Since this is a convolution, the Fourier transformation of ρ ρ TOT ( r ) is a product ρ ρ ~ ~ TOT ( k ) = L ~ ~ ( k ) ρ ρ ~ ~ u c ( k ) where the Fourier transform of the lattice function is another sum over delta functions L ~ ~ ( k ) = ( 2 π π ) 3 Ω Ω ∑ ∑ m 1 , m 2 , m 3 δ δ ( k − − m 1 b 1 − − m 2 b 2 − − m 3 b 3 ) where the reciprocal space vectors are defined b 1 = d e f 2 π π a 2 × × a 3 Ω Ω (and cyclic permutations) where Ω Ω = d e f a 1 ⋅ ⋅ ( a 2 × × a 3 ) is the volume of the central unit cell (if it is geometrically a parallelepiped, which is often but not necessarily the case). Note that both L ( r ) and L ~ ~ ( k ) are real, even functions.