International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 95100

Since the origins of Xray crystal structure analysis, the calculation of crystallographic Fourier series has been closely associated with the process of refinement. Fourier coefficients with phases were obtained for all or part of the measured reflections in the basis of some trial model for all or part of the structure, and Fourier syntheses were then used to complete and improve this initial model. This approach is clearly described in the classic paper by Bragg & West (1929), and was put into practice in the determination of the structures of topaz (Alston & West, 1929) and diopside (Warren & Bragg, 1929). Later, more systematic methods of arriving at a trial model were provided by the Patterson synthesis (Patterson, 1934, 1935a,b; Harker, 1936) and by isomorphous replacement (Robertson, 1935, 1936c). The role of Fourier syntheses, however, remained essentially unchanged [see Robertson (1937) for a review] until more systematic methods of structure refinement were introduced in the 1940s. A particularly good account of the processes of structure completion and refinement may be found in Chapters 15 and 16 of Stout & Jensen (1968).
It is beyond the scope of this section to review the vast topic of refinement methods: rather, it will give an account of those aspects of their development which have sought improved power by exploiting properties of the Fourier transformation. It is of more than historical interest that some recent advances in the crystallographic refinement of macromolecular structures had been anticipated by Cochran and Cruickshank in the early 1950s.
Hughes (1941) was the first to use the already well established multivariate leastsquares method (Whittaker & Robinson, 1944) to refine initial estimates of the parameters describing a model structure. The method gained general acceptance through the programming efforts of Friedlander et al. (1955), Sparks et al. (1956), Busing & Levy (1961) and others.
The Fourier relations between and F (Section 1.3.4.2.2.6) are used to derive the `observational equations' connecting the structure parameters to the observations comprising the amplitudes and their experimental variances for a set of unique reflections.
The normal equations giving the corrections δu to the parameters are thenwhereTo calculate the elements of A, write:hence
In the simple case of atoms with realvalued form factors and isotropic thermal agitation in space group P1,where being a fractional occupancy.
Positional derivatives with respect to are given byso that the corresponding subvector of the righthand side of the normal equations reads:
The setting up and solution of the normal equations lends itself well to computer programming and has the advantage of providing a thorough analysis of the accuracy of its results (Cruickshank, 1965b, 1970; Rollett, 1970). It is, however, an expensive task, of complexity , which is unaffordable for macromolecules.
It was the use of Fourier syntheses in the completion of trial structures which provided the incentive to find methods for computing 2D and 3D syntheses efficiently, and led to the Beevers–Lipson strips. The limited accuracy of the latter caused the estimated positions of atoms (identified as peaks in the maps) to be somewhat in error. Methods were therefore sought to improve the accuracy with which the coordinates of the electrondensity maxima could be determined. The naive method of peakshape analysis from densities recalculated on a grid using highaccuracy trigonometric tables entailed 27 summations per atom.
Booth (1946a) suggested examining the rapidly varying derivatives of the electron density rather than its slowly varying values. Ifthen the gradient vector of at can be calculated by means of three Fourier summations from the vector of Fourier coefficientsSimilarly, the Hessian matrix of at can be calculated by six Fourier summations from the unique elements of the symmetric matrix of Fourier coefficients:
The scalar maps giving the components of the gradient and Hessian matrix of will be called differential syntheses of 1st order and 2nd order respectively. If is approximately but not exactly a maximum of , then the Newton–Raphson estimate of the true maximum is given by:This calculation requires only nine accurate Fourier summations (instead of 27), and this number is further reduced to four if the peak is assumed to be spherically symmetrical.
The resulting positions are affected by seriestermination errors in the differential syntheses. Booth (1945c, 1946c) proposed a `backshift correction' to eliminate them, and extended this treatment to the acentric case (Booth, 1946b). He cautioned against the use of an artificial temperature factor to fight seriestermination errors (Brill et al., 1939), as this could be shown to introduce coordinate errors by causing overlap between atoms (Booth, 1946c, 1947a,b).
