The relation between space group symmetry and the properties of direct solutions based on the Σ<sub>2</sub>formula. A method for using centrosymmetric Σ<sub>2</sub>solutions for solving non-centrosymmetric structures

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Likewise, the elimination of the noise of the refine-'ment of the mean structure from the A matrix may be performed in a similar manner as before. The pseudosymmetry operator which gives high covariance between parameters need not always allow a sensible selection of combination of parameters. An example is the pseudosymmetry operators (-}+x, y, k-z), (2-+x, y, ~--z). A refinement procedure for a crystal structure with these pseudosymmetry operators is currently being investigated, and will be published elsewhere.We have seen earlier that a full-matrix solution tends to overweight the changes in the 'difference' structure whereas a block diagonal approach overweights the changes in the 'mean' structure but underweights the 'difference' structure, and so other refinement procedures may be considered.As .,tated in the Introduction, the refinement of the 'difference' structure is only possible as (sin 0)/2 increases. Thus the simplest method to reduce covariance between highly covarying parameters is to use highangle data or to weight data according to (sin 0)/2. Similarly, if there is an index condition so that only the 'difference' structure contributes to certain reflexions then the weight of these reflexions can be increased.A combination of a full-matrix approach and a diagonal approach can be considered. This is simply brought about by multiplying the diagonal elements au of the A matrix by a constant, kl, greatei than 1, and the elements of the column matrix B by a constant k2. The constant kz should be less than kl, otherwise the refinement of the 'mean' structure will be overweighted. A down weighting of the refinement of the 'difference' structure is brought about by a reduction of the apparent covariance between parameters. The shifts of highly covarying parameters are reduced more than the shifts in parameters which are less covariant. A read-out/read-in interruption to the leastsquares refinement program prior to the inversion of the A matrix allows any number of modifications to be tested without re-evaluating the A matrix each time.This final method has a lot to recommend it. Consider our simple two-variable example again. The equation~'/~ varies from eo for 6 = 0 to 1 for 6 = 1. Also (2-6)1~ varies from ,2 for ~ = 0 to 1 for ~ = 1, whereas o)zlo)l = (2-3')/6' varies from (kl + 1)/(kl-1) for ~ = 0 to 1 for 6=1.Thus we have a built-in discrimination against those parameters of the 'difference' structure which are least accurately determined. If we choose kl as 1.4 say, then COz/O91 varies between 6 and 1. This procedure will also work when a larger set of covarying variables is considered.Acta Cryst. (1973 Phase relations between quartets of reflexions can be found either directly from a generalized Hughes expression or by elimination of the phase of a reflexion common to two Y2 relationships. By combining the information from both types, strengthened quartet relationships (SQR) can be constructed, which are comparable to the Y z relations in reliability and number. Most of the reliable SQR's involve ...

“…For (23) and (24) refinement criteria can be found in the same way as for (21) and (22) (Schenk, 1972c). The criterion of (23) is:…”

Section: Non-centrosymmetric Structuresmentioning

confidence: 99%

“…The forms of (25) and (26) show that all trivialities resulting from space-group symmetry (Schenk, 1972c) also occur with the quartet relationships.…”

Section: Non-centrosymmetric Structuresmentioning

confidence: 99%

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The use of phase relationships between quartets of reflexions

Schenk¹

1973

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“…Indeed, each quartet in NQEST involves the pair ~h+ ~k--~,+k and ~0~+ ~m+ ~h+k having [Eh+d small. On the contrary, NTREST exploits triplets estimated negative in spite of the Since (Schenk, 1972) Owing to the well known tendency of this crystal structure to lose the enantiomorph, no satisfactory solution may be expected among the 14 sets. Accordingly the program warns the user that the enantiomorph is probably lost (ENANT calculated from published phases is >35°).…”

Section: Di(g)= Ii(g)/io(g) and II Is The Modifiedmentioning

confidence: 99%

Figures of merit in direct methods: a new point of view

Cascarano¹,

Giacovazzo²,

Viterbo³

1987

The probabilistic approach has been extensively used for analysing the statistical meanings of some traditional figures of merit. New figures of merit have also been introduced; some exploit one-phase and twophase structure seminvariants of the first rank, the expected negative and/or enantiomorph-sensitive triplet invariants, and the expected negative and/or enantiomorph-sensitive quartets. Other figures of merit exploit the distribution of statistical parameters connected with PSI(0) and active triplets. A new combined figure of merit is shown to be a powerful tool for selecting the correct solutions among the various sets output by multisolution methods.

“…Some of these difficulties have been dealt with by Schenk (1972), who indicated that they can be ascribed to the influence of the space group symmetry and the properties of the Y2 relation.…”

Section: Introductionmentioning

confidence: 99%

Application of quartet and quintet invariants in phase determination

Putten¹,

Schenk²

1979

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Recent derivations of probability expressions for quartet and quintet structure invariants allow the reliable estimation of the phase sums of quartets and quintets in the range 0 to zt. A new quartet figure of merit, ENQUAC and a new quintet figure of merit, ENQUIC, based on these estimates are described, which are particularly useful in non-centrosymmetric symmorphic and polar space groups. An adapted tangent-refinement procedure employing selected triplets with a phase sum of 0, together with the quartet and quintet phase estimates, enables enantiomorphspecific phase refinement. The way in which the figures of merit and the refinement technique can be used in various practical procedures is demonstrated and applications to two structures in space group P1 are presented.