Least-squares minimization and the residual density

The relation between the least-squares minimization and the residual density follows from the Fourier convolution theorem (Arfken 1970). It states that the Fourier transform of a convolution is the product of the Fourier transforms of the individual functions F(f g) = F(f)F(g). If G(y) is the Fourier transform of 9(x)- 

Since Ap is the Fourier transform of AF, Eq. (5.12) implies that minimization of J (Fobs - Pcaic )2 dr and of J (Fobs - Fcalc)2 dS are equivalent. Thus, the structure factor least-squares method also minimizes the features in the residual density. Since the least-squares method minimizes the sum of the squares of the discrepancies in reciprocal space, it also minimizes the features in the difference density. The flatness of residual maps, which in the past was erroneously interpreted as the insensitivity of X-ray scattering to bonding effects, is an intrinsic result of the least-squares technique. If an inadequate model is used, the resulting parameters will be biased such as to produce a flat Ap(r). 

It is of importance that expression (5.12) holds even when /(x) is known only in part of space, as is the case in a crystallography experiment at finite resolution determined by Hmax. Using the Fourier convolution theorem, we may write (Dunitz and Seiler 1973) 

On the left is the Fourier transform of the even function (AF)2 on the right 

This result is equivalent to Eq. (5.12), except that the left-hand side is no longer an integral over all space, but a summation up to the limit of resolution. 

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