We can intuitively understand that electrons will repel one another in their spin paired configuration. In fact, the idea of electron screening is crucial since we will need to find a way to decompose the N-body multi-eletron problem into a collection of one electron problems. This can be done by picturing each electron experiencing the potential energy due to all of the other electrons as they screen the nucleus. Moreover, screening is important for parameterization of atoms when we use atomic orbitals as basis sets. We need accurate calculation of the screening in order to accurately calculate the atomic radius. We can take the first step towards understanding these issues by calculating the screening in helium. Helium has only two electrons, but this means that there is an electron-electron repulsion term. Therefore, the energy of the He atom cannot be solved exactly. We will provide the details of an approximate procedure using the variational principle.
The variational principle gives us a route to determining the screening experienced by an electron in any atom. In the linear combination of atomic orbitals (LCAO) approach we need accurate values for the atomic orbitals in order to obtain reasonable agreement with electronic structure and energetics. We need a basis set that includes an accurate estimate of atomic size effects. This means that we must do exactly the type of calculations we have shown here for the atom He. We must account for the screening of the nuclear for each type of electron in the atom. This is a great amount of effort and the scientists who work on the parameterization of basis sets spend years optimizing the screening parameters using the variational principle. Of course, the application of the variational principle to electronic screening will also depend on the type of radial function used, i.e. on whether we use Gaussians or exponentials.
