# Numerical solution to SE - variational method, many electrons

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## Main Question or Discussion Point

Hi everyone,

I am trying to find electron wavefunction of a system I am working in. Numerical method I choose is the Variational method (VM). This method is convenient to find the ground state of the system. More details are available here.

Problem I have can be explained on a very simple system like an infinitely deep square quantum well, $U_0$ .

Let's say we have two electrons, and we are thinking how the wavefunction of the electron in the higher state looks like. We would also like to take into account electric potential these two electrons have on each other.

Now my question is, would the following procedure make sense:

1. we first find the ground state and its wavefunction with the VM of the infinite well with no electrons, $E_1$ and $\psi_1$ .
2. then we "fill" that state with an electron.
3. we now have a new "structure" which consists of a quantum well plus the electron. We construct a new potential $U_{new}=U_0-e|\psi_1|^2$.
4. then we find the ground state and its wavefunction, $E_2$ and $\psi_2$, which corresponds to the potential $U_{new}$.

What do you think, does $\psi_2$ corresponds to the real value of the wavefunction of the second electron?

I understand that in this procedure it is only taken into account influence of the first electron to the second, and there is no influence of the second electron to the first. This is a clear drawback of this procedure, but I cannot estimate if this would be a big problem or not.

Is there anyone who has experience with this kind of problems? I assume this is a routine problem in quantum chemistry.

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