  We calculate the screened electron-electron interaction for the
charge-disproportionated insulator CaFeO$_3$ using the constrained random-phase
approximation (cRPA). While in many correlated materials, the formation of a
Mott-insulating state is driven by a large local Coulomb repulsion, represented
by the Hubbard $U$, several cases have been identified more recently where $U$
is strongly screened and instead the Hund's interaction $J$ dominates the
physics. Our results confirm a strong screening of the local Coulomb repulsion
$U$ in CaFeO$_3$ whereas $J$ is much less screened and can thus stabilize a
charge-disproportionated insulating state. This is consistent with the case of
the rare-earth nickelates where similar behavior has been demonstrated. In
addition, we validate some common assumptions used for parametrizing the local
electron-electron interaction in first-principles calculations based on
density-functional theory (DFT), assess the dependence of the interaction on
the choice of correlated orbitals, and discuss the use of the calculated
interaction parameters in DFT+$U$ calculations of CaFeO$_3$. Our work also
highlights certain limitations for the direct use of cRPA results in DFT-based
first-principles calculations, in particular for systems with strong
entanglement between the correlated and uncorrelated bands.
