Calculate the energy of the electron in a non-H like atom

[jorgeha]

Hello mates. I was doing some research about Rydberg atoms, and I came up with this question: what's the energy of an electron in n energy level in an atom which is NOT hydrogen-like, that is, an atom with more than 1 electron? How can we calculate it?
What if the electron we are studying is in a much higher energy level (Rydberg energy level) and the others are in the lowest posible? What if we have an excited electron apart of the one we are studying?

Thank you in advance.

 

[Vanadium 50]

How can we calculate it?

Numerically or approximately. It does not have an exact solution. Note that you have 3 or more bodies here, and that doesn't even have a classical solution.

 

[jorgeha]

Numerically or approximately. It does not have an exact solution. Note that you have 3 or more bodies here, and that doesn't even have a classical solution.

I was asking for a numerical calculation. I guessed it could only be an approximate answer but I didn't know to what extent. I'd like the most exact approach possible, if you could lead me to some articles or books about these calculations I would be grateful. Thanks.

 

[DrDu]

http://www.springer.com/br/book/9783642065033#otherversion=9783540256441

 

[hilbert2]

If one of the electrons has a position distribution where it's at a much longer average distance from the nucleus than the others, I think it will effectively see the nucleus as a point charge of +1e because of the screening by the other electrons. Another way to obtain the same effect is to make a "helium atom" where one of the orbiting particles is an electron and the other a muon (the muon will have a much smaller "orbit radius" because of its large mass compared to the electron).

 

[DrDu]

If one of the electrons has a position distribution where it's at a much longer average distance from the nucleus than the others, I think it will effectively see the nucleus as a point charge of +1e because of the screening by the other electrons. Another way to obtain the same effect is to make a "helium atom" where one of the orbiting particles is an electron and the other a muon (the muon will have a much smaller "orbit radius" because of its large mass compared to the electron).

Yes, the core is taken into account then via some "quantum defect": https://en.wikipedia.org/wiki/Quantum_defect

 

[hilbert2]

I remember some perturbation calculations in a quantum chemistry homework where we had to estimate the effect of the finite size of the nucleus by assuming that the nucleus is a small sphere that contains a constant positive charge density. Then the potential inside the nucleus was calculated with Gauss's law. A core that contains both positive and negative charge is probably not very different.