First-principles method justifying the Dieke diagram and beyond

We present a method to determine the model Hamiltonians to treat rare-earth multiplets in solids from the results of the quasiparticle self-consistent $\mathit{\text{GW}}$ (QSGW) method. We apply the method to trivalent Eu compounds EuCl${}_3$, EuN, and Eu-doped GaN after examining free rare-earth ions. We solve the model Hamiltonian by the exact diagonalization. Our results justify applying the Dieke diagram to ions in solid, while its limitation is clarified. In particular, we show that the crystal fields cause sizable breaking of the Russell-Saunders coupling.