Cruickshank was able to derive estimates for the standard uncertainties of the atomic coordinates obtained in this way (Cox & Cruickshank, 1948; Cruickshank, 1949a,b) and to show that they agreed with those provided by the leastsquares method.
The calculation of differential Fourier syntheses was incorporated into the crystallographic programs of Ahmed & Cruickshank (1953b) and of Sparks et al. (1956).
Having defined the now universally adopted R factors (Booth, 1945b) as criteria of agreement between observed and calculated amplitudes or intensities, Booth proposed that R should be minimized with respect to the set of atomic coordinates by descending along the gradient of R in parameter space (Booth, 1947c,d). This `steepest descents' procedure was compared with Patterson methods by Cochran (1948d).
When calculating the necessary derivatives, Booth (1948a, 1949) used the formulae given above in connection with least squares. This method was implemented by Qurashi (1949) and by Vand (1948, 1951) with parameterrescaling modifications which made it very close to the leastsquares method (Cruickshank, 1950; Qurashi & Vand, 1953; Qurashi, 1953).
Cochran (1948b,c, 1951a) undertook to exploit an algebraic similarity between the righthand side of the normal equations in the leastsquares method on the one hand, and the expression for the coefficients used in Booth's differential syntheses on the other hand (see also Booth, 1948a). In doing so he initiated a remarkable sequence of formal and computational developments which are still actively pursued today.
Let be the electrondensity map corresponding to the current atomic model, with structure factors ; and let be the map calculated from observed moduli and calculated phases, i.e. with coefficients . If there are enough data for to have a resolved peak at each model atomic position , thenwhile if the calculated phases are good enough, will also have peaks at each :It follows thatwhere the summation is over all reflections in or related to by spacegroup and Friedel symmetry (overlooking multiplicity factors!). This relation is less sensitive to seriestermination errors than either of the previous two, since the spectrum of could have been extrapolated beyond the data in by using that of [as in van Reijen (1942)] without changing its righthand side.
Cochran then used the identityin the formto rewrite the previous relation as(the operation [] on the first line being neutral because of Friedel symmetry). This is equivalent to the vanishing of the subvector of the righthand side of the normal equations associated to a leastsquares refinement in which the weights would beCochran concluded that, for equalatom structures with for all j, the positions obtained by Booth's method applied to the difference map are such that they minimize the residualwith respect to the atomic positions. If it is desired to minimize the residual of the ordinary leastsquares method, then the differential synthesis method should be applied to the weighted difference mapHe went on to show (Cochran, 1951b) that the refinement of temperature factors could also be carried out by inspecting appropriate derivatives of the weighted difference map.
This Fourier method was used by Freer et al. (1976) in conjunction with a stereochemical regularization procedure to refine protein structures.
Cruickshank consolidated and extended Cochran's derivations in a series of classic papers (Cruickshank, 1949b, 1950, 1952, 1956). He was able to show that all the coefficients involved in the righthand side and normal matrix of the leastsquares method could be calculated by means of suitable differential Fourier syntheses even when the atoms overlap. This remarkable achievement lay essentially dormant until its independent rediscovery by Agarwal in 1978 (Section 1.3.4.4.7.6).
To ensure rigorous equivalence between the summations over (in the expressions of leastsquares righthand side and normal matrix elements) and genuine Fourier summations, multiplicitycorrected weights were introduced by:where Gh denotes the orbit of h and its isotropy subgroup (Section 1.3.4.2.2.5). Similarly, derivatives with respect to parameters of symmetryunique atoms were expressed, via the chain rule, as sums over the orbits of these atoms.
Let be the label of a parameter belonging to atoms with label j. Then Cruickshank showed that the pth element of the righthand side of the normal equations can be obtained as , where is a differential synthesis of the formwith a polynomial in (h, k, l) depending on the type of parameter p. The correspondence between parameter type and the associated polynomial extends Booth's original range of differential syntheses, and is recapitulated in the following table.Unlike Cochran's original heuristic argument, this result does not depend on the atoms being resolved.
