  The half-filled $t_{2g}$ shell of the $t_{2g}^3$ configuration usually, in LS
coupling, hosts a S = 3/2 ground state with quenched orbital moment. This state
is not Jahn-Teller active. Sufficiently large spin-orbit coupling $\zeta$ has
been predicted to change this picture by mixing in orbital moment, giving rise
to a sizable Jahn-Teller distortion. In $5d^3$ K$_2$ReCl$_6$ we study the
electronic excitations using resonant inelastic x-ray scattering (RIXS) and
optical spectroscopy. We observe on-site intra-$t_{2g}$ excitations below 2 eV
and corresponding overtones with two intra-$t_{2g}$ excitations on adjacent
sites, the Mott gap at 2.7 eV, $t_{2g}$-to-$e_g$ excitations above 3 eV, and
charge-transfer excitations at still higher energy. The intra-$t_{2g}$
excitation energies are a sensitive measure of $\zeta$ and Hund's coupling
$J_H$. The sizable value of $\zeta \approx$ 0.29 eV places K$_2$ReCl$_6$ into
the intermediate coupling regime, but $\zeta/J_H \approx 0.6$ is not
sufficiently large to drive a pronounced Jahn-Teller effect. We discuss the
ground state wavefunction in a Kanamori picture and find that the S = 3/2
multiplet still carries about 97 % of the weight. However, the finite admixture
of orbital moment allows for subtle effects. We discuss small
temperature-induced changes of the optical data and find evidence for a
lowering of the ground state by about 3 meV below the structural phase
transitions.
