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,