  Ilmenites $AB$O$_3$ provide a platform for electron correlation and magnetism
on alternatively stacked honeycomb layers of edge-sharing $A$O$_6$ or $B$O$_6$
octahedra. When $A$ and $B$ are $3d$ transition metals, strong electron
correlation makes the systems Mott insulators showing various magnetic
properties, while when $B$ is Ir with $5d$ electrons, competition between
electron correlation and spin-orbit coupling realizes a spin-orbital coupled
Mott insulator as a potential candidate for quantum spin liquids. Here we
theoretically investigate intermediate $4d$ ilmenites, $A$RuO$_3$ with $A$=Mg
and Cd, which were recently synthesized and shown to be metallic, unlike the
$3d$ and $5d$ cases. By using first-principles calculations, we optimize the
lattice structures and obtain the electronic band structures. We show that
MgRuO$_3$ exhibits strong dimerization on RuO$_6$ honeycomb layers, leading to
the formation of bonding and anti-bonding bands for one of three $t_{2g}$
orbitals; the lattice symmetry is lowered from $R\bar{3}$ to $P\bar{1}$, and
the Fermi surfaces are composed of the other two $t_{2g}$ orbitals. In
contrast, we find that CdRuO$_3$ has a lattice structure close to $R\bar{3}$,
and all three $t_{2g}$ orbitals contribute almost equally to the Fermi
surfaces. Comparison of our results with other Ru honeycomb materials such as
Li$_2$RuO$_3$ indicates that the metallic ruthenium ilmenites stand on a subtle
balance among electron correlation, spin-orbit coupling, and electron-phonon
coupling.