Cruickshank (1952) also considered the elements of the normal matrix, of the formassociated with positional parameters. The block for parameters and may be writtenwhich, using the identitybecomes(Friedel's symmetry makes redundant on the last line). Cruickshank argued that the first term would give a good approximation to the diagonal blocks of the normal matrix and to those offdiagonal blocks for which and are close. On this basis he was able to justify the `nshift rule' of Shoemaker et al. (1950). Cruickshank gave this derivation in a general space group, but using a very terse notation which somewhat obscures it. Using the symmetrized trigonometric structurefactor kernel of Section 1.3.4.2.2.9 and its multiplication formula, the above expression is seen to involve the values of a Fourier synthesis at points of the form .
Cruickshank (1956) showed that this analysis could also be applied to the refinement of temperature factors.
These two results made it possible to obtain all coefficients involved in the normal equations by looking up the values of certain differential Fourier syntheses at or at . At the time this did not confer any superiority over the standard form of the leastsquares procedure, because the accurate computation of Fourier syntheses was an expensive operation. The modified Fourier method was used by Truter (1954) and by Ahmed & Cruickshank (1953a), and was incorporated into the program system described by Cruickshank et al. (1961). A more recent comparison with the leastsquares method was made by Dietrich (1972).
There persisted, however, some confusion about the nature of the relationship between Fourier and leastsquares methods, caused by the extra factors which make it necessary to compute a differential synthesis for each type of atom. This led Cruickshank to conclude that `in spite of their remarkable similarities the leastsquares and modifiedFourier methods are fundamentally distinct'.
Agarwal (1978) rederived and completed Cruickshank's results at a time when the availability of the FFT algorithm made the Fourier method of calculating the coefficients of the normal equations much more economical than the standard method, especially for macromolecules.
As obtained by Cruickshank, the modified Fourier method required a full 3D Fourier synthesis
Agarwal disposed of the latter dependence by pointing out that the multiplication involved is equivalent to a realspace convolution between the differential synthesis and , the standard electron density for atom type j (Section 1.3.4.2.1.2) smeared by the isotropic thermal agitation of that atom. Since is localized, this convolution involves only a small number of grid points. The requirement of a distinct differential synthesis for each parameter type, however, continued to hold, and created some difficulties at the FFT level because the symmetries of differential syntheses are more complex than ordinary spacegroup symmetries. Jack & Levitt (1978) sought to avoid the calculation of difference syntheses by using instead finite differences calculated from ordinary Fourier or difference Fourier maps.
In spite of its complication, this return to the Fourier implementation of the leastsquares method led to spectacular increases in speed (Isaacs & Agarwal, 1978; Agarwal, 1980; Baker & Dodson, 1980) and quickly gained general acceptance (Dodson, 1981; Isaacs, 1982a,b, 1984).
Lifchitz [see Agarwal et al. (1981), Agarwal (1981)] proposed that the idea of treating certain multipliers in Cruickshank's modified differential Fourier syntheses by means of a convolution in real space should be applied not only to , but also to the polynomials which determine the type of differential synthesis being calculated. This leads to convoluting with the same ordinary weighted difference Fourier synthesis, rather than with the differential synthesis of type p. In this way, a single Fourier synthesis, with ordinary (scalar) symmetry properties, needs be computed; the parameter type and atom type both intervene through the function with which it is convoluted. This approach has been used as the basis of an efficient generalpurpose leastsquares refinement program for macromolecular structures (Tronrud et al., 1987).
This rearrangement amounts to using the fact (Section 1.3.2.3.9.7) that convolution commutes with differentiation. Letbe the inversevariance weighted difference map, and let us assume that parameter belongs to atom j. Then the Agarwal form for the pth component of the righthand side of the normal equations iswhile the Lifchitz form is
A very simple derivation of the previous results will now be given, which suggests the possibility of many generalizations.
The weighted difference map has coefficients which are the gradients of the global residual with respect to each :By the chain rule, a variation of each by will result in a variation of R by withThe operation is superfluous because of Friedel symmetry, so that may be simply written in terms of the Hermitian scalar product in :If is the transform of , we have also by Parseval's theoremWe may therefore writewhich states that is the functional derivative of R with respect to .
