  Distortions of the oxygen octahedra influence the fundamental electronic
structure of perovskite oxides, such as their bandwidth and exchange
interactions. Utilizing a fully ab-initio methodology based on density
functional theory plus dynamical mean field theory (DFT+DMFT), we study the
crystal and magnetic structure of SrMoO$_3$. Comparing our results with DFT+$U$
performed on the same footing, we find that DFT+$U$ overestimates the
propensity for magnetic ordering, as well as the octahedral rotations, leading
to a different ground state structure. This demonstrates that structural
distortions can be highly sensitive to electronic correlation effects, and to
the considered magnetic state, even in a moderately correlated metal such as
SrMoO$_3$. Moreover, by comparing different downfolding schemes, we demonstrate
the robustness of the DFT+DMFT method for obtaining structural properties,
highlighting its versatility for applications to a broad range of materials.
