  Quasimolecular orbitals in cluster Mott insulators provide a route to tailor
exchange interactions, which may yield novel quantum phases of matter. We
demonstrate the cluster Mott character of the lacunar spinel GaTa$_4$Se$_8$
using resonant inelastic x-ray scattering (RIXS) at the Ta $L_3$ edge.
Electrons are fully delocalized over Ta$_4$ tetrahedra, forming quasimolecular
$J_{\rm tet}$=3/2 moments. The modulation of the RIXS intensity as function of
the transferred momentum q allows us to determine the cluster wavefunction,
which depends on competing intracluster hopping terms that mix states with
different character. This mixed wavefunction is decisive for the macroscopic
properties since it affects intercluster hopping and exchange interactions and
furthermore renormalizes the effective spin-orbit coupling constant. The
versatile wavefunction, tunable via intracluster hopping, opens a new
perspective on the large family of lacunar spinels and cluster Mott insulators
in general.
