The independent electron approximation is used in both the free electron model and the nearly-free electron model. In this approximation we do not consider electron-electron interaction in a crystal. It is more difficult to treat electron-electron interactions than ion-electron interactions because:
- We are not aware of the wavefunctions of every electron.
- The potential due to electron-electron interactions is not periodic.
- We need to consider the dynamics of all of the electrons at once.
Fortunately, electron-electron interactions are often weaker than ion-electron interactions due to the following:
- Electrons with parallel spins stay away from each other due to the Pauli exclusion principle.
- Electrons with opposite spins stay away from each other in order to have the least energy for the system.
One major effect of electron-electron interactions is that electrons distribute around the ions so that they screen the ions in the lattice from other electrons.
Electron-electron interactions may be very important for certain properties in materials. For example, the theory covering much of superconductivity is BCS theory, in which the attraction of pairs of electrons to each other, termed "Cooper pairs", is the mechanism behind superconductivity.
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