In structure determination, the process of improving the parameters of an approximate (trial) structure until the best fit is achieved between an observed diffraction pattern and that calculated by Fourier transformation from the numerically parameterized trial structure.
The most common approach in the determination of inorganic or small-molecule structures is to minimize a function
where Yo represents the strength of an observed diffraction spot or reflection from a lattice plane of the crystal,Yc is the value calculated from the structural model for the same reflection, and w is an assigned weight reflecting the importance that this reflection makes to the sum. The weights usually represent an estimate of the precision of the measured quantity. The sum is taken over all measured reflections.
The structural model describes a collection of scattering centers (atoms), each located at a fixed position in the crystal lattice, and with some degree of mobility or extension around that locus. In adjusting the structural model to improve the fit between calculated and observed diffraction patterns, the crystallographer may vary these and other parameters. Refinable parameters are those that may be varied in order to improve the fit. Usually they comprise atomic coordinates, atomic displacement parameters, a scale factor to bring the observed and calculated amplitudes or intensities to the same scale. They may also include extinction parameters, occupancy factors, twin component fractions, and even the assigned space group. Relations between the refinable parameters may be expressed as constraints or restraints that modify the function to be minimized.
A constraint is an exact mathematical relationship that reduces the number of free parameters in a model. For example, the position of an atom on a general position is specified by three coordinates, all of which may be varied independently. However, an atom sitting on a special symmetry position has one or more positional coordinates determined by the symmetry (for example, an atom on an inversion centre in the unit cell has all three coordinates fixed). Constraints are rigid mathematical rules which must be adhered to during the refinement; they reduce the number of refinable parameters. A constrained refinement is one that includes constraints other than those arising from space group symmetry (since these are necessarily always present).
A restraint is an additional condition that the model parameters must meet to satisfy some additional piece of knowledge appropriate to the structure. For example, if the chemical identities of certain atoms within a molecule are known, their intermolecular distance may be fit to a target value characteristic of bond lengths in other known chemical species of the same type. Restraints are therefore treated as if they were additional experimental observations, and have the effect of increasing the number of refinable parameters.
Refinement against F, F2 or I?
The function to minimize in least-squares refinement was given above in the general form
and the quantity Y was referred to as a measure of the strength of a reflection. In practice, Y, sometimes known as the structure-factor coefficient, may be either I, the intensity of the measured reflection, | F | , the magnitude of the structure factor, or F2, the square of the structure factor.
Refinement against I, the measured intensities, has the merit of using the raw measurements directly, although it requires the incorporation in the refinement of the correction factors (scale factor, Lorentz–polarization and absorption) that are applied during standard data reduction. There are, however, problems of high statistical correlation when refining absorption parameters against anisotropic displacement parameters.
Refinement against | F | involves mathematical problems with very weak reflections or reflections with negative measured intensities. There are also difficulties in estimating standard uncertainties σ(F) from the σ(F2) values for weak or zero measured intensities.
Refinement against F2 avoids these difficulties, and also reduces the probability of the refinement iterations settling into a local minimum. It also simplifies the treatment of twinned and non-centrosymmetric structures. For these reasons, it is probably currently the most frequently used technique, although it does rely heavily on the assignment of reasonable weights to individual reflections.
The principle of maximum likelihood formalizes the idea that the quality of a model is judged by its consistency with the observations. If a model is consistent with an observation, then -- if the model were correct -- there would be a high probability of making an observation with that value. For a set of relevant observations, the probability of generating such a set is an excellent measure of the quality of the model. For independent observations, the joint probability of making the set of observations is the product of the probabilities of making each independent observation.
In crystallography, let P( | Fo | ; | Fc | ) represent the probability of obtaining an observed structure factor Fo given a calculated value Fc. The joint probability is the likelihood function L:
|L =||∏||P( | Fo | ; | Fc | )|
Since it is more convenient to work with sums than products, one typically works with the negative logarithm of the likelihood function
The mathematical procedure for determining maximum likelihood then becomes that of minimizing .
Least squares. E. Prince and P. T. Boggs. International Tables for Crystallography (2006). Vol. C, ch. 8.1, pp. 678-688 doi:10.1107/97809553602060000609
Other refinement methods. E. Prince and D. M. Collins. International Tables for Crystallography (2006). Vol. C, ch. 8.2, pp. 689-692 doi:10.1107/97809553602060000610