Reciprocal-space phase refinement or expansion (shaking)

Sheldrick, G. M.; Hauptman, H. A.; Weeks, C. M.; Miller, R.; Usón, I.

doi:10.1107/97809553602060000689

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 16.1, pp. 335-336 | 1 | 2 |

Section 16.1.4. Reciprocal-space phase refinement or expansion (shaking)

G. M. Sheldrick,^c H. A. Hauptman,^b C. M. Weeks,^b ^* R. Miller^b and I. Usón^a

^a Institut für Anorganisch Chemie, Universität Göttingen, Tammannstrasse 4, D-37077 Göttingen, Germany,^bHauptman–Woodward Medical Research Institute, Inc., 73 High Street, Buffalo, NY 14203-1196, USA, and ^cLehrstuhl für Strukturchemie, Universität Göttingen, Tammannstrasse 4, D-37077 Göttingen, Germany
Correspondence e-mail: weeks@orion.hwi.buffalo.edu

16.1.4. Reciprocal-space phase refinement or expansion (shaking)

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Once a set of initial phases has been chosen, it must be refined against the set of structure invariants whose values are presumed known. In theory, any of a variety of optimization methods could be used to extract phase information in this way. However, so far only two (tangent refinement and parameter-shift optimization of the minimal function) have been shown to be of practical value.

16.1.4.1. The tangent formula

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The tangent formula, $[\tan (\varphi_{\bf H}) = {-\textstyle\sum_{\bf K}\displaystyle |E_{\bf K} E_{-{\bf H}-{\bf K}}| \sin (\varphi_{\bf K} + \varphi_{-{\bf H}-{\bf K}}) \over \textstyle\sum_{\bf K}\displaystyle |E_{\bf K} E_{-{\bf H}-{\bf K}}| \cos (\varphi_{\bf K} + \varphi_{-{\bf H}-{\bf K}})}, \eqno(16.1.4.1)]$ (Karle & Hauptman, 1956), is the relationship used in conventional direct-methods programs to compute $[\varphi_{\bf H}]$ given a sufficient number of pairs ( $[\varphi_{\bf K}, \varphi_{-{\bf H}-{\bf K}}]$ ) of known phases. It can also be used within the phase-refinement portion of the dual-space Shake-and-Bake procedure (Weeks, Hauptman et al., 1994; Sheldrick & Gould, 1995). The variance associated with $[\varphi_{\bf H}]$ depends on $[\sum_{\bf K} E_{\bf H} E_{\bf K} E_{-{\bf H}-{\bf K}}/N^{1/2}]$ and, in practice, the estimate is only reliable for $[|E_{\bf H}|\gg 1]$ and for structures with a limited number of atoms (N). If equation (16.1.4.1) is used to redetermine previously known phases, the phasing process is referred to as tangent-formula refinement; if only new phases are determined, the phasing process is tangent expansion.

The tangent formula can be derived using the assumption of equal resolved atoms. Nevertheless, it suffers from the disadvantage that, in space groups without translational symmetry, it is perfectly fulfilled by a false solution with all phases equal to zero, thereby giving rise to the so-called `uranium-atom' solution with one dominant peak in the corresponding Fourier synthesis. In conventional direct-methods programs, the tangent formula is often modified in various ways to include (explicitly or implicitly) information from the so-called `negative' quartet invariants (Schenk, 1974; Hauptman, 1974; Giacovazzo, 1976) that are dependent on the smallest as well as the largest E magnitudes. Such modified tangent formulas do indeed largely overcome the problem of pseudosymmetric solutions for small N, but because of the dependence of quartet-term probabilities on [1/N] , they are little more effective than the normal tangent formula for large N.

16.1.4.2. The minimal function

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Constrained minimization of an objective function like the minimal function, $[R(\Phi) = \textstyle\sum\limits_{{\bf H},{\bf K}}\displaystyle A_{\bf HK} \left\{\cos \Phi_{\bf HK} - [I_{1} (A_{\bf HK})/ I_{0}(A_{\bf HK})]\right\}^{2} \Big/ \textstyle\sum\limits_{{\bf H},{\bf K}}\displaystyle A_{\bf HK} \eqno(16.1.4.2)]$ (Debaerdemaeker & Woolfson, 1983; Hauptman, 1991; DeTitta et al., 1994), provides an alternative approach to phase refinement or phase expansion. $[R(\Phi)]$ is a measure of the mean-square difference between the values of the triplets calculated using a particular set of phases and the expected values of the same triplets as given by the ratio of modified Bessel functions. The minimal function is expected to have a constrained global minimum when the phases are equal to their correct values for some choice of origin and enantiomorph (the minimal principle). Experimentation has thus far confirmed that, when the minimal function is used actively in the phasing process and solutions are produced, the final trial structure corresponding to the smallest value of $[R(\Phi)]$ is a solution provided that $[R(\Phi)]$ is calculated directly from the atomic positions before the phase-refinement step (Weeks, DeTitta et al., 1994). Therefore, $[R(\Phi)]$ is also an extremely useful figure of merit. The minimal function can also include contributions from higher-order (e.g. quartet) invariants, although their use is not as imperative as with the tangent formula because the minimal function does not have a minimum when all phases are zero. In practice, quartets are rarely used in the minimal function because they increase the CPU time while adding little useful information for large structures. The cosine function in equation (16.1.4.2) can also be replaced by other functions of the phases giving rise to alternative minimal functions. In particular, an exponential expression has been found to give superior results for several P1 structures (Hauptman et al., 1999).

16.1.4.3. Parameter shift

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In principle, any minimization technique could be used to minimize $[R(\Phi)]$ by varying the phases. So far, a seemingly simple algorithm, known as parameter shift (Bhuiya & Stanley, 1963), has proven to be quite powerful and efficient as an optimization method when used within the Shake-and-Bake context to reduce the value of the minimal function. For example, a typical phase-refinement stage consists of three iterations or scans through the reflection list, with each phase being shifted a maximum of two times by 90° in either the positive or negative direction during each iteration. The refined value for each phase is selected, in turn, through a process which involves evaluating the minimal function using the original phase and each of its shifted values (Weeks, DeTitta et al., 1994). The phase value that results in the lowest minimal-function value is chosen at each step. Refined phases are used immediately in the subsequent refinement of other phases. It should be noted that the parameter-shift routine is similar to that used in ψ-map refinement (White & Woolfson, 1975) and XMY (Debaerdemaeker & Woolfson, 1989).

References

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International Tables for Crystallography (2006). Vol. F. ch. 16.1, pp. 335-336