Data for "Between Mott and cluster Mott: spin-orbit entangled dimer singlets in Ba3CeRu2O9"

The hexagonal 4d ruthenates Ba3MRu2O9 host structural dimers and exhibit a delicate balance of competing interactions. Hund's coupling, trigonal crystal-field splitting, 
and hopping for a1g and eg\pi orbitals all fall within a narrow energy window. This yields a series of possible ground states, ranging from the localized Mott limit with (anti-)ferromagnetic exchange coupling via orbital-selective behavior to the cluster Mott limit with quasimolecular orbitals that are delocalized over the two dimer sites. 
Using resonant inelastic x-ray scattering, we show that Ba3CeRu2O9 with four holes per dimer resides in the intricate crossover regime between the localized Mott case 
and the quasimolecular limit. The spin-orbit entangled singlet ground state predominantly shows a Mott-like charge distribution with two holes per Ru site, but at the same time a dominant fraction of the holes occupies bonding orbitals. Furthermore, spin and orbital occupation contradict an exchange-based Mott scenario but agree with a cluster Mott approach. A quasimolecular trial wave function describes more than 70% of the ground state. In this crossover regime, small changes of, e.g., the crystal field 
may strongly affect the character of electronic states. In Ba3CeRu2O9, both the crystal field and hopping lower the a1g orbitals. For spin-orbit coupling zeta=0, Hund's coupling favors an S=0 ground state for small hopping but S=1 for realistic larger hopping. Finite zeta, even though small, finally yields a non-magnetic J=0 state.