Absence of mixed valency for Pr in pristine and hole-doped PrNiO$_2$

Infinite-layer nickelates ($R$NiO$_2$) exhibit some distinct differences as compared to cuprate superconductors, leading to a debate concerning the role of rare-earth ions ($R$=La, Pr, Nd) in the low-energy many-body physics. Although rare-earth $4f$ orbitals are typically treated as inert `core' electrons in studies, this approximation has been questioned. An active participation of $4f$ states is most likely for PrNiO$_2$ based on an analogy to cuprates where Pr cuprates differ significantly from other cuprates. Here, we adopt density functional plus dynamical mean field theory (DFT+DMFT) to investigate the role of Pr $4f$ orbitals and more generally the correlated electronic structure of PrNiO$_2$ and its hole-doped variant. We find that the Pr $4f$ states are insulating and show no evidence for either a Kondo resonance or Zhang-Rice singlet formation as they do not have any hybridization channels near the Fermi energy. The biggest effects of hole doping are to shift the Pr $5d$ and $4f$ states further away from the Fermi energy while enhancing the Ni $3d$ - O $2p$ hybridization, thus reducing correlation effects as the O $2p$ states get closer to the Fermi energy. We again find no evidence for either Kondo or Zhang-Rice physics for the $4f$ states upon hole doping. We conclude by commenting on implications for other reduced valence nickelates.