There is more to $d$-electrons than Hubbard $U$ and Hund's rule $J$

Multi-band Hubbard models including all $d$-bands are central for the description of many interesting correlated materials, e.g., the Iron based High-T$_c$ materials. In this work we compare two prevailing spin and angular momentum rotationally invariant models for the local $dd$-interaction, the generalized Kanamori interaction, and the Slater-Condon atomic Coulomb interaction, and establish how the first can be mapped to a very special case of the former. Using the recently developed multi-band Gutzwiller approximation solver, we show that the partial localization of orbital moments in the intermediately correlated regime of the paramagnetic state, is poorly described by the Kanamori model containing only Hubbard and Hund's rule interactions. In fact, for some integer fillings it differs qualitatively compared to the Slater-Condon interaction.