The righthand side of the normal equations has for its pth element, and this may be writtenIf belongs to atom j, thenhenceBy the identity of Section 1.3.2.4.3.5, this is identical to Lifchitz's expression . The present derivation in terms of scalar products [see Brünger (1989) for another presentation of it] is conceptually simpler, since it invokes only the chain rule [other uses of which have been reviewed by Lunin (1985)] and Parseval's theorem; economy of computation is obviously related to the good localization of compared to . Convolutions, whose meaning is less clear, are no longer involved; they were a legacy of having first gone over to reciprocal space via differential syntheses in the 1940s.
Cast in this form, the calculation of derivatives by FFT methods appears as a particular instance of the procedure described in connection with variational techniques (Section 1.3.4.4.6) to calculate the coefficients of local quadratic models in a search subspace; this is far from surprising since varying the electron density through a variation of the parameters of an atomic model is a particular case of the `free' variations considered by the variational approach. The latter procedure would accommodate in a very natural fashion the joint consideration of an energetic (Jack & Levitt, 1978; Brünger et al., 1987; Brünger, 1988; Brünger et al., 1989; Kuriyan et al., 1989) or stereochemical (Konnert, 1976; Sussman et al., 1977; Konnert & Hendrickson, 1980; Hendrickson & Konnert, 1980; Tronrud et al., 1987) restraint function (which would play the role of S) and of the crystallographic residual (which would be C). It would even have over the latter the superiority of affording a genuine secondorder approximation, albeit only in a subspace, hence the ability of detecting negative curvature and the resulting bifurcation behaviour (Bricogne, 1984). Current methods are unable to do this because they use only firstorder models, and this is known to degrade severely the overall efficiency of the refinement process.
The impossibility of carrying out a fullmatrix leastsquares refinement of a macromolecular crystal structure, caused by excessive computational cost and by the paucity of observations, led Diamond (1971) to propose a realspace refinement method in which stereochemical knowledge was used to keep the number of free parameters to a minimum. Refinement took place by a leastsquares fit between the `observed' electrondensity map and a model density consisting of Gaussian atoms. This procedure, coupled to iterative recalculation of the phases, led to the first highly refined protein structures obtained without using fullmatrix least squares (Huber et al., 1974; Bode & Schwager, 1975; Deisenhofer & Steigemann, 1975; Takano, 1977a,b).
Realspace refinement takes advantage of the localization of atoms (each parameter interacts only with the density near the atom to which it belongs) and gives the most immediate description of stereochemical constraints. A disadvantage is that fitting the `observed' electron density amounts to treating the phases of the structure factors as observed quantities, and to ignoring the experimental error estimates on their moduli. The method is also much more vulnerable to seriestermination errors and accidentally missing data than the leastsquares method. These objections led to the progressive disuse of Diamond's method, and to a switch towards reciprocalspace least squares following Agarwal's work.
The connection established above between the Cruickshank–Agarwal modified Fourier method and the simple use of the chain rule affords a partial refutation to both the premises of Diamond's method and to the objections made against it:
The calculation of the inner products from a sampled gradient map D requires even more caution than that of structure factors via electrondensity maps described in Section 1.3.4.4.5, because the functions have transforms which extend even further in reciprocal space than the themselves. Analytically, if the are Gaussians, the are finite sums of multivariate Hermite functions (Section 1.3.2.4.4.2) and hence the same is true of their transforms. The difference map D must therefore be finely sampled and the relation between error and sampling rate may be investigated as in Section 1.3.4.4.5. An examination of the sampling rates commonly used (e.g. one third of the resolution) shows that they are insufficient. Tronrud et al. (1987) propose to relax this requirement by applying an artificial temperature factor to (cf. Section 1.3.4.4.5) and the negative of that temperature factor to D, a procedure of questionable validity because the latter `sharpening' operation is ill defined [the function exp does not define a tempered distribution, so the associativity properties of convolution may be lost]. A more robust procedure would be to compute the scalar product by means of a more sophisticated numerical quadrature formula than a mere grid sum.
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