The joint probability distribution function of structure factors with rational indices. V. The estimates

1999 ◽  
Vol 55 (3) ◽  
pp. 525-532
Author(s):  
Carmelo Giacovazzo ◽  
Dritan Siliqi ◽  
Cristina Fernández-Castaño ◽  
Giuliana Comunale
Keyword(s):  
Probability Distribution Function ◽  
Joint Probability ◽  
Experimental Tests ◽  
Joint Probability Distribution ◽  
Integral Index ◽  
Acta Cryst ◽  
Reliability Factor ◽  
Practical Applications ◽  
Structure Factors ◽  
Joint Probability Distribution Function

The probabilistic formulae [Giacovazzo, Siliqi & Fernández-Castaño (1999). Acta Cryst. A55, 512–524] relating standard and half-integral index reflections are modified for practical applications. The experimental tests prove the reliability of the probabilistic relationships. The approach is further developed to explore whether the moduli of the half-integral index reflections can be evaluated in the absence of phase information; i.e. by exploiting the moduli of the standard reflections only. The final formulae indicate that estimates can be obtained, even though the reliability factor is a constant.

The probability distribution function of structure factors with non-integral indices. III. The joint probability distribution in the P1¯ case

1999 ◽  
Vol 55 (2) ◽  
pp. 322-331 ◽  
Author(s):  
Carmelo Giacovazzo ◽  
Dritan Siliqi ◽  
Angela Altomare ◽  
Giovanni Luca Cascarano ◽  
Rosanna Rizzi ◽  
...  
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Function Method ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Conditional Distributions ◽  
Structure Factors ◽  
Joint Probability Distribution Function ◽  
Atomic Parameters

The joint probability distribution function method has been developed in P1¯ for reflections with rational indices. The positional atomic parameters are considered to be the primitive random variables, uniformly distributed in the interval (0, 1), while the reflection indices are kept fixed. Owing to the rationality of the indices, distributions like P(F p 1 , F p 2 ) are found to be useful for phasing purposes, where p 1 and p 2 are any pair of vectorial indices. A variety of conditional distributions like P(|F p 1 | | |F p 2 |), P(|F p 1 | |F p 2 ), P(\varphi_{{\bf p}_1}|\,|F_{{\bf p}_1}|, F_{{\bf p}_2}) are derived, which are able to estimate the modulus and phase of F p 1 given the modulus and/or phase of F p 2 . The method has been generalized to handle the joint probability distribution of any set of structure factors, i.e. the distributions P(F 1, F 2,…, F n+1), P(|F 1| |F 2,…, F n+1) and P(\varphi1| |F|1, F 2,…, F_{n+1}) have been obtained. Some practical tests prove the efficiency of the method.

The joint probability distribution function of structure factors with rational indices. VI. Cases with reduced dimensionality

1999 ◽  
Vol 55 (6) ◽  
pp. 984-990
Author(s):  
Carmelo Giacovazzo ◽  
Dritan Siliqi ◽  
Cristina Fernández-Castaño ◽  
Giovanni Luca Cascarano ◽  
Benedetta Carrozzini
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Mixed Type ◽  
Three Dimensional ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Reduced Dimensionality ◽  
Structure Factors ◽  
Joint Probability Distribution Function

The probabilistic formulas relating standard and mixed type reflections (these last show integral and half-integral indices) are derived. It is shown that probabilistic estimates can be obtained by using particular sections of the three-dimensional reciprocal space. The concept of structure invariant is extended to define the wider class of structure quasi-invariant. Their statistical behaviour is briefly discussed with the help of some practical tests.

The method of joint probability distribution functions, neighbourhoods, and representations

2013 ◽  
Author(s):  
Carmelo Giacovazzo
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Joint Probability ◽  
Distribution Functions ◽  
Joint Probability Distribution ◽  
Phase Problem ◽  
Additional Information ◽  
Structure Factors ◽  
Joint Probability Distribution Function

Wilson statistics, described in Chapter 2, aims at calculating the distribution of the structure factor P(F) ≡ P(|F|, φ) when nothing is known about the structure; the positivity and atomicity of the electron density (both promoted by the positive nature of the atomic scattering factors fj) are the only necessary assumptions. Wilson results may be synthesized as follows: . . . the modulus R = |E| is distributed according to equations (2.7) or (2.8), while no prevision is possible about φ, which is distributed with constant probability 1/(2π). . . . In other words, knowledge of the R moduli does not provide information about a phase; this agrees well with Section 3.3, according to which experimental data only allow an estimate of s.i. (and also s.s. if the algebraic form of the symmetry operators has been fixed). Let us now consider P(Fh1 , Fh2 ) ≡ P(|Fh1 |, |Fh2 |, φh1 , φh2 ), the joint probability distribution function of two structure factors. If the two structure factors are uncorrelated (i.e. no relation is expected between their moduli and between their phases), P will coincide with the product of two Wilson distributions (2.7) or (2.8), say, . . . P(Fh1 , Fh2 ) ≡ P(|Fh1 |, φh1 ) · P(|Fh2 |, φh2 ) = 1/4π2 P(|Fh1 |)P(|Fh2 |), . . . which is useless (because the two Wilson distributions are useless) for solving the phase problem; indeed, the relation does not provide any phase information. The question is now: if two structure factors are correlated, may their joint probability distribution function be used for solving the phase problem? Let us first use a simple example to show how much additional information (i.e. that is not present in the two elementary distributions) may be stored in a joint probability distribution function; then we will answer the question. Let us suppose that the human population of a village has been submitted to statistical analysis to define how weight and height are distributed.

About difference electron densities and their properties

2017 ◽  
Vol 73 (6) ◽  
pp. 460-473 ◽  
Author(s):  
Maria Cristina Burla ◽  
Benedetta Carrozzini ◽  
Giovanni Luca Cascarano ◽  
Carmelo Giacovazzo ◽  
Giampiero Polidori
Keyword(s):  
Crystal Structure ◽  
Electron Density ◽  
Probability Distribution Function ◽  
Strong Support ◽  
Joint Probability ◽  
The Other ◽  
Joint Probability Distribution ◽  
Electron Densities ◽  
Structure Solution ◽  
Joint Probability Distribution Function

Difference electron densities do not play a central role in modern phase refinement approaches, essentially because of the explosive success of the EDM (electron-density modification) techniques, mainly based on observed electron-density syntheses. Difference densities however have been recently rediscovered in connection with theVLD(Vive la Difference) approach, because they are a strong support for strengthening EDM approaches and forab initiocrystal structure solution. In this paper the properties of the most documented difference electron densities, here denoted asF−Fp,mF−FpandmF−DFpsyntheses, are studied. In addition, a fourth new difference synthesis, here denoted as {\overline F_q} synthesis, is proposed. It comes from the study of the same joint probability distribution function from which theVLDapproach arose. The properties of the {\overline F_q} syntheses are studied and compared with those of the other three syntheses. The results suggest that the {\overline F_q} difference may be a useful tool for making modern phase refinement procedures more efficient.

The phantom derivative method when a structure model is available: about its theoretical basis

2017 ◽  
Vol 73 (3) ◽  
pp. 218-226 ◽  
Author(s):  
Maria Cristina Burla ◽  
Giovanni Luca Cascarano ◽  
Carmelo Giacovazzo ◽  
Giampiero Polidori
Keyword(s):  
Theoretical Calculations ◽  
Joint Probability ◽  
Experimental Tests ◽  
General Distribution ◽  
Joint Probability Distribution ◽  
Random Structure ◽  
Structure Model ◽  
Target Structure ◽  
Density Maps ◽  
Structure Factors

This study clarifies why, in the phantom derivative (PhD) approach, randomly created structures can help in refining phases obtained by other methods. For this purpose the joint probability distribution of target, model, ancil and phantom derivative structure factors and its conditional distributions have been studied. Since PhD may usenphantom derivatives, withn≥ 1, a more general distribution taking into account all the ancil and derivative structure factors has been considered, from which the conditional distribution of the target phase has been derived. The corresponding conclusive formula contains two components. The first is the classical Srinivasan & Ramachandran term, relating the phases of the target structure with the model phases. The second arises from the combination of two correlations: that between model and derivative (the first is a component of the second) and that between derivative and target. The second component mathematically codifies the information on the target phase arising from model and derivative electron-density maps. The result is new, and explains why a random structure, uncorrelated with the target structure, adds useful information on the target phases, provided a model structure is known. Some experimental tests aimed at checking if the second component really provides information on φ (the target phase) were performed; the favourable results confirm the correctness of the theoretical calculations and of the corresponding analysis.

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