  Using ab initio wave-function-based calculations, we provide valuable
insights with regard to the magnetic exchange in 5$d$ and 4$d$ oxides with
face-sharing ligand octahedra, BaIrO$_3$ and BaRhO$_3$. Surprisingly strong
antiferromagnetic Heisenberg interactions as large as 400 meV are computed for
idealized iridate structures with 90$^{\circ}$ Ir-O-Ir bond angles and in the
range of 125 meV for angles of 80$^{\circ}$ as measured experimentally in
BaIrO$_3$. These estimates exceed the values derived so far for corner-sharing
and edge-sharing systems and motivate more detailed experimental investigations
of quantum magnets with extended 5$d$/4$d$ orbitals and networks of
face-sharing ligand cages. The strong electron-lattice couplings evidenced by
our calculations suggest rich phase diagrams as function of strain and
pressure, a research direction with much potential for materials of this type.
