While restraints are used for the same purpose as constraints, to fit models that are more complex than what can be managed with the limited set of observations allowed by our powder diffraction measurement(s). However, constraints and restraints work in completely different ways. Constraints remove degrees of freedom from the model. If our model structure has an asymmetric unit with five atoms, where none are on special positions and with no vacancies, then we have 20 structural parameters (\(5\times 3\) coordinates plus 5 \(\rm U_{iso}\) values). If we create a constraint that groups 4 of the atoms to have the same \(\rm U_{iso}\) value, then we replace 4 parameters with 1 and now have only 17 structural parameters, which is a significantly less complex model. The model will also have other non-structural parameters, but they will not directly change reflection intensities, so they are not as directly coupled to the number of observed reflections.
Restraints function where you provide a target value for something that can be determined from the model, and there is a penalty function that makes the fit worse the further the model refines from that target. Along with the target value for the restraint, you will also provide both an uncertainty and a weight for the restraint values. As was noted in Chapter 7, the goal of our optimization is to minimize the \(\chi ^2\) function, \[\chi ^2 = \sum _j^N w_j(y_{obs,j}-y_{calc,j})^2\] where \(y_{obs,j}\) and \(y_{calc,j}\) are the observed and computed powder diffraction intensities, respectively, and \(w_j\) are the weights for each intensity, where optimally \(w_j = 1/\sigma _j^2\). When we add restraints to this, we add an additional set of terms for the restraints, which can be expressed as: \[\chi ^2 = \sum _j^N w_j(y_{obs,j}-y_{calc,j})^2 + w_{R} \sum _k^M \left ( \frac {R_k - R_{calc,k}}{\sigma _{R,k}}\right )^2\] where there are \(M\) restraints, \(R_k\) are the restraint values, \(R_{calc,k}\) is the value we compute for \(R_{k}\) from the structural model, \(\sigma _{R,k}\) is the uncertainty on the observation and \(w_{R}\) is the weight on the restraints. It is worth noting that in practice, other than weighting, the \(R_k\) values are handled no differently than diffraction observations – as far as the math goes. However, in fact restraints are not something we measure. They are values that we believe are true about our material. I will discuss in §25.9, below, thoughts on how to test if what we think we know ain’t so.
Why does one use restraints? More complex structural models can have the problem that the fit is just not that sensitive to some of aspects of the structure, so that if the coordinates or occupancies are refined freely, the resulting structural model is implausible in some fashion. A good example for this would be the fitting of zeolites to powder data.
As a bit of background on zeolites: They are constructed from tetrahedral networks of silicon and aluminum atoms linked by oxygen atoms. Each aluminum will be accompanied by a charge-balancing cation. For some materials, the net scattering from the cations may exceed the net scattering from the framework (Si, Al and O) atoms. Zeolites often have some intrinsic disorder that limits the range of observable reflections, so the range of observed reflections is often limited. It is common that if one tries to fit the coordinates for a zeolite, against even high quality synchrotron or neutron diffraction data, the coordinates will refine to give Si-O and Al-O bond distances that are well outside the expected values and/or O-Si-O/O-Al-O bond angles that are far from tetrahedral. We know well from both plenty of crystal structures, as well as high-level ab initio theory that it takes a lot of energy to force Si-O and Al-O bond distances from their preferred values (Si-O 1.62 Å, Al-O 1.74 Å) by more than a few hundredths of an Å. This is not seen in these materials. Bending the bonding angles takes less energy, but I would expect them to be within 5-10\(^\circ \) of the tetrahedral angle of 109.5\(^\circ \) unless the framework is very strained.
When we fit a zeolite structure to powder diffraction data, we want not only a structural model that fits the data, we also that our model will have reasonable bond distances and angles. In one of my early zeolitic material structures, back when I was an industrial scientist, I measured both high resolution synchrotron and neutron powder diffraction data on Ca LSX (LSX is a synthetic form of the zeolite faujasite, where Al and Si atoms strictly alternate; this is one of the most simple of all zeolite structures). For reasons that I do not remember, the structure was initially fitted to each set of data separately, before a combined refinement was performed with both together. There were serious inaccuracies with each model when the data sets were handled separately. With the x-ray dataset alone, the O atoms did not refine to positions at reasonable distances from the Si and Al atoms. With the neutron dataset there were problems with the Ca atoms. The problem with use of the x-ray data alone was not that the fit produced unreasonable Si-O and Al-O distances and angles, it was that there were a continuous range of structural models that all produced about the same quality of fit to the diffraction data, but many unreasonable bonding geometries. Had only the x-ray data had been available, we would want a model that not only fits the data well, but also has a plausible geometry for the zeolite framework atoms. Use of restraints on the Si-O and Al-O distances, and possibly the O-Si-O and O-Al-O bond angles would have provided such a model. Without the restraints, the details of the framework geometry would be dictated by the starting point for the refinement and perhaps minor statistical scatter in the data. For what it is worth, the combined fit to both sets of data was excellent and no restraints were needed.
In the GSAS-II GUI, access to restraints is found as a child data tree entry, named for each phase, under Restraints. However, if a project has only one phase, clicking on the top level Restraints data tree entry will also open the restraints for that phase. GSAS-II provides seven types of restraints. Note that GSAS-II allows you to provide a separate \(w_R\) (weight) value for each type of restraint. The seven types of restraints are discussed individually, below. You select between the different types of restraints by clicking on the notebook tab at the top of the data window. Note that each restraint type has a “Use” checkmark button and a “Restraint weight factor” entry value. How to select the weight is discussed in §25.8, below. If “Use” is not checked, the restraint settings are ignored. If the weight is set to 0, the restraints are evaluated, but do not affect the refinement.