  Infinite-layer nickelate superconductors have recently been discovered to
share both similarities and differences with cuprate superconductors. Notably,
the incorporation of hydrogen (H) through topotactic reduction has been found
to play a critical role in their electronic structure and, consequently, their
superconductivity. In this study, we utilized a theoretical approach combining
density-functional theory and impurity approximation to design three
characteristic multi-orbital Hubbard models representing low, moderate, and
high concentrations of topotactic-hydrogen. Consistent with experimental
findings, our simulations revealed that both low and high concentrations of
topotactic-hydrogen induce high-spin states ($S$=1) that are composed by holes
at $d_{x^2-y^2}$ and $d_{z^2}$ orbitals and consequently the emergent
inter-site hopping between $d_{z^2}$ to $d_{x^2-y^2}$ is unfavorable for
superconductivity. Conversely, an optimal concentration of 25\% H aligns with
the single Ni-$d_{x^2-y^2}$ band picture of superconductivity in infinite-layer
nickelates, demonstrating its beneficial effect on promoting superconducting
behavior.
