Reciprocal Space Refinement Least Squares

It is wise to remember that the term refinement means exactly what it says it means the minimization of small errors. Refinement is most certainly not a stage of structure determination as is sometimes supposed. The initial model must be a good approximation to the truth. If it is distant from the actual structure, either overall or locally, refinement is unlikely to correct those errors and result in a correct structure. Thus it is important, in practice, not to build a sloppy or careless model, or a model based on a wing and a prayer, and then hope that refinement will find the right solution or fix the problems. 

We know, however, that in reality, for each xj at which yj is measured, the observed value for yj must contain some experimental error, ej. According to Lagrange, the best line will be that for which e2 has a minimum value. For any measurement ej = yj — axj — b, where axj - b represents the true value of yj predicted by the model. For all measurements yj, then 

We know from differential calculus (try to remember) that all functions assume a minimum when their first derivatives are zero hence J2j is minimized when the first derivatives of 

We will not carry this any further, except to say that when the experimental values, the observed points, are inserted into the equation and the derivatives are set to zero, two linear equations in two unknowns are obtained. These can easily be solved to yield the slope a and the intercept b of the best straight line. 

The same idea can equally well be applied to linear equations having more than two variables, that is, where the observed or predicted result q is described by a model of the form 

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Campbell, P.J.S. and S. Amott. 1978. LAIS A linked-atom least-squares reciprocal-space refinement system incorporating stereochemical restraints to supplement sparse diffraction. Acta Crystallographica. Section A 34 3-11. 

The conclusion on the equivalence of direct-space and reciprocal-space minimization is not completely flawless, because weights are assigned to the observations in the least-squares refinement, so a weighted difference density is minimized. 

The Fermi surface plays an important role in the theory of metals. It is defined by the reciprocal-space wavevectors of the electrons with largest kinetic energy, and is the highest occupied molecular orbital (HOMO) in molecular orbital theory. For a free electron gas, the Fermi surface is spherical, that is, the kinetic energy of the electrons is only dependent on the magnitude, not on the direction of the wavevector. In a free electron gas the electrons are completely delocalized and will not contribute to the intensity of the Bragg reflections. As a result, an accurate scale factor may not be obtainable from a least-squares refinement with neutral atom scattering factors. 

As part of a study of biscyclic dibenzacridines, the crystal structure of 1,2-8,9-dibenzacridine (72) has been investigated by Mason (1957, 1960). The space group was shown to be Pna2v thus, with four molecules in the unit cell, no molecular symmetry is required. The structure was determined from an examination of the weighted reciprocal lattice and trial and error methods, refinement being by two-dimensional least-squares procedures. At the conclusion of the analysis,... 

Subsequently, the baboon a-lactalbumin structure was refined at 1.7-A resolution by Acharya et al. (1989). Using the structure of domestic hen egg white lysozyme as the starting model, preliminary refinement was made using heavily constrained least-squares minimization in reciprocal space. Further refinement was made using stereochemical restraints at 1.7-A resolution to a conventional crystallographic residual of 0.22 for 1141 protein atoms. 

It is wise to remember that every operation, mathematical or physical, in real or reciprocal space has an equivalent operation in the other. Often these are not obvious, but they are always there. Molecular replacement, for example, can in theory be carried out in either space, though our computational tools for doing so are much more powerful in reciprocal space. Similarly, structure refinement may be carried out using least squares procedures in reciprocal space or, equivalently, difference Fourier methods in real space. 

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