Direct methods: the identification of space-group-specific inconsistent three-phase structure invariants

Han, Fu-son; Langs, D. A.

doi:10.1107/s010876738800306x

Cited by 3 publications

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References 4 publications

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“…Table 2 provides an answer as to why this particular structure is so troublesome. Of the thousands of triples used to solve this structure, a small percentage are inconsistent (Han & Langs, 1988). The two strongest Y.2 triple indications for this structure, with A values of 3.31, represent an inconsistent pair (see Table 2).…”

Section: Discussion Of Resultsmentioning

confidence: 99%

“…Table 6 lists a group of 33 zonal reflections for the GRAMA structure that can be abstracted from the phase-invariant list of Table 5 and expressed in terms of five symbols. These 33 phases together with 9 additional unrestricted phases, expressed as magic integers, were actually used to solve this structure (Langs, 1988), which contains more than 300 independent atoms in the unit cell. It sould be cautioned that results such as these are sensitive to the accuracy with which the data have been measured and scaled, as well as to the degree to which the scaled [El values model the ideal pointatom structures upon which probabilistic direct methods have been derived (Langs, 1993).…”

Section: P212121mentioning

confidence: 99%

“…That is, if one of the two Y~ estimates is strongly 548 COSINE INVARIANTS OF THREE-PHASE RELATIONSHIPS indicated to be positive, the other must exhibit a strong negative estimate to be consistent. Similar constructions to unmask and actively utilize aberrant 5-' -2 triples' phase information exist in the form of inconsistent quadrupoles (Viterbo & Woolfson, 1973) and triples (Han & Langs, 1988). …”

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Frequency statistical method for evaluating cosine invariants of three-phase relationships

Langs¹

1993

Acta Cryst Sect A

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A new variation on the established procedure to evaluate three-phase structure invariants through quadrupole relationships is described. This method differs from earlier algebraic formulations in that the cosine-invariant estimates are based on a conditional observed frequency distribution of I EI magnitudes for the quadrupole, rather than on the values of the magnitudes themselves. Successful applications of this method to a number of structures that ranged in size from 84 to 317 independent non-hydrogen light atoms are given. IntroductionThe three-phase crystallographic structure invariants play a central role in the determination of crystal structures by direct-phasing methods. Tangentformula methods for small-molecule determinations have traditionally relied on the 0 (modulo 27r) probability estimate for these 'triples' (Karle & Hauptman, 1956). Efforts to extend these techniques to larger structures have required more precise estimates to be obtained for these phase invariants, though use of algebraic formulae (Karle & Hauptman, 1957;Vaughan, 1958;Hauptman, 1964;Hauptman, Fisher, Hancock & Norton, 1969;Karle, 1970;Duax, Weeks & Hauptman, 1972;, determinantal joint probability distributions (Tsoucaris, 1970;Messager & Tsoucaris, 1972;Giacovazzo, 1976Giacovazzo, , 1977aKarle, 1979Karle, , 1980 or probabilistic formulae, as applied to isomorphous-replacement or anomalous-dispersion data (Hauptman, 1982;Giacovazzo, 1983;Fortier, Moore & Frazer, 1985) and to the extended neighborhoods or phasing shells of data that define higher-order relationships into which these triples have been suitably embedded (Hauptman, 1975;Giacovazzo, 1977b;Karle, 1982). This report describes a new method to estimate three-phase 0108-7673/93/030545-13 $06.00 invariants based on examination of the frequency distribution of ILl magnitudes that complete a family of conditionally constructed quadrupoles that are common to the evaluated triple. BackgroundOne of the earliest strategies in direct-methods research was the development of formulae to evaluate crystal-structure phase invariants and semi-invariants as a means to determine crystal structures. This work was initiated about the same time that the rules for origin and enantiomorph specification and phaseextension techniques were being developed. Formulae to estimate the single-phase structure invariants had an immediate application; they provided a means to reduce the number of algebraic symbols that had to be permuted and tested for a selected starting group of phases. But algebraic formulae that were developed for the determination of the cosine values of the three-phase structure invariants, for example, for P1 symmetry (Karle & Hauptman, 1957), ~-N-'/2(IEhl 2 + IEkl 2 + [Ek_h] 2-2)-t-½ N3/2< (I E,I 2-1) (I E,-k] 2-1)(I E,-hl 2-1))i,however, did not have an immediate impact on phasing practices. Firstly, these formulae were computationally demanding; the average of a product of ILl 2-1 magnitudes had to be computed over a range of I that sampled the whole of reciprocal sp...

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Section: Discussion Of Resultsmentioning

confidence: 99%

Section: P212121mentioning

confidence: 99%

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confidence: 99%

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Frequency statistical method for evaluating cosine invariants of three-phase relationships

Langs¹

1993

Acta Cryst Sect A

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AbstractIn a recent paper [Han& Langs (1988). Acta Crysr A44, 563-566], the 230 space groups were examined to identify conditions which permit symmetry-inconsistent triplets.

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About symmetry-inconsistent three-phase structure invariants

Giacovazzo¹

1989

Acta Cryst Sect A

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In a recent paper [Han& Langs (1988). Acta Crysr A44, 563-566], the 230 space groups were examined to identify conditions which permit symmetry-inconsistent triplets. A careful analysis of the paper shows that results are incomplete, several wrong statements have been made and that a great deal of literature on the subject has been missed. In the present paper new conditions allowing the existence of symmetryconsistent and -inconsistent triplets are also given.

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The Method of Representations of Structure Seminvariants I: The General Background

Giacovazzo

1991

Direct Methods of Solving Crystal Structures

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Direct methods: the identification of space-group-specific inconsistent three-phase structure invariants

Cited by 3 publications

References 4 publications

Frequency statistical method for evaluating cosine invariants of three-phase relationships

Frequency statistical method for evaluating cosine invariants of three-phase relationships

About symmetry-inconsistent three-phase structure invariants

The Method of Representations of Structure Seminvariants I: The General Background

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