we report results of ab initio many-body electronic
structure calculations and establish that the nearest-neighbor exchange is
strongly anisotropic with a dominant ferromagnetic Kitaev part, whereas the
Heisenberg contribution is significantly weaker and antiferromagnetic. The
calculations further reveal a strong sensitivity to tiny structural details
such as the bond angles. In addition to the large spin-orbit interactions, this
strong dependence on distortions of the Ir$_2$O$_2$ plaquettes singles out the
honeycomb 5$d^5$ oxides as a new playground for the realization of
unconventional magnetic ground states and excitations in extended systems.
