Allow me to explain how easy it is to come up with an equation that is far simpler than Mills's equation (10.48) and is slightly more accurate (assuming the values given in Table 10.1 really do come from equation (10.48), as you have repeatedly assured us they do).
First, let's note that equation (10.48) specifies the ionization energy as a function of Z. if you trace back through the equations giving the value of E(Electric), you'll find that Z not only appears linearly in equation (10.43) but also appears in equation (10.42), the equation for r
3, which appears in equation (10.43). You will notice that equation (10.43) is not a simple polynomial in Z; although the value of r
3 is an algebraic function of Z, that function is far more complex than a simple polynomial function of Z.
The following cubic function of Z is therefore a great deal simpler than the function given by equation (10.48):
f(Z) = 0.005035826300931423 Z3 + 3.2268883952560006 Z2 - 8.67775469504161 Z - 1.1090053331352037
Not only is that formula simpler than the formula given by Mills, it is slightly more accurate (as judged by the RMS relative error) for the ionization energies given in Table 10.1, excluding only the first two rows. (Mills used a different function for the first row, and got the value given in that row wrong, so I think it's fair for me to come up with a slightly different function for the first two rows.)
I can assure you that the method I used to come up with the cubic formula given above can be applied to all of the tables in Mills's Big Book of Boo-Boos. (Mills uses different formulas for each table, so it's only fair to let me come up with different formulas for each table.)
Please note that I am not claiming my formula is based on quantum mechanics, or that it has any scientific value. The only thing I'm trying to demonstrate here is that it's actually quite easy to improve upon the simplicity and accuracy of Mills's formulas.
That fact suggests you should bring other criteria to bear when you start to compare Mills's Big Book of Boo-Boos against quantum mechanics.