Phase determination for the estriol structure

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...

Section: ~-N-'/2(iehl 2 + Iekl 2 + [Ek_h] 2-2)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Langs¹

1993

“…We can see that this imitative function is practically identical with the function minimized by the program YZARC (Baggio, Woolfson & Germain, 1978;Wright, 1983). Hauptman, Fisher, Hancock & Norton (1969) …”

Section: Appendix 2 Examples Of Imitative Functionsmentioning

confidence: 99%

The direct method based on a fitting of distributions of semi-invariants

Kríz¹

1989

A new direct method (called TRYMIN) for phaseproblem solution is described. At first a methodical procedure is presented for the construction of a direct method. The aim of the calculation is formulated as a consistency between theoretical and calculated distributions of invariants. A minimized function is obtained and an algorithm for its minimization is proposed. The algorithm is based on the partial decomposition of phases into three subsets and on the cyclical improvement of the estimate of local minima. The efficiency of the method has been tested on 23 structures and a short evaluation of the results of computer experiments with test structures is presented.0108-7673/89/070456-08503.00 Definition of the problemThe direct method for the solution of the phase problem is based on a theory that gives for a great number of functions Y, of phases ~ a forecast of their values for the correct phases ~*. For arbitrary values l~, u~ the theory affords the probability that the theoretically correct value Y* = Yi(~0*) satisfies l, < Y* < u,.Implementing this theory we can formulate the following task: to construct an algorithm for the generation of a limited number of sets q3 for which the values 17"~ = Y~(~) will satisfy inequalities l; < ~' < u~ (for a priori given l~, u~) with frequency in correspondence with the theory.O 1989 International Union of Crystallography VACLAV Kl~i2; 457We can presume that some of the sets ~3 will be close to the correct value ~*. This assumption is acceptable, because the power of direct methods resides in the fact that the problem is overdetermined. For some number of unknown phases the theory gives typically a twenty times greater number of functions Y~.We shall use only functions Y~ of the form Y~(~) = rsd (~ ku~j-P,), where ko are equal to 1, -1 or 0, and most of them are zero; p~ is chosen so that the theoretical probability density function of the absolute values of Y~ has a maximum at zero; and rsd is the real function of the real argument defined as rsd (x) = z if and only if -Tr -z < 7r, and an integer value j(x) exists that satisfies x--z + 2rrj(x).We shall use the symbol K for the matrix (ku), n for the number of functions Y~ and rn for the number of unknown phases. Phases are chosen to fix the origin and their valuesprojected into terms pi. Now we can suppose that K'K is a non-singular matrix.In the examples we shall use only the triple-phase relationships as they are generated by the MUL-TAN80 SIGMA2 routine (Main, Fiske, Hull, Lessinger, Germain, Declercq & Woolfson, 1980) for phase determination. We shall use the symbol K~ for the parameter of their distribution functions. If Y~ is derived from triple-phase relationships then at most three coefficients k~ for each i = 1,..., m are non-zero and p~ contains the phase shift and the values of the phases fixing the origin. Fitting the distributionThe principle of crystal structure solution using distribution fitting methods was used by Ha~ek (1974). However, a general description of these methods was presented later (...

“…A lot of research has been undertaken to modify the B3.o formula (e.g. Hauptman, 1964;Hauptman, Fisher, Hancock & Norton, 1969;Karle, 1970;Fisher, Hancock & Hauptman, 1970). All these approaches tried to calculate the exact value of the 0108-7673/89/070468-04503.00 cosine invariant rather than giving a statistical interpretation of it.…”

Section: Introductionmentioning

confidence: 99%

A statistical interpretation for theB_3,0formula inP1

Brosius¹

1989

We expect that method A will work best for invariants like quintets and higher, since then a lot of correlation terms of order N -3/2 can be taken into account. Method B is expected to work fine for large m values (m ~-N-1/2). We think that a combination of method A and method B will give still better results, since then the correlation terms that normally appear in method A will be boosted. We shall explore this in a forthcoming paper.Finally, let us notice that one can also use another approach for calculating triplet phase invariants; in this approach one considers the atomic position vectors to be fixed and the structure factors to be random variables of the reciprocal-lattice vectors. For more on this we refer to Hauptman (1985) and Gilmore & Hauptman (1985 [Acta Cryst. (1957), 10, 515-524] for the space group P1. The main idea is to use a suitable probability measure for the interatomic position vectors and to 'linearize' the triplet phase invariant. As a result of the model a statistical formula is given using a 'first neighborhood' of random variables.