A solution of this form can be plugged into the Schrödinger equation, resulting in the central equation:
where is the total energy, and the kinetic energy is characterized by
which, after dividing by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \psi_{\mathbf{k}}(\mathbf{r})}, reduces to
if we assume that is almost constant and
The reciprocal parameters and are the Fourier coefficients of the wave function and the screened potential energy , respectively:
The vectors are the reciprocal lattice vectors, and the discrete values of are determined by the boundary conditions of the lattice under consideration.
Before doing the perturbation analysis, let us first consider the base case to which the perturbation is applied. Here, the base case is , and therefore all the Fourier coefficients of the potential are also zero. In this case the central equation reduces to the form
This identity means that for each , one of the two following cases must hold:
,
If is a non-degenerate energy level, then the second case occurs for only one value of , while for the remaining , the Fourier expansion coefficient is zero. In this case, the standard free electron gas result is retrieved:
If is a degenerate energy level, there will be a set of lattice vectors with . Then there will be independent plane wave solutions of which any linear combination is also a solution:
Now let be nonzero and small. Non-degenerate and degenerate perturbation theory, respectively, can be applied in these two cases to solve for the Fourier coefficients of the wavefunction (correct to first order in ) and the energy eigenvalue (correct to second order in ). An important result of this derivation is that there is no first-order shift in the energy in the case of no degeneracy, while there is in the case of degeneracy (and near-degeneracy), implying that the latter case is more important in this analysis. Particularly, at the Brillouin zone boundary (or, equivalently, at any point on a Bragg plane), one finds a twofold energy degeneracy that results in a shift in energy given by:[clarification needed]
.
This energy gap between Brillouin zones is known as the band gap, with a magnitude of .
In this model, the assumption is made that the interaction between the conduction electrons and the ion cores can be modeled through the use of a "weak" perturbing potential. This may seem like a severe approximation, for the Coulomb attraction between these two particles of opposite charge can be quite significant at short distances. It can be partially justified, however, by noting two important properties of the quantum mechanical system:
The force between the ions and the electrons is greatest at very small distances. However, the conduction electrons are not "allowed" to get this close to the ion cores due to the Pauli exclusion principle: the orbitals closest to the ion core are already occupied by the core electrons. Therefore, the conduction electrons never get close enough to the ion cores to feel their full force.
Furthermore, the core electrons shield the ion charge magnitude "seen" by the conduction electrons. The result is an effective nuclear charge experienced by the conduction electrons which is significantly reduced from the actual nuclear charge.