On the equivalence between a commonly used correlation coefficient and a least-squares function

Many objective functions have been proposed in X-ray crystallography to solve the molecular replacement (MR) problem and other optimization problems. This paper establishes the equivalence of optimizing two of these target functions, a commonly used correlation coefficient and a least-squares function. This equivalence may exist only in the neighborhoods about the global optima or the entire MR variable space depending on whether the mean values of the observed and calculated data are subtracted from the data. In addition, an argument is presented that the correlation coefficient between structure-factor magnitudes is likely to perform better than the correlation coefficient between intensities, especially when low-resolution data are used. This prediction was tested during coarse grid searches at low resolution using the MR program SOMoRe.