Given
1 α - 64 π - 1 + 1 18 π-1 + 1 64 0
Solving for α
α 1152 π - 1 + 1 64 1152 π - 1 2 + π - 1 - 1
Multiplying by 8
α 2 8 18 32 π - 1 + 8 8 2 2 8 18 32 π - 1 2 + 8 π - 1 - 8

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

α ≈ 0.00729735256808166