  For 4$d^1$ and 5$d^1$ spin-orbit-coupled electron configurations, the notion
of nonmagnetic j=3/2 quartet ground state discussed in classical textbooks is
at odds with the observed variety of magnetic properties. Here we throw fresh
light on the electronic structure of 4$d^1$ and 5$d^1$ ions in molybdenum- and
osmium-based double-perovskite systems and reveal different kinds of on-site
many-body physics in the two families of compounds: while the sizable magnetic
moments and $g$ factors measured experimentally are due to both metal
$d$-ligand $p$ hybridization and dynamic Jahn-Teller interactions for 4$d$
electrons, it is essentially $d$-$p$ covalency for the 5$d^1$ configuration.
These results highlight the subtle interplay of spin-orbit interactions,
covalency and electron-lattice couplings as the major factor in deciding the
nature of the magnetic ground states of 4$d$ and 5$d$ quantum materials. Cation
charge imbalance in the double-perovskite structure is further shown to allow a
fine tuning of the gap between the $t_{2g}$ and $e_g$ levels, an effect of much
potential in the context of orbital engineering in oxide electronics.
