What is the potential field of an ion near the Bohr radius?

[Ganesh Ujwal]

I figure that at large enough distances, the potential field of an ion is just the Coulomb potential for its net charge. But what happens at scales comparable to the ion's Bohr radius? Could there be, for example, some sort of screening effect from the electron shell that changes the potential? (depending on what the test charge is, like if you dropped a single electron near an ion)

I'm a bit rusty on quantum mechanics, but I do remember that the math for atoms that aren't hydrogen gets complicated. Is there a known good way to approximate this potential? Or is my best bet to go download some quantum chemistry software?