$\text{Given}$
$\frac{1}{\alpha }-64\left(\pi -1\right)+\frac{1}{18\left(\mathrm{\pi -1}\right)+\frac{1}{64}}\approx 0$
$\text{Solving for α}$
$\alpha \approx \frac{1152\left(\pi -1\right)+1}{64\left[1152{\left(\pi -1\right)}^{2}+\left(\pi -1\right)-1\right]}$
$\text{Multiplying by 8}$
$\alpha \approx \frac{\left(2\circ 8\circ 18\circ 32\right)\left(\pi -1\right)+8}{{8}^{2}\left[\left(2\circ 8\circ 18\circ 32\right){\left(\pi -1\right)}^{2}+8\left(\pi -1\right)-8\right]}$

Uses the maximum number of electrons in the first four principal energy levels

α ≈ 0.00729735256808166