We investigate the normal state of the superconducting compound PuCoGa$_5$ using the combination of density functional theory (DFT) and dynamical mean field theory (DMFT), with the continuous time quantum Monte Carlo (CTQMC) and the vertex-corrected one-crossing approximation (OCA) as the impurity solvers. Our DFT+DMFT(CTQMC) calculations suggest a strong tendency of Pu-5$f$ orbitals to differentiate at low temperatures. The renormalized 5$f_{5/2}$ states exhibit a Fermi-liquid behavior whereas one electron in the 5$f_{7/2}$ states is at the edge of a Mott localization. We find that the orbital differentiation is manifested as the removing of 5$f_{7/2}$ spectral weight from the Fermi level relative to DFT. We corroborate these conclusions with DFT+DMFT(OCA) calculations which demonstrate that 5$f_{5/2}$ electrons have a much larger Kondo scale than the 5$f_{7/2}$.
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