Modeling the chemical shift of lanthanide4felectron binding energies

Lanthanides in compounds can adopt the tetravalent [Xe]$4{f}^{n\ensuremath{-}1}$ (like Ce${}^{4+}$, Pr${}^{4+}$, Tb${}^{4+}$), the trivalent [Xe]$4{f}^{n}$ (all lanthanides), or the divalent [Xe]$4f{\phantom{\rule{0.16em}{0ex}}}^{n+1}$ configuration (like Eu${}^{2+}$, Yb${}^{2+}$, Sm${}^{2+}$, Tm${}^{2+}$). The $4f$-electron binding energy depends on the charge $Q$ of the lanthanide ion and its chemical environment $A$. Experimental data on three environments (i.e., the bare lanthanide ions where $A=\text{vacuum}$, the pure lanthanide metals, and the lanthanides in aqueous solutions) are employed to determine the $4f$-electron binding energies in all divalent and trivalent lanthanides. The action of the chemical environment on the $4f$-electron binding energy will be represented by an effective ambient charge ${Q}_{A}=\ensuremath{-}Q$ at an effective distance from the lanthanide. This forms the basis of a model that relates the chemical shift of the $4f$-electron binding energy in the divalent lanthanide with that in the trivalent one. Eu will be used as the lanthanide of reference, and special attention is devoted to the $4f$-electron binding energy difference between Eu${}^{2+}$ and Eu${}^{3+}$. When that difference is known, the model provides the $4f$-electron binding energies of all divalent and all trivalent lanthanide ions relative to the vacuum energy.