Some Beq are more equivalent than others

Crystallographic structural models for macromolecules have typically included an isotropic displacement parameter B-iso for each atom. In cases where the structural model instead includes anisotropic displacement parameters U-ij, the derived quantity B-eq can be substituted for B-iso for many purposes. B-eq is not, however, the best predictor of the value B-iso that would hypothetically have been obtained by direct refinement of an isotropic model. A new entity B-est is proposed that represents an estimate for B-iso that minimizes the Kullback-Leibler divergence from a paired anisotropic model. In general B-est/B-eq < 1, with the difference between the two values becoming larger for atoms that are more anisotropic. Although this difference does not affect direct refinement of either isotropic or anisotropic models, it is relevant to any analysis that compares isotropic and anisotropic models of the same underlying structure. In particular, it may lead to improved selection of multi-group TLS models based on analysis of an initial isotropic refinement.