The minimal function

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, p. 335 | 1 | 2 |

Section 16.1.4.2. The minimal function

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

16.1.4.2. 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).

References

DeTitta, G. T., Weeks, C. M., Thuman, P., Miller, R. & Hauptman, H. A. (1994). Structure solution by minimal-function phase refinement and Fourier filtering. I. Theoretical basis. Acta Cryst. A50, 203–210.Google Scholar

Debaerdemaeker, T. & Woolfson, M. M. (1983). On the application of phase relationships to complex structures. XXII. Techniques for random phase refinement. Acta Cryst. A39, 193–196.Google Scholar

Hauptman, H. A. (1991). A minimal principle in the phase problem. In Crystallographic computing 5: from chemistry to biology, edited by D. Moras, A. D. Podjarny & J. C. Thierry, pp. 324–332. Oxford: International Union of Crystallography and Oxford University Press.Google Scholar

Hauptman, H. A., Xu, H., Weeks, C. M. & Miller, R. (1999). Exponential Shake-and-Bake: theoretical basis and applications. Acta Cryst. A55, 891–900.Google Scholar

Weeks, C. M., DeTitta, G. T., Hauptman, H. A., Thuman, P. & Miller, R. (1994). Structure solution by minimal-function phase refinement and Fourier filtering. II. Implementation and applications. Acta Cryst. A50, 210–220.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 16.1, p. 335