 Solution

Far from a neutral bound system, the slowest-decaying electron-removal amplitude is controlled by the first ionization energy $I=E_{N-1}-E_N>0$. Its asymptotic wavefunction satisfies a free-particle-like equation at negative energy $-I$, giving a decay exponent $\kappa =\sqrt{2m_{e}I}/\hslash$. Density is quadratic in the amplitude, soor $a=2\sqrt{2I}$ in atomic units. Power-law radial factors and angular dependence do not alter this leading exponential coefficient. It is the ionization energy, not the electron affinity, that controls this tail.