Single-Electron Energies, Many-Electron Effects, and the Renormalized-Atom Scheme as Applied to Rare-Earth Metals

A systematic investigation of certain electronic properties of the rare-earth metals is reported. Calculations are performed within the framework of the renormalized-atom method in which Hartree-Fock free-atom solutions, with electronic configurations appropriate to the metal, are initially computed; the wave functions are then renormalized to the Wigner-Seitz sphere and used to construct l-dependent Hartree-Fock-Wigner-Seitz crystal potentials. The following results are obtained: (i) Recent spectral information together with the free-atom solutions permits us to estimate the change in neutral-atom correlation energy associated with changing the $4f$ electron count; contrary to expectation, we find that correlation effects are more significant in a configuration with one fewer $4f$ and one more $5d$ electron. (ii) Band extrema and Fermi levels are placed. (iii) The positions of occupied and unoccupied $4f$ levels are estimated in both a one-electron approach and a multielectron method taking screening and relaxation effects into account in a definite way. The one-electron approximation for the $4f$ levels fails badly in reproducing the results of recent photoemission experiments, while the multielectron calculations are in surprisingly good accord with experiment. The effective Coulomb-interaction energy between two $4f$ electrons at the same site, the familiar U, is reduced from the single-particle value of approximately 27 eV to about 7 eV with the inclusion of multielectron effects. (iv) Hartree-Fock values for the $4s$- and $5s$-shell exchange splittings are compared with soft-x-ray photoemission studies of the rare-earth fluorides and oxides; the calculated $4s$ splittings are roughly twice as large as experiment while, unexpectedly, the $5s$ results are in almost precise agreement.