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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?

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Yes the maths is a bit complicated. When I was doing this stuff about thirty five (!!) years ago we used a Hartree Fock/self consistent field calculation. You wouldn't want to program this up by hand, but since software was widely available 35 years ago I'm sure it still is.

As I recall the changes are surprisingly subtle, to the point where to the eye there is not much change. However the electron density shrinks in a bit i.e. the ion is significantly smaller that the unionised atom. That's because the missing electron, whatever atomic orbital it came from, has a finite probability of being near the nucleus so it partially shields all the other electrons. When the electron is removed all the other electrons feel a slightly greater effective positive charge.

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You're right, but the effect you're interested in is not manifest as a change in the Coulomb potential. Basically you define an interaction potential which is a function of time and space during the reaction and that changes accordingly.

The Coulomb potential is a purely radial potential defined over all distances from the centre of the atomic system, including within the Bohr radius. If two atoms collide with each other with enough kinetic energy such that they are within a Bohr radius of each other and their electron clouds overlap, the interaction becomes extremely messy and is further complicated by the fact that classical physics is no longer appropriate to describe the system. I believe there are numerical simulations of atomic collisions that take into account this sort of thing, but it's mainly a statistical/energy-related problem. Collisions between atoms are currently treated in a kind of "add two ingredients to a box, shake them up and open the box to see the final result".

You might need to be careful not to confuse this with "electron screening", which is the change in the interaction between particles caused by an overarching electron cloud in a plasma. This screening is manifest usually as a change to the equation of state of plasmas or by using "screening factors" to nuclear reaction rates. It is a macroscopic phenomenon.

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