  Combining density functional theory (DFT) and embedded dynamical mean-field
theory (DMFT) methods, we study the metal-insulator transition in
$R_2$Ir$_2$O$_7$ ($R$=Y, Eu, Sm, Nd, Pr, and Bi) and the topological nature of
the insulating compounds. Accurate free energies evaluated using the charge
self-consistent DFT+DMFT method reveal that the metal-insulator transition
occurs for an A-cation radius between that of Nd and Pr, in agreement with
experiments. The all-in-all-out magnetic phase, which is stable in the Nd
compound but not the Pr one, gives rise to a small Ir$^{4+}$ magnetic moment of
$\approx 0.5\mu_B$ and opens a sizable correlated gap. We demonstrate that
within this state-of-the-art theoretical method, the insulating bulk pyrochlore
iridates are topologically trivial.
