We write the Hamiltonian of an atom with N electrons, under the approximation of an immobile punctual nucleus, with $m_N/m_e\rightarrow \infty$. \begin{equation} \hat{H}=\frac{-\hbar^2}{2m}\sum_{i=1}^{N}\nabla_{i}^2+\sum_{i=1}^ {N}\frac{Ze^2}{r_i}+\sum_{i=1}^{N}\sum_{j>i}^{N}\frac{e^2}{r_{ij}} \end{equation}
The first term contains the kinetic energy for the N electrons.
The second term is the potential energy of electron-nucleus interaction.
The third addend takes into account the interelectronic interactions (repulsions between electrons). The constraint i>j prevents adding the same interaction twice, $\frac{e^2}{r_{12}}=\frac{e^2}{r_{21}}$.
The terms $\frac{e^2}{r_{ij}}$ make the Schrödinger equation not separable and approximate methods have to be used to solve it. In the absence of the third term (bielectronic terms) the Schrödinger equation is separable into N independent equations, one for each electron.
Solving each of these equations gives us the energy of the electron and its wave function. It is the so-called orbital approximation, which neglects the interaction between electrons, transforming the polyelectronic atom into an atom formed by N hydrogenoid electrons.
The total energy is the sum of the energies of each electron and the wave function is the product of hydrogen wave functions.