In the nearly free electron approximation, interactions between electrons are completely ignored. This approximation allows use of Blochs Theorem which states that electrons in a periodic potential have wavefunctions and energies which are periodic in wavevector up to a constant phase shift between neighboring reciprocal lattice vectors. The consequences of periodicity are described mathematically by the Blochs theorem, which states that the eigenstate wavefunctions have the form
where the Bloch function  is periodic over the crystal lattice, that is,
Here index  refers to the -th energy band, wavevector  is related to the direction of motion of the electron,  is the position in the crystal, and  is the location of an atomic site.
The NFE model works particularly well in materials like metals where distances between neighbouring atoms are small. In such materials the overlap of atomic orbitals and potentials on neighbouring atoms is relatively large. In that case the wave function of the electron can be approximated by a (modified) plane wave. The band structure of a metal like aluminium even gets close to the empty lattice approximation.