  We show how an accurate first-principles treatment of the
canted-antiferromagnetic ground state of Sr$_2$IrO$_4$, a prototypical $5d$
correlated spin-orbit coupled material, can be obtained without invoking any
free parameters such as the Hubbard U or tuning the spin-orbit coupling
strength. Our theoretically predicted iridium magnetic moment of 0.250 $\mu_B$,
canted by 12.6$^{\circ}$ off the a-axis, is in accord with experimental
results. By resolving the magnetic moments into their spin and orbital
components, we show that our theoretically obtained variation of the magnetic
scattering amplitude $<M_{m}>$ as a function of the polarization angle is
consistent with recent non-resonant magnetic x-ray scattering measurements. The
computed value of the band gap (55 meV) is also in line with the corresponding
experimental values. A comparison of the band structure to that of the cuprates
suggests the presence of incommensurate charge-density wave phases in
Sr$_{2}$IrO$_{4}$.
