Electronic, magnetic and optical properties of Sr$_{n+1}$Ir$_n$O$_{3n+1}$ ($n$=1, 2, and $\infty$)*

We have studied the crossover between metallic/insulating and non-magnetic/magnetic phases in Ruddlesden-Popper series of iridates Sr$_{n+1}$Ir$_n$O$_{3n+1}$ ($n$=1, 2, and $\infty$) by means of density functional theory including an on-site Hubbard $U$ correction and many-body first principles methods. By systematically investigating the evolution of the orbital and spin properties as a function of $U$, spin-orbit coupling (SOC) strength, and $n$ we have constructed detailed phase diagrams of the metal-insulator transition (MIT) which provide clear evidence for the crucial role played by SOC and $U$ in establishing a relativistic Mott-Hubbard insulating state in the $n$=1 and 2 compounds. Optical spectra computed within a model Bethe-Salpeter scheme show the typical double peak structure observed in experiments and capture well the progressive shrinking of the band-gap and the widening of the bandwidth going from $n$=1 to $n$=$\infty$. Finally, we clarify the origin of the canted magnetic ground state of Sr$_2$IrO$_4$ as due to the the synergistic effect of structural distortions (rotation and tetragonal distortion of IrO$_6$ octahedral) and to the competition between exchange and Dzyaloshinskii-Moriya interactions.