Refinement of R₂ and R₁: a physical interpretation

Least-squares refinement of R₂ = Σ| I₁ - I₂|², where I₁ and I₂ are, for example, the observed and calculated intensities for a crystal structure, is equivalent to obtaining a least-squares fit between the Patterson syntheses for the observed intensities and the model structure. Least-squares refinement of R'₁ = Σ|F₁ - F₂|² is equivalent to obtaining a least-squares fit between the calculated electron density and that obtained from a Fourier series based on the observed structure factors; R'₁ approximates to the usual residual R₁ = Σ| |F₁| - |F₂| | ² when the model is good and refinement is approaching completion. Inclusion of weights in the residuals is equivalent to distorting the Patterson or electron-density synthesis to be fitted by modifying the amplitudes of the Fourier coefficients by factors proportional to the square roots of the weights.