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I have read in a textbook (Modern Quantum Chemistry Szabo and Ostlund) that the Coulomb operator of the form \begin{equation} \mathcal{J}_{j}\left(\mathbf{x}_{1}\right)=\int d \mathbf{x}_{2}\left|\chi_{j}\left(\mathbf{x}_{2}\right)\right|^{2}r_{12}^{-1} \end{equation} describes the average local potential at $\mathbf{x}_{1}$ arising from the charge distribution from an electron in $\chi_{j}$. And therefore the term $\sum_{j \neq i}\left[\int d \mathbf{x}_{2}\left|\chi_{j}\left(\mathbf{x}_{2}\right)\right|^{2} r_{12}^{-1}\right] \chi_{i}\left(\mathbf{x}_{1}\right)$ for an $N$-electron system gives the total averaged potential of $N-1$ electrons in other spin orbitals on the electron in $\chi_{i}$.

However, if one expands the sum
\begin{equation} \sum_{j \neq i}\left[\int d \mathbf{x}_{2}\left|\chi_{j}\left(\mathbf{x}_{2}\right)\right|^{2} r_{12}^{-1}\right] \chi_{i}\left(\mathbf{x}_{1}\right) = \left(\int d \mathbf{x}_{2}\left|\chi_{1}\left(\mathbf{x}_{2}\right)\right|^{2}r_{12}^{-1} + \int d \mathbf{x}_{2}\left|\chi_{2}\left(\mathbf{x}_{2}\right)\right|^{2}r_{12}^{-1} +...\right) \chi_{i}\left(\mathbf{x}_{1}\right) \end{equation} it is indeed a sum over the spin orbital but not of the electrons in the system. My interpretation for the sum is as follows: the total averaged potential acting on electron 1 due to the charge distribution of electron 2 in $N-1$ spin orbitals. I think it has something to do with the fact that electrons are indistinguishable so one cannot assign an electron to an orbital. However, I think the Hartree Fock integro-differential equation \begin{equation} h\left(\mathbf{x}_{1}\right) \chi_{i}\left(\mathbf{x}_{1}\right)+\sum_{j \neq i}\left[\int d \mathbf{x}_{2}\left|\chi_{j}\left(\mathbf{x}_{2}\right)\right|^{2} r_{12}^{-1}\right] \chi_{i}\left(\mathbf{x}_{1}\right)-\sum_{j \neq i}\left[\int d \mathbf{x}_{2} \chi_{j}^{*}\left(\mathbf{x}_{2}\right) \chi_{i}\left(\mathbf{x}_{2}\right) r_{12}^{-1}\right] \chi_{j}\left(\mathbf{x}_{1}\right)=\epsilon_{i} \chi_{i}\left(\mathbf{x}_{1}\right) \end{equation} was derived for hydrogen molecule. Is it the same for an $N$-electron system?

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Short answer: the source of the confusion seems to be that you think the variable $x_2$ refers to the second electron, but that's not true.

My interpretation for the sum is as follows: the total averaged potential acting on electron 1 due to the charge distribution of electron 2 in 𝑁−1 spin orbitals.

No, the variable $x_2$ is just some dummy integration variable. To avoid confusion, it should have been called something like $y$ or $r$.

The logic of the approximation being made is as follows. We are trying to find the wavefunction for some particular orbital $i$. So instead of trying to solve the full multi-electron Schrodinger equation, we will just replace all the other orbitals with a classical distribution of charge.

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