where Φ Φ ~ ~ ℓ ℓ r and ρ ρ ~ ~ ( k ) represent the Fourier transforms of the potential and the charge density (this is the Ewald part). Since both summations converge quickly in their respective spaces (real and Fourier), they may be truncated with little loss of accuracy and great improvement in required computational time. To evaluate the Fourier transform ρ ρ ~ ~ ( k ) of the charge density field efficiently, one uses the fast Fourier transform, which requires that the density field be evaluated on a discrete lattice in space (this is the mesh part).