In compounds containing lanthanoid ions, the 4f orbitals are sufficiently localized that interactions with orbitals on neighboring atoms can be neglected, hence W ≈ 0. In this limit, the Hubbard U can be experimentally determined from photoelectron spectra.

(a) Use ionization energies for gas-phase ions to estimate U for Pr³⁺ and Gd³⁺ (the third and fourth ionization energies of Pr are 21.6 eV and 39.0 eV, respectively, those for Gd are 20.6 eV and 44.0 eV).

(b) Measured values of U for these two ions in solids are estimated as 6 eV for Pr³⁺ and 12 eV for Gd³⁺. Why are the estimated values from part (a) higher than the measured values?

(c) Why is U so much higher for Gd³⁺ than it is for Pr³⁺?

(d) By analogy, would you expect a higher U for TiO or MnO?