The rigid-bond criterion (Hirshfeld, 1976
) plays a key role in validating, understanding and restraining atomic displacement parameters (ADPs) in both small-molecule and macromolecular crystal structures. Its validity was established by Rosenfield et al.
), Dunitz, Maverick & Trueblood (1988
), Dunitz, Schomaker & Trueblood (1988
), Bürgi (1989
) and many others. It plays an important part in the validation of crystal structures using PLATON
) and the IUCr CheckCIF suite. Rollett (1970
) was probably the first to apply rigid-bond restraints
in the least-squares refinement of crystal structures by means of additional observational equations. This rigid-bond restraint [DELU in SHELXL
) or RBON in REFMAC
(Murshudov et al.
)] is useful both for the treatment of positional disorder in small-molecule structures and for enabling anisotropic refinement of macromolecules at relatively good resolution.
The rigid-bond criterion states that the mean-square displacement amplitudes of bonded atoms are equal in the direction of the bond joining them, and it is often applied to 1,3-distances (involving two atoms that are both bonded to a common atom) as well. When applied as a restraint, the standard deviation is usually set to a value in the range 0.01 to 0.001 Å2
, a value that is consistent with the deviations observed in structures in which an anisotropic refinement is possible without restraints. However, it only provides about one restraint per atom, which is not enough to compensate for the six degrees of freedom per atom associated with the anisotropic displacement parameters U
. In practice it needs to be supplemented by other ADP restraints. By also applying it to 1,3-bonded atoms, the number of restraints can be increased to an average of about two per atom. It is usually necessary to apply further restraints, for example that the atom is approximately isotropic (ISOR in SHELXL
) and that the U
values are equal to the corresponding U
values of spatially close atoms (the similarity restraint SIMU). Since these restraints are much less justified by theory and experimental evidence, they are given large estimated standard deviations, but at least they add about six restraints (SIMU) or five (ISOR) per atom. The similarity restraints enable a stable refinement of severely overlapping disorder components, but in practice they are rather approximate descriptions of the real atomic motion. Either these restraints are made too tight, and the R
factors are high, or (typically for macromolecular refinements) the restraints are too slack and the gap between the R
values of the working set and the test set (the free R
; Brünger, 1992
) becomes large, indicating over-refinement.
In this paper a simple extension of the rigid-bond concept will be discussed.