Expansion of scalar validation criteria to three dimensions: the R tensor

Assessment of quality in crystal structure determination entails analysis of global statistics. In data reduction, quality is assessed using R_merge and mean Iσ(I). Progress in structure solution and refinement is checked by the goodness of fit, variants of the R index, R_cryst, and its cross-validation counterpart, R_free. These statistics are useful and provide a convenient means of comparison but their scalar nature renders them unable to capture the essence of three-dimensional entities such as diffraction patterns and molecular models. A simple general method to quantify spatial variations in scalar statistics has been developed. In it, a symmetric matrix, the R tensor, is used to represent the local average residual as a function of diffraction geometry. An effective value of the statistic in question can then be found for any direction in reciprocal space. Differences between these effective R indices for individual reflections or groups of reflections can help to steer refinement strategy and assess the final structure.