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

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Large starting sets of random phases are in general inconsistent with positivity and atomicity of the electron density. A correct solution can be achieved from them because the tangent formula is a process which maximizes entropy under physical constraints involving the positivity and atomicity of electron density. Starting sets which are themselves tendentially maximally entropical can be created by associating phase shifts A, generated according to the yon Mises distribution of each triplet phase, to some (from 50 to 300) triplet invariants. Then the phases generated via these perturbed triplets are samples of the phase population expected to be maximally entropical on 0108-7673/87/060797-06501.50 the basis of the prior information. Experimental tests show that the method may be a useful alternative to other conventional multisolution methods.

Section: Experimental Results and Conclusionmentioning

confidence: 99%

A multisolution procedureviatendentially maximally entropical starting sets

Burla¹,

Giacovazzo²,

Polidori³

1987

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“…when no prior information is available, and it is only necessary that the fractional part of hr be uniformly distributed in the interval (0, 1)]. On the other hand prior information often breaks down the symmetry of the roles, so that the two distributions are deeply different [for example, when a structural fragment has a known position (Giacovazzo, 1983), or when superstructural effects are present (Cascarano, Giacovazzo & Lui6, 1985, 1987]. Symmetrization of the roles is still possible but some supplementary conditions on the reciprocal vectors are needed if they are used as primitive random va.riables.…”

Section: H+k+l=0 (2)mentioning

confidence: 99%

On integrating the techniques of direct methods and isomorphous replacement. A new probabilistic formula for triplet invariants

Giacovazzo¹,

Cascarano²,

Chao-De³

1988

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“…Some FOMs, like the well known MABS, ABSFOM, R,~ and NALF (Declercq, Germain & Woolfson, 1979;Karle & Karle, 1966;Cascarano, Giacovazzo & Viterbo, 1987) check if the distribution of the O~3,h'S fitS the expected distribution. On the same principle are based the so-called psi(0) FOMs (Cochran & Douglas, 1957;Cascarano, Giacovazzo & Guagliardi, 1992) while the negative-quartet criterion (DeTitta, Edmonds, Langs & Hauptman, 1975;Cascarano, Giacovazzo & Viterbo, 1987) relies on the expectation that some quartet invariants are negative.…”

Section: Violation Of the Positivity Condition: Its Effect On Tripletmentioning

confidence: 99%

Triplet and quartet relationships and the `positivity' postulate

Altomare¹,

Giacovazzo²,

Guagliardi³

et al. 1994

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311For the cytochrome css0.PtCl42-structure, the reported atomic coordinates were used to calculate normalized structure factors and form the sets of six TPSI for the 676 Friedel pairs with ]EI---1.5 in the resolution range 8 >_ d___ 2 A at Cr Ka and Cu Ka wavelengths. A typical result for the six true and estimated TPSI for a strong ]E I -3 reflection is illustrated in Fig. 4 and the essentially perfect behavior on the average over the 676 Friedel pairs is shown in Fig. 5. Over the 676 Friedel pairs, the percentage incidences of true TPSI with negative sign were 8.3, 8.1, 11.2, 8.0, 8.0 and 10.9% for the TPSI types to~ to 096, respectively. The scatter in the correlation between the true and estimated mixedwavelength TPSI is illustrated in Figs. 6 and 7, which show that the range of true TPSI values is considerably wider than the range of the estimated values.The incidence of true TPSI with negative sign and the scatter between the true and estimated TPSI magnitudes are, unsurprisingly, worse for the protein than for the small molecule. Still, the incidence of negative TPSI is only -10% for the protein Friedel pairs with ]E I >_ 1.5 and the errors in the estimated TPSI magnitudes are no worse for the mixedwavelength estimates (4.3) and (4.4) than for the single-wavelength estimates (4.1) and (4.2). This indicates that the simple extension of the probabilistic TPSI-estimation theory from the one-wavelength case to the two-wavelength case, as expressed in the results (4.1)-(4.4), should be practically useful for structure determinations with two-wavelength anomalous-dispersion data.We are grateful for the support of our work by grants nos. GM46733 and GM34073 from the National Institutes of Health. AbstractPositivity and atomicity are traditionally considered as basic conditions for direct methods. While the role of atomicity is well understood, the theoretical and practical consequences for a direct-methods procedure generated by the violation of the positivity criterion are not quite clear. Triplet and quartet