  Despite similar chemical compositions, LiOsO$_3$ and NaOsO$_3$ exhibit
remarkably distinct structural, electronic, magnetic, and spectroscopic
properties. At low temperature, LiOsO$_3$ is a polar bad metal with a
rhombohedral $R3c$ structure without the presence of long-range magnetic order,
whereas NaOsO$_3$ is a $G$-type antiferromagnetic insulator with an
orthorhombic $Pnma$ structure. By means of comparative first-principles DFT+$U$
calculations with the inclusion of the spin-orbit coupling, we ($i$) identify
the origin of the different structural ($R3c$ vs. $Pnma$) properties using a
symmetry-adapted soft mode analysis, ($ii$) provide evidence that all
considered exchange-correlation functionals (LDA, PBE, PBEsol, SCAN, and HSE06)
and the spin disordered polymorphous descriptions are unsatisfactory to
accurately describe the electronic and magnetic properties of both systems
simultaneously, and ($iii$) clarify that the distinct electronic (metallic vs.
insulating) properties originates mainly from a cooperative steric and magnetic
effect. Finally, we find that although at ambient pressure LiOsO$_3$ with a
$Pnma$ symmetry and NaOsO$_3$ with a $R\bar{3}c$ symmetry are energetically
unfavorable, they do not show soft phonons and therefore are dynamically
stable. A pressure-induced structural phase transition from $R3c$ to $Pnma$ for
LiOsO$_3$ is predicted, whereas for NaOsO$_3$ no symmetry change is discerned
in the considered pressure range.
