Fourier series

 

Bloch's theorem:

$$\psi({\bf r}) = \exp( {\rm i} {\bf k} \cdot {\bf r}) u_{\bf k}({\bf r})$$

where $u_{\bf k}({\bf r})$ is cell-periodic i.e. $u_{\bf k}({\bf r} + {\bf R}) = u_{\bf k}({\bf r})$ for any lattice vector ${\bf R}$.

$\Rightarrow$

Expand $u_{\bf k}({\bf r})$ as a Fourier series:

$$u_{\bf k}({\bf r}) = \sum_{\bf G} c_{\bf k}({\bf G}) \exp( {\rm i} {\bf G} \cdot {\bf r})$$

where ${\bf G}$ denotes a reciprocal lattice vector.

 

Fourier inversion theorem gives:

$$c_{\bf k}({\bf G}) = {1 \over V_{\rm cell}} \int_{\rm cell} {... ...r} u_{\bf k}({\bf r}) \exp( -{\rm i} {\bf G} \cdot {\bf r})$$