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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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From the labels ($i,j$), I am guessing that $\chi$ is a molecular orbital (MO). When formulating the equations for molecular orbitals, one assumes that there are as many (spin-)orbitals as there are electrons. In any event, when writing down the Coulomb operator in the MO representation for the ground state, only occupied orbitals appear.

Therefore, I believe that the equations you are showing here are for the general problem, not just $\ce{H2}$. In some sense, the equations are the same for either, it is just a question of how many summands there are. Finally, note that the last equation you showed is valid for any system, but that it is actually part of a set of such equations, one for each MO.

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