a*nl(r, l')an(r, l) dr = ^£exp[-/(k • l - k • l')]

2.4.1. Effective Hamiltonian

We will now look for the solution of the nonstationary Schrodinger equation for one electron in the crystal, under an external potential; that is,

where H is the Hamiltonian of the electron in the periodic field of crystal and U is the perturbation potential associated with the external field.

Let us look for a solution of the form t) = EEfn(0 t)an(r, l)

n i where fn(l, t) is denominated envelope function of the nth band. Notice that the solution involves all the bands.

Substituting the Eq. (13) in (12), after multiplying by a*n-(r - l') and integrating, we get

Considering the normalization condition of the Wannier functions and making the sum in the right side of the equation we get

Remembering that for the nonperturbed electron

we apply the H operator to the Wannier functions [Eq. (6)] and taking into account Eq. (5) we get

0 0

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