FFT convention:

\[ f({\bf r}) = \sum_{{\bf G}} e^{i{\bf G}{\bf r}} f({\bf G}) \]

is a backward transformation from a set of pw coefficients to a function.

\[ f({\bf G}) = \frac{1}{\Omega} \int e^{-i{\bf G}{\bf r}} f({\bf r}) d {\bf r} = \frac{1}{N} \sum_{{\bf r}_j} e^{-i{\bf G}{\bf r}_j} f({\bf r}_j) \]

is a forward transformation from a function to a set of coefficients.

We use plane waves in two different cases: a) plane waves (or augmented plane waves in the case of APW+lo method) as a basis for expanding Kohn-Sham wave functions and b) plane waves are used to expand charge density and potential. When we are dealing with plane wave basis functions it is convenient to adopt the following normalization:

\[ \langle {\bf r} |{\bf G+k} \rangle = \frac{1}{\sqrt \Omega} e^{i{\bf (G+k)r}} \]

such that

\[ \langle {\bf G+k} |{\bf G'+k} \rangle_{\Omega} = \delta_{{\bf GG'}} \]

in the unit cell. However, for the expansion of periodic functions such as density or potential, the following convention is more appropriate:

\[ \rho({\bf r}) = \sum_{\bf G} e^{i{\bf Gr}} \rho({\bf G}) \]

where

\[ \rho({\bf G}) = \frac{1}{\Omega} \int_{\Omega} e^{-i{\bf Gr}} \rho({\bf r}) d{\bf r} = \frac{1}{\Omega} \sum_{{\bf r}_i} e^{-i{\bf Gr}_i} \rho({\bf r}_i) \frac{\Omega}{N} = \frac{1}{N} \sum_{{\bf r}_i} e^{-i{\bf Gr}_i} \rho({\bf r}_i) \]

i.e. with such convention the plane-wave expansion coefficients are obtained with a normalized FFT.