The Hamiltonian for a many-electron atom has three parts — electronic kinetic energy, electron-nucleus attraction, and electron-electron repulsion — and the question is to identify the middle one. Three features single it out.

The sign must be negative. The nucleus is positive and the electrons negative, so the interaction is attractive, and attractive potential energy is negative by convention. This immediately eliminates the positive terms.

The distance must be \(r_i\), not \(r_{ij}\). The subscript matters: \(r_i\) is the distance of electron i from the nucleus, whereas \(r_{ij}\) is the distance between two electrons i and j. Any term containing \(r_{ij}\) describes electron-electron repulsion, not electron-nucleus attraction.

The charges must be \(Ze^{2}\). Coulomb's law multiplies the two charges: the nuclear charge \(+Ze\) and the electronic charge \(-e\), giving \(Ze^{2}\). A term carrying only \(Ze\) has the wrong dimensions for an energy.

The sum runs over all electrons, since every electron is attracted to the nucleus.

Putting these together gives \(-\sum_{i}\frac{Ze^{2}}{r_{i}}\).

It is worth noting that it is the other term, the \(\sum_{i \lt j}\frac{e^{2}}{r_{ij}}\) repulsion, that makes the many-electron Schrödinger equation unsolvable exactly, because it couples the coordinates of different electrons and prevents separation of variables. That is precisely why approximate methods such as Hartree-Fock, which replace it by an averaged field, are needed.

Hence the electron-nucleus term is proportional to \(-\sum_{i}\frac{Ze^{2}}{r_{i}}\).