On the method of least squares as applied to the refinement of crystal structures

Abstract

The effect of off-diagonal terms of the normal equations of least-squares on the refinement of parameters (both positional and thermal) is discussed critically. It is shown that the usual procedure of neglecting the off-diagonal terms during refinement of positional parameters in a three-dimensional structure, or in a projection without overlap of atoms, is valid only if the structure or the projection concerned is centrosymmetric and the axes are orthogonal. If the structure is non-centrosymmetric and contains atoms or groups of atoms which are centrosymmetric and which contribute predominantly to the structure factor, then a new type of overlap termed 'inverse overlap' has to be taken into account. The inverse overlap is particularly significant since it can occur in three dimensions. Finally it is shown that in the refinement of thermal parameters the linear approximation formula is not valid and all the off-diagonal terms involving the B_if's which occur for the particular symmetry have to be included in the refinement.