Section 25.2.10.4.4. Refinement of anisotropic displacement parameters

The motion of macromolecules is clearly anisotropic, but the data-to-parameter ratio rarely permits the refinement of the six independent anisotropic displacement parameters (ADPs) per atom; even for small molecules and data to atomic resolution, the anisotropic refinement of disordered regions requires the use of restraints. SHELXL employs three types of ADP restraint (Sheldrick 1993; Sheldrick & Schneider, 1997). The rigid bond restraint, first suggested by Rollett (1970), assumes that the components of the ADPs of two atoms connected via one (or two) chemical bonds are equal within a specified standard deviation. This has been shown to hold accurately (Hirshfeld, 1976; Trueblood & Dunitz, 1983) for precise structures of small molecules, so it can be applied as a `hard' restraint with small estimated standard deviation. The similar ADP restraint assumes that atoms that are spatially close (but not necessarily bonded, because they may be different components of a disordered group) have similar U^ij components. An approximately isotropic restraint is useful for isolated solvent molecules. These two restraints are only approximate and so should be applied with low weights, i.e. high estimated standard deviations.

The transition from isotropic to anisotropic roughly doubles the number of parameters and almost always results in an appreciable reduction in the R factor. However, this represents an improvement in the model only when it is accompanied by a significant reduction in the free R factor (Brünger, 1992b). Since the free R factor is itself subject to uncertainty because of the small sample used, a drop of at least 1% is needed to justify anisotropic refinement. There should also be a reduction in the goodness of fit, and the resulting displacement ellipsoids should make chemical sense and not be `non-positive-definite'!