  In this work we examine critically the electronic structure of the rare-earth
elements by use of the so-called Hubbard I approximation. From the theoretical
side all measured features of both occupied and unoccupied states are
reproduced, without significant deviations between observations and theory. We
also examine cohesive properties like the equilibrium volume and bulk modulus,
where we find, in general, a good agreement between theory and measurements. In
addition we have reproduced the spin and orbital moments of these elements, as
they are reflected from measurements of the saturation moment. We have also
employed the Hubbard I approximation to extract the interatomic exchange
parameters of an effective spin Hamiltonian for the heavy rare earths. We show
that the Hubbard I approximation gives results which are consistent with
calculations where $4f$ electrons are treated as core states for Gd. The latter
approach was also used to address the series of the heavy/late rare-earths. Via
Monte Carlo simulations we obtained ordering temperatures which reproduce
measurements within about $20\%$. We have further illustrated the accuracy of
these exchange parameters by comparing measured and calculated magnetic
configurations for the heavy rare earths and the magnon dispersion for Gd. The
Hubbard I approximation is compared to other theories of the electronic
structure, and we argue that it is superior. We discuss the relevance of our
results in general, and how this makes it possible to treat the electronic
structure of materials containing rare-earth elements, such as permanent
magnets, magnetostrictive compounds, photovoltaics, optical fibers, topological
insulators, and molecular magnets.
