ALBERT

Notes: A new direct-methods computer program, MITHRIL, is described. Written in a neutral subset of Fortran IV, it is built around a heavily modified MULTAN80 system. It incorporates many recent theoretical developments in direct methods including the use of quartet and quintet invariants, a new method for estimating triplets, YZARC and MAGEX, and random-phase tangent refinement. It can be run as a menu-driven interactive real-time package or in the more conventional batch mode. Several levels of user-program interaction are provided.

Notes: Owing to a printer's error, the key reference of the paper by Gilmore & Brown [J. Appl. Cryst. (1988), 21, 571–572] was omitted. The reference is here given in full.

Notes: A least-squares technique for extracting individual phase angles from a set of quartet invariants is described. For symmorphic space groups, this procedure offers the advantages of stability and, in noncentrosymmetric cases, a systematic way of defining the enantiomorph, in contrast with traditional direct methods employing triplets. The application to a phthalic anhydride derivative C26H16O5 in space group P1 is described. The method is readily extended to other space groups.

Notes: A method of using quartets and quintets in the direct-methods program MULTAN is described with several successful applications.

Notes: The system of ten linear equations derived in the previous paper [Hauptman (1985), Acta Cryst. A41, 454-456, equations (3.21)-(3.30)] contains a triple phase cosine as an unknown parameter, which can be calculated via standard techniques. The viability of this set of equations and the accuracy with which cosine invariants may be calculated is assessed with reference to a number of variable parameters in the equations. The assessment is performed on both ideal and real data sets; in both cases the method is capable of identifying a well determined subset of invariants for use in direct methods.

Notes: The use of a likelihood criterion associated with maximum-entropy (ME) extrapolation for selecting phase sets as part of a new multisolution phasing strategy, already applied to solving small crystal structures from single-crystal data [Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297–308] and X-ray powder diffraction data [Gilmore, Henderson & Bricogne (1991). Acta Cryst. A47, 830–841], has been tested on the small protein avian pancreatic polypeptide (APP) with 301 non-H atoms in the asymmetric unit in space group C2. A collection of 50 phase sets for APP were provided by Woolfson & Yao. They had been generated from random starting phases by the SAYTAN procedure [Woolfson & Yao (1990). Acta Cryst. A46, 409–413] using data to a resolution of 0.98 Å. Six of these had an unweighted mean absolute phase error, 〈|Δφ|〉, of less than 50°, the remainder having phase errors of 60° or more. However, none of the conventional figures of merit were able to identify these preferred sets. Each phase set was subjected to our standard procedure of entropy maximization and of evaluation of the log-likelihood gain resulting from the associated ME extrapolation. With only a small subset of data (to 2 Å resolution), the likelihood criterion identified unambiguously the phase sets with 〈|Δφ|〉 less than 50°. In contrast, conventional figures of merit showed no such ability. We conclude that, although SA YTAN is at present much more efficient than our program (MICE) at generating large trial phase sets, the latter is clearly superior in discerning the best of these phase sets. We therefore expect that our current efforts to make MICE run faster should result in better prospects of ab initio phasing for small macromolecules.

Notes: The multisolution method of phase determination combing entropy maximization and likelihood evaluation, previously developed for and applied to single-crystal X-ray studies [Bricogne & Gilmore (1990). Acta Cryst. A46, 284–297; Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297–308], is here extended to permit structure determination from X-ray powder diffraction data using the formulae derived in the previous paper [Bricogne (1991). Acta Cryst. A47, 803–829]. Traditionally, structures are difficult to solve ab initio from powder diffraction data because of peak overlaps, which arise accidentally or are imposed by point-group symmetry. Overlaps reduce both the effective sampling of reciprocal space and the resolution of the data; this makes the application of traditional direct methods difficult. In the method of combined entropy maximization and likelihood evaluation described here, the intensity data are normalized using both the overlapped and non-overlapped reflections by means of a suitably modified version of the MITHRIL computer program [Gilmore (1984). J. Appl. Cryst. 17, 42–46; Gilmore & Brown (1988). J. Appl. Cryst. 22, 571–572]. The data are then passed to the maximum-entropy program MICE [Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297–308]. Following origin and enantiomorph definition (if relevant), this builds a phasing tree in which nodes of the tree represent phase permutations of basis-set reflections which are used as constraints in entropy maximization. The nodes of this tree are ranked according to a likelihood criterion evaluated by a new expression capable of using the combined intensities of overlapped reflections. Successive nodes are built via continuing phase permutation, keeping only those solutions for which the likelihoods are large. Centroid maps are used to determine atomic positions. The method is applied to two data sets from known structures: KAlP2O7 [McMurdie, Morris, Evans, Paretzkin, Wong-Ng, Ettlinger & Hubbard (1986). Powder Diffr. 1(2), 64–77] collected using a conventional X-ray source; and the Sigma-2 clathrasil [McCusker (1988). J. Appl. Cryst. 21, 305–310] which is a synchrotron-derived data set. In both cases the structures are solved routinely and show even some of the light-atom positions in the final maps. The Sigma-2 map even shows the positions of some C and N atoms in the disordered 1-aminoadamantane molecule present in the cavity. The phasing tree for KAlP2O7 reveals the structure after 29 nodes have been computed whilst Sigma-2 shows a complete structure after 17 nodes. Furthermore, the inclusion of overlapped reflections in the likelihood calculations turns out to be essential – nodes which cannot be distinguished when overlaps are not present are readily and correctly ranked when the latter are included. The centroid maps computed with the inclusion of overlapped reflections show a significant improvement in signal-to-noise ratio over those in which overlapped reflections are omitted. We conclude that because of its stability at any resolution range, this method has the potential to be the most powerful technique available for solving structures from their powder diffraction data.

Notes: A survey of practical applications of the maximum entropy method to problems in crystallography is presented along with the related field of Bayesian statistical inference. Included in this review are problems of the processing of intensity data, the crystallographic phase problem, accurate electron-density studies and algorithms for the constrained maximization of entropy. The discussion of the phase problem reviews the solution of structures from powder and single-crystal diffraction data sets, as well as applications to electron and macromolecular crystallography. It is believed that the methodology has a secure future and a sound statistical basis, but it is still not a technique that is regularly employed by crystallographers and the reasons for this are also examined.