  The recent discovery of superconductivity in oxygen-reduced monovalent
nickelates has raised a new platform for the study of unconventional
superconductivity, with similarities and differences with the cuprate high
temperature superconductors. In this paper we investigate the family of
infinite-layer nickelates $R$NiO$_2$ with rare-earth $R$ spanning across the
lanthanide series, introducing a new and non-trivial "knob" with which to tune
nickelate superconductivity. When traversing from La to Lu, the out-of-plane
lattice constant decreases dramatically with an accompanying increase of Ni $
d_{x^2-y^2}$ bandwidth; however, surprisingly, the role of oxygen charge
transfer diminishes. In contrast, the magnetic exchange grows across the
lanthanides which may be favorable to superconductivity. Moreover, compensation
effects from the itinerant $5d$ electrons present a closer analogy to Kondo
lattices, indicating a stronger interplay between charge transfer, bandwidth
renormalization, compensation, and magnetic exchange. We also obtain the
microscopic Hamiltonian using Wannier downfolding technique, which will provide
the starting point for further many-body theoretical studies.
