Racah materials: role of atomic multiplets in intermediate valence systems

We address the long-standing mystery of the nonmagnetic insulating state of the intermediate valence compound SmB$_6$. Within a combination of the local density approximation (LDA) and an exact diagonalization (ED) of an effective discrete Anderson impurity model, the intermediate valence ground state with the $f$-shell occupation $\langle n_{4f} \rangle=5.6$ is found for the Sm atom in SmB$_6$. This ground state is a singlet, and the first excited triplet state $\sim 3$ meV higher in the energy. SmB$_6$ is a narrow band insulator already in LDA, with the direct band gap of $\sim 10$ meV. The electron correlations increase the band gap which now becomes indirect. Thus, the many-body effects are relevant to form the indirect band gap, crucial for the idea of ``topological Kondo insulator" in SmB$_6$. Also, an actinide analog PuB$_6$ is considered, and the intermediate valence singlet ground state is found for the Pu atom. We propose that [Sm,Pu]B$_6$ belong to a new class of the intermediate valence materials with the multi-orbital ``Kondo-like" singlet ground-state. Crucial role of complex spin-orbital $f^n$-$f^{n+1}$ multiplet structure differently hybridized with ligand states in such Racah materials is discussed.