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We review the electronic structure of nickelate superconductors with and without effects of electronic correlations. As a minimal model, we identify the one-band Hubbard model for the Ni 3dx2−y2 orbital plus a pocket around the A-momentum. The latter, however, merely acts as a decoupled electron reservoir. This reservoir makes a careful translation from nominal Sr-doping to the doping of the one-band Hubbard model mandatory. Our dynamical mean-field theory calculations, in part already supported by the experiment, indicate that the Γ pocket, Nd 4f orbitals, oxygen 2p, and the other Ni 3d orbitals are not relevant in the superconducting doping regime. The physics is completely different if topotactic hydrogen is present or the oxygen reduction is incomplete. Then, a two-band physics hosted by the Ni 3dx2−y2 and 3d3z2−r2orbitals emerges. Based on our minimal modeling, we calculated the superconducting Tc vs. Sr-doping x phase diagram prior to the experiment using the dynamical vertex approximation. For such a notoriously difficult to determine quantity as Tc, the agreement with the experiment is astonishingly good. The prediction that Tc is enhanced with pressure or compressive strain has been confirmed experimentally as well. This supports that the one-band Hubbard model plus an electron reservoir is the appropriate minimal model.

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Superconductivity has entered the nickel age marked by enormous experimental and theoretical efforts. Notwithstanding, synthesizing nickelate superconductors remains extremely challenging, not least due to incomplete oxygen reduction and topotactic hydrogen. Here, we present density-functional theory calculations for nickelate superconductors with additional topotactic hydrogen or oxygen, namely La1−xSrxNiO2Hδ and LaNiO2+δ. We identify a phonon mode as a possible indication for topotactic hydrogen and discuss the charge redistribution patterns around oxygen and hydrogen impurities.