The independent electron approximation is used in both the free electron model

Free electron model

In solid-state physics, the free electron model is a simple model for the behaviour of valence electrons in a crystal structure of a metallic solid. It was developed principally by Arnold Sommerfeld who combined the classical Drude model with quantum mechanical Fermi-Dirac statistics and hence it...

 and the nearly-free electron model

Nearly-free electron model

In solid-state physics, the nearly-free electron model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual Empty Lattice Approximation...

. 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:

1. We are not aware of the wavefunction
   Wavefunction
   Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
   
   s of every electron.
2. The potential due to electron-electron interactions is not periodic.
3. 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:

1. Electrons with parallel spins stay away from each other due to the Pauli exclusion principle
   Pauli exclusion principle
   The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
   
   .
2. 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

Superconductivity

Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...

 is BCS theory

BCS theory

BCS theory — proposed by Bardeen, Cooper, and Schrieffer in 1957 — is the first microscopic theory of superconductivity since its discovery in 1911. The theory describes superconductivity as a microscopic effect caused by a "condensation" of pairs of electrons into a boson-like state...

, in which the attraction of pairs of electrons to each other, termed "Cooper pair

Cooper pair

In condensed matter physics, a Cooper pair or BCS pair is two electrons that are bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper...

s", is the mechanism behind superconductivity.