  In this study, we investigated theoretically the Mott-insulating phase of a
deficient spinel chalcogenide GaV$_4$S$_8$, which is known to form a
tetrahedral V$_4$S$_4$ cluster unit that results in molecular orbitals (MOs)
with a narrow bandwidth in the noninteracting limit. We used a cluster
extension of charge self-consistent embedded dynamical mean-field theory to
study the impact of strong intra-cluster correlations on the spectral
properties as well as the structural degrees of freedom of the system. We found
that the strong tetrahedral clustering renders the atomic Mott picture
ineffective, and that the resulting MO picture is essential to describe the
Mott phase. It was also found that, while the spectral properties can be
qualitatively described by the truncation of the Hilbert space down to the
lowest-energy MO, a proper description of the structural degrees of freedom
requires the inclusion of multi-MO correlations that span a larger energy
window. Specifically, we found that the lowest-energy MO description
overemphasizes the clustering tendency, while the inclusion of the Hund's
coupling between the lower- and higher-energy MOs corrects this tendency,
bringing the theoretically predicted crystal structure into good agreement with
the experiment.
