Advantages of the new FFT method

• Exploits localization in real-space: no need to do the initial 3D FFT on subcells not overlapping a function in the FFT box.
• Can store the intermediate stage of the FFT in the same amount of memory as in real-space e.g. for calculating the density:
  
  $$n({\bf r}) = 2 \sum_{\alpha \beta} \phi_{\alpha}({\bf r}) K^{\alpha \beta} \phi_{\beta}({\bf r})$$
  
  
  where the functions $\{ \phi_{\alpha}({\bf r}) \}$ must be Fourier interpolated before their product is calculated.