TheSIRprogram. I. Use of negative quartets

Abstract

The present paper deals with the finding and the estimation of negative quartets. The use of symmetry is also discussed and the results obtained for four structures of increasing size are reported. Conditional probability distributions using magnitudes contained in the second representation of the quartets provides improved estimates of their phases. IntroductionDuring recent years, direct methods have been developed to allow the crystal-structure determination of a great number of large molecules. Most of the currently available direct-method programs are based on the use of the single statistical triplet relation (sin (~0 H + ~0 K + ~0h-~))~ ~_ 0, from which the so-called tangent formula is derived.The estimate of the triplet cosine invariant by statistical formulae [e.g. MDKS (Hauptman, 1972;Giacovazzo, 1976a)] is a more accurate way of using the information provided by the triplet invariants. More recently, a modification of the classical tangent formula, making use of the estimated phases of the triplet invariants (Busetta, 1976), provided a good way of determining a structure in those cases where enantiomorph discrimination is difficult. However, in order to deal with large molecules -or with very troublesome small ones -it is necessary to use the whole information provided by the complete set of structure *Present address: Laboratoire de Cristallographie (LA 144-CNRS), Universit6 de Bordeaux I, 351, Cours de la Lib6ration, 33405 Talence, France.0567-7394/80/010068-07501.00 invariants and seminvariants. This is the basic idea behind the SIR program. All the (sem)invariant phase relationships are estimated following the Sem Invariant Representation theory (Giacovazzo, 1977). The present available version of the program uses triplets, negative quartets and one-phase seminvariants.The present paper, the first of a series, deals with negative quartets. The positive quartets provide information which is strongly correlated to the information contained in triplet relations. Their use requires a more complicated approach and will be analysed in a future paper. Here we intend to develop different points concerned with finding and estimating negative quartets. Until now, only the four basis vectors and the three related cross vectors have been used in their estimation (Hauptman, 1974; Schenk, 1974); we will show that the use of symmetry (Giacovazzo, 1976b) and of the second representation can improve these estimates.In the estimation of quartets, seven vectors are necessary, the basis vectors H1, H 2, H 3 and H a whose sum is zero and the cross vectors H a + H 2, H~ + H a and H~ + H 4. Of these seven vectors only three are independent. It is possible to search in different ways for negative quartets.(1) Starting from three basis vectors with large magnitudes, H l, HE, H3, we may search for the fourth basis vectors H 4 and the three corresponding cross vectors, checking if they have weak magnitudes.(2) A second way is to start from three cross vectors U, V, W of weak magnitudes and to search for the four bas...

The theory of representations: its application to the estimation of the two-phase variants and seminvariants

“…But in the group we have a great number of well estimated and a number of badly estimated phases in such a way that the averaged experimental e.s.d, is almost equal to the theoretical deviation. The situation is very similar to that described by Busetta et al (1980) for negative quartets.…”

“…In particular we shall estimate q> via the first representation of 7". According to the symmetry class, 7" can depend on more than three cross magnitudes (Giacovazzo, 1976;Busetta, Giacovazzo, Burla, Nunzi, Polidori & Viterbo, 1980). To involve all the available cross magnitudes some algebraic considerations are useful, which are described in 8 3.…”

Practical aspects and applications of some probabilistic formulae which estimate one-phase seminvariants

“…However, Overbeek & Schenk (1976) made full use of the space-group symmetry for the Y~ criterion, while they applied to monoclinic structures the usual negative-quartet criterion as derived in the triclinic system. On the other hand, a recent paper by Busetta et aL (1980) shows that the effectiveness of the negative-quartet criteria strongly increases when full use of the space-group symmetry is made [see Giacovazzo (1976) for the theoretical background]. The results previously mentioned strongly suggest that K~ and negative-quartet criteria are two of the most effective figures of merit for selecting the correct solution from a large number of possible candidates often produced by multisolution techniques.…”

Triplet and quartet relations: their use in direct procedures

“…This effect can be minimized if maximum use of the symmetry for estimating quartets is made (Giacovazzo, 1976b). In fact, even if quartets with more than three cross magnitudes in the first phasing shell are a small percentage of the total number of quartets, they can constitute a large percentage of the most reliable quartets (Busetta et al, 1980). These considerations suggest that quartets should not be used alone in phasing procedures, but always associated with triplets.…”