# Interpretation of Coulomb operator in Hartree-Fock equation

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

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.
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.