The application of evolution strategies to disordered structures

1999 ◽  
Vol 32 (5) ◽  
pp. 902-910 ◽  
Author(s):  
Karsten Knorr ◽  
Fritz Mädler
Keyword(s):  
High Pressure ◽  
Maximum Entropy ◽  
Electron Density ◽  
Structural Model ◽  
Evolution Strategies ◽  
Qualitative Model ◽  
Density Distributions ◽  
Disordered Structures ◽  
Structure Factors ◽  
Complex Anions

Evolution strategies are applied to refine structural fragments, like molecules or complex anions, of orientationally disordered crystals. Optimal geometric embedding of the fragments into electron density distributions, resulting from maximum-entropy (ME) reconstructions, are performed. The evolution paradigm is found to be also applicable for the refinement against structure factors, for which the structural model is carefully selected from the ME densities. Suitably modified, the method is used successfully to compute reorientation pathways and to predict disordered high-pressure configurations in a non-classical qualitative model.

The maximum entropy distribution consistent with observed structure amplitudes

1989 ◽  
Vol 45 (2) ◽  
pp. 200-203 ◽  
Author(s):  
E. Prince
Keyword(s):  
Maximum Entropy ◽  
Electron Density ◽  
Direct Methods ◽  
Positive Definite ◽  
Sufficient Information ◽  
Large Set ◽  
Unique Structure ◽  
Structure Factors ◽  
The Matrix ◽  
Maximum Entropy Distribution

It is shown that an electron density distribution of the formρk= exp [Σfj(rk)xj] has maximum entropy under the constraint that the expected values of a set of functions,fj(r), are constant. For a Fourier map the functionsfj(r) are the magnitudes of the structure factors for a set of reflectionshjincludingF(000). The values of the parametersxjfor which [(exp (2πihj. r))] = [Fobs(hj)[ for an arbitrarily large set of reflections may be found by an iterative algorithm in whichxi+ 1=xi+Hi-1Δi, where the matrixHis positive definite. Because the distributionρ(r) is everywhere positive, if non-negativity of electron density is sufficient information to determine a unique structure by direct methods, it follows that the maximum entropy procedure must lead to the same unique structure. Maximum entropy is thus an efficient way of expressing the phase implications of a large set of structure amplitudes.

Extracting charge density distributions from diffraction data: a model study on urea

2000 ◽  
Vol 56 (1) ◽  
pp. 118-123 ◽  
Author(s):  
R. Y. de Vries ◽  
D. Feil ◽  
V. G. Tsirelson
Keyword(s):  
Density Distribution ◽  
Electron Density ◽  
Diffraction Data ◽  
Experimental Situation ◽  
Original Structure ◽  
Random Errors ◽  
Density Distributions ◽  
Structure Factors ◽  
Multipole Refinement ◽  
Interaction Density

The quality of the extraction of electron density distributions by means of a multipole refinement method is investigated. Structure factors of the urea crystal have been obtained from an electron density distribution (EDD) resulting from a density function calculation with the CRYSTAL95 package. To account for the thermal motion of the atoms, the stockholder-partioned densities of the atoms have been convoluted with thermal smearing functions, which were obtained from a neutron diffraction experiment. A POP multipole refinement yielded a good fit, R = 0.6%. This disagreement factor is based on magnitudes only. Comparison with the original structure factors gave a disagreement of 0.8% owing to differences in magnitude and phase. The fitted EDD still showed all the characteristics of the interaction density. After random errors corresponding to the experimental situation were added to the structure factors, the refinement was repeated. The fit was R = 1.1%. This time the resulting interaction density was heavily deformed. Repetition with another set of random errors from the same distribution yielded a widely different interaction density distribution. The conclusion is that interaction densities cannot be obtained from X-ray diffraction data on non-centrosymmetric crystals.

scholarly journals A chemical interpretation of protein electron density maps in the worldwide protein data bank

2019 ◽  
Author(s):  
Sen Yao ◽  
Hunter N.B. Moseley
Keyword(s):  
Protein Data Bank ◽  
Electron Density ◽  
Structural Model ◽  
Data Bank ◽  
X Ray ◽  
New Methods ◽  
Link Type ◽  
Density Maps ◽  
Structure Factors ◽  
Python Package

AbstractHigh-quality three-dimensional structural data is of great value for the functional interpretation of biomacromolecules, especially proteins; however, structural quality varies greatly across the entries in the worldwide Protein Data Bank (wwPDB). Since 2008, the wwPDB has required the inclusion of structure factors with the deposition of x-ray crystallographic structures to support the independent evaluation of structures with respect to the underlying experimental data used to derive those structures. However, interpreting the discrepancies between the structural model and its underlying electron density data is difficult, since derived electron density maps use arbitrary electron density units which are inconsistent between maps from different wwPDB entries. Therefore, we have developed a method that converts electron density values into units of electrons. With this conversion, we have developed new methods that can evaluate specific regions of an x-ray crystallographic structure with respect to a physicochemical interpretation of its corresponding electron density map. We have systematically compared all deposited x-ray crystallographic protein models in the wwPDB with their underlying electron density maps, if available, and characterized the electron density in terms of expected numbers of electrons based on the structural model. The methods generated coherent evaluation metrics throughout all PDB entries with associated electron density data, which are consistent with visualization software that would normally be used for manual quality assessment. To our knowledge, this is the first attempt to derive units of electrons directly from electron density maps without the aid of the underlying structure factors. These new metrics are biