Formula for calculating the spin orbit coupling of atoms with $Z$ atomic number

Is it possible to use this formula $$Delta E=frac{E_n}{n}(Zalpha)^2left(frac{1}{j+1/2}-frac{3}{4n}right)$$ to determine the spin orbit coupling of an atom with atomic number $Z$ at energy level $n $? $alpha=1/137$ is the fine structure constant.

The formula comes from the fine structure article in wikipedia and takes into account the relativistic correction: https://en.wikipedia.org/wiki/Fine_structure#Total_effect ?

I would like to find the energy splitting between the levels $3p^{1/2}$ and $3p^{3/2}$ of Na which experimentally is found to be $0.0021$ eV. However when I used the formula $E_n = -E_0 Z^2/n^2$ with $E_0 = 13.6$ eV and $Z = 11$, $j=3/2$ and $n=3$ for Na $3p^{1/2}$ I don’t get the right order of magnitude: $Delta E_{SO} approx -2$ eV.

I am not even sure if the formula $E_n = -E_0*Z^2/n^2$ gives realistic values for the energy levels because they are far away from the binding energies measured with XPS or XAS. Example $E_1(text{Na}) = 1645$ eV but Na $1s = 1070.8$ eV and $E_2(text{Na}) = 411$ eV where Na $2s = 63$ eV.