How to deal with the Coulomb interactions between particles is one of the more difficult questions in physics. Without their presence, N-particle problems can generally be turned into N one-particle problems. An exact treatment of the Coulomb interaction requires knowledge of the positions of the particles in space. This would not be a problem if the particles were not moving or the velocities of particles differed by orders of magnitudes. The last assumption is generally valid for the Coulomb interaction between electrons and the nuclei. Since in most systems the nuclei are more or less fixed (say, in a solid or a molecule), we can replace the effect of the nuclei on the electrons by an effective potential. For the interaction between the electrons, we can separate the electrons into two types. Core electrons are strongly bound to the nuclei and generally have a binding energy of tens to thousands of electronvolts. If the atomic shells of these electrons are full, we can include their effect in the effective potential of the nucleus. On the other hand, this approach often fails for the valence electrons, a term that we loosely use to describe electrons in states that have a relatively low binding energy, such as the highest-occupied and lowest-unoccupied molecular orbitals (HOMO and LUMO, respectively) in a molecule, the states close to the Fermi level in a metal, or the valence and conduction bands in a semiconductor. However, even for these electrons, treating their interaction in terms of an effective potential often works surprisingly well.

The most commonly-used approach is the local-density approximation used in density-functional theory. However, for many systems, this theory has serious deficiencies. In Chapter 5, we looked at Coulomb multiplets for an atom/ion, which are often clearly visible in X-ray spectroscopy. These effects cannot be described within an effective independent-particle framework. For solids, materials that are known to be insulating are often predicted to be metallic when the interactions between particles are described in terms of an effective potential.