Simple relationships between the ionization energies of one-, two- and three-electron ions of consecutive elements

The ionization energy of one-electron ions can be calculated from a well-known equation that is based on quantum mechanics and on the Bohr model, but no theoretically justified equation is available for the calculation of the ionization energies of multi-electron ions. I report here simple empirical relationships between the ionization energies of one-, two- and three-electron ions of elements whose atomic numbers are Z, Z + 1 and Z + 2. On the basis of these relationships, an equation was constructed for the calculation of the ionization energies of two- and three-electron ions (IE2el(Z) and IE3el(Z), respectively) as a function of Z only: $${\text{IE}}_{{N{\text{el}}}} {\left( Z \right)} = \frac{{2{\left( {Z - N + 1} \right)}^{2} + 3{\left( {Z - N + 1} \right)}^{3} }}{{{\left( {N - 1} \right)}^{2} \times {\left[ {2^{2} \times 3{\left( {Z - N + 1} \right)} - 1} \right]}}} \times 2^{2} \times E_{0} {\left( H \right)}$$ where N = the number of electrons, i.e. 2 or 3. For N = 3, this equation is only valid when Z > N, being inaccurate for the neutral Li atom. Graphs of the difference between calculated and experimental values of the ionization energies as a function of Z reveal inaccurate experimental results that are impossible to detect by inspection of the ionization energy itself. On the basis of the present results, more accurate values can be predicted for these ionization energies. A striking example is the inaccuracy of the traditional handbook value of IE3el(Fe).