Section 25.2.10.4.2. Least-squares refinement algebra

The original SHELX refinement algorithms were modelled closely on those described by Cruickshank (1970). For macromolecular refinement, an alternative to (blocked) full-matrix refinement is provided by the conjugate-gradient solution of the least-squares normal equations as described by Hendrickson & Konnert (1980), including preconditioning of the normal matrix that enables positional and displacement parameters to be refined in the same cycle. The structure-factor derivatives contribute only to the diagonal elements of the normal matrix, but all restraints contribute fully to both the diagonal and non-diagonal elements, although neither the Jacobian nor the normal matrix itself are ever generated by SHELXL. The parameter shifts are modified by comparison with those in the previous cycle to accelerate convergence whilst reducing oscillations. Thus, a larger shift is applied to a parameter when the current shift is similar to the previous shift, and a smaller shift is applied when the current and previous shifts have opposite signs.

SHELXL refines against F² rather than F, which enables all data to be used in the refinement with weights that include contributions from the experimental uncertainties, rather than having to reject F values below a preset threshold; there is a choice of appropriate weighting schemes. Provided that reasonable estimates of σ(F²) are available, this enables more experimental information to be employed in the refinement; it also allows refinement against data from twinned crystals.