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

International Tables for Crystallography (2012). Vol. F, ch. 16.1, pp. 417-418   | 1 | 2 |

## Section 16.1.4.2. The minimal function

G. M. Sheldrick,a C. J. Gilmore,b H. A. Hauptman,c C. M. Weeks,c* R. Millerc and I. Usónd

aLehrstuhl für Strukturchemie, Georg-August-Universität Göttingen, Tammannstrasse 4, D-37077 Göttingen, Germany,bDepartment of Chemistry, University of Glasgow, Glasgow G12 8QQ, UK,cHauptman–Woodward Medical Research Institute, Inc., 700 Ellicott Street, Buffalo, NY 14203–1102, USA, and dInstitució Catalana de Recerca i Estudis Avançats at IBMB-CSIC, Barcelona Science Park. Baldiri Reixach 15, 08028 Barcelona, Spain
Correspondence e-mail:  weeks@hwi.buffalo.edu

#### 16.1.4.2. The minimal function

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Constrained minimization of an objective function like the minimal function, (Debaerdemaeker & Woolfson, 1983; Hauptman, 1991; DeTitta et al., 1994), provides an alternative approach to phase refinement or phase expansion. 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 is a solution provided that is calculated directly from the atomic positions before the phase-refinement step (Weeks, DeTitta et al., 1994). Therefore, 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. Examples include an exponential expression that has been found to give superior results for several P1 structures (Hauptman et al., 1999). In addition, substructure determination using a very simple and computationally efficient modified minimal function,(where I is an arbitrary interval [−r, r], NI is the number of triplets whose values lie in I and NT is the total number of triplets), has been reported (Xu & Hauptman, 2004, 2006; Xu et al., 2005) and incorporated into the BnP software (see Section 16.1.12.4).

### References

Xu, H. & Hauptman, H. A. (2004). Statistical approach to the phase problem. Acta Cryst. A60, 153–157.
Xu, H. & Hauptman, H. A. (2006). Recent advances in direct phasing for heavy-atom substructure determination. Acta Cryst. D62, 897–900.
Xu, H., Weeks, C. M. & Hauptman, H. A. (2005). Optimizing statistical Shake-and-Bake for Se-atom substructure determination. Acta Cryst. D61, 976–981.
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.
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.
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.
Hauptman, H. A., Xu, H., Weeks, C. M. & Miller, R. (1999). Exponential Shake-and-Bake: theoretical basis and applications. Acta Cryst. A55, 891–900.
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.