Characteristics of the measured first ionization potentials of lanthanides and actinides

The experimentally determined atomic ionization potentials $\mathrm{IP}_1$ of lanthanides and actinides are analyzed. A semiclassical method is used to determine the dependence of the orbital binding energies on the atomic number $Z$ and on quantum numbers. Features revealed for chemical elements with the regularly filled $4f$ and $5f$ shells allow approximating the dependences $\mathrm{IP}_1(Z)$ with an error less than $1\%$ using three fitting parameters for lanthanides and one for actinides, respectively. The revealed features for valence electrons justify a general atomic number scaling law for orbital binding energies, which is valid for both outer and inner electron shells with the regular filling.