What is residual Coulomb interaction and why does it exist?

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Sonko
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Towards the end of my atomic and quantum course my lecturer makes a note of residual Coulomb interaction and does a few graphs to show the concequences of it are (such as some states not being allowed do to symmetry) and that they favour larger values of angular momentum, but doesn't really explain very well what it acually is and where it comes from. I was wondering if somebody could acually explain to me then what residual Coulomb interaction is and why it exists?
my first thoughts are that it may be due to the fluctuations of charge around the nucleus of an atom as the Hartree-Fock approximation ignores this and so that theory doesn't match what we see but that's the limit to my understanding.

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I'm not sure what your teacher means either, since "residual coulomb interaction" could mean any neglected part of a Coulomb interaction in any context.

Without knowing the context, I can't say. If he's talking about Hartree-Fock, he could mean the neglected dynamical effects you mention, but that's usually termed the "correlation energy". If he's talking about many-body perturbation theory, then it's something else. Or he could be talking about the Breit interaction, which is the relativistic effect caused by the coulomb potential acting at a finite speed.