Screened Coulomb interaction calculations: cRPA implementation and applications to dynamical screening and self-consistency in uranium dioxide and cerium

We report an implementation of the constrained Random Phase Approximation (cRPA) method within the Projector Augmented-Wave framework. It allows for the calculation of the screened interaction in the same Wannier orbitals as our recent DFT+$U$ and DFT+DMFT implementations. We present calculations of the dynamical Coulomb screened interaction in uranium dioxide and $\alpha$ and $\gamma$ cerium on Wannier functions. We show that a self-consistent calculation of the static screened interaction in DFT+$U$ together with a consistent Wannier basis is mandatory for $\gamma$ cerium and uranium dioxide. We emphasize that a static approximation for the screened interaction in $\alpha$ cerium is too drastic.