The nearly free particle model

In the tight binding model we start from electronic states localized on individual atoms and explore the consequence of coupling between these atomic centers. Here our starting point is the free electron, and the periodic lattice potential enters as a small perturbation. Thus, writing 

How can we use this to simplify our problem in the present context Consider one of the eigenfunctions of the unperturbed Hamiltonian Hq 

We have found that the perturbation U couples each such eigenfunction to other zero-order wavefunctions according to (cf. Eq. (4.81)) 

Inserting (4.104) into the Schrodinger equation, = Eilr find that the coefficients Cr are the solutions of 

Here Qpc is the volume of the primitive cell. Note that we can take Uq = 0 without loss of generality. This just means that we have taken the average lattice potential to be zero, that is, = 0. Equation (4.105) represents a set of coupled 

Here is the volume of the primitive cell. Note that we can take Uo = 0 without 

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