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An Investigation of the Empirical Relationships for the Ionization Potentials

The Bohr model of a hydrogen-like atom indicates that the energy required to remove an electron, called the ionization potential, should follow the form

IE = RZ²/n²

where R is the Rydberg constant (approximately 13.6 electron Volts (eV), Z is the net charge experienced by the electron and n is the principal quantum number, effectively the shell number. The value of Z is the number of protons in the nucleus #p less the shielding by the electrons in inner shells and also in the same shell. Thus the ionization potential would be

IE = (R/n²)(#p−ε)²
which can be put in the form
IE = (R/n²)((#p)² − 2(#p)ε + ε²)

A regression equation of the form

IE = c0 + c1(#p) + c2(#p)²

Such a form gives a very good fit to the data. The coefficient of determination goes as high as 0.9999998+. The value of ε is found as

ε = −½c1/c2

However it also should be that c0/c2 should be ε² and thus equal to the square of the value found from c1 and c2. The regression coefficients are not constrained to achieve that equality. Thus effectively the form assumed for the relationship for ionization potential is

IE = (R/n²)[(#p−ε)² + ζ]

where R is an empirical value, rather than necessarily being the Rydberg constant, and ζ is a constant. The values of R are howver notably close to the Rydberg constant. The values for some cases are given below are for the first electrons in several shells.


The value for the fourth shell is not close to the Rydberg constant.

The value of ε, as was indicated above, is found as −½c1/c2. The values for the first few shells are given below.


There can be only two electrons in the first shell, eight in the second and third shells and eighteen in the fourth shells. The standard model presumes that ε is zero for the first shell, two for the second shell, ten for the third shell and eighteen for the fourth shell. The results indicate that there is only partial shielding by the electrons in inner shells.

Now for the values of ζ. For the fourth shell the value of epsilon; is 17.57065433 and this squared is 308.7278935. On the other hand the ratio of c0 to c2 is 309.3893215, only 0.661427979 more than the value of ε². This is remarkably close. The value of ζ for case of the first electron in the fourth shell is thus 0.661427979. The values of ζ for the other shells and this case are given in the table below.


The regression coefficients can be constrained to make ζ equal to zero but that would significantly degrade the empirical fit.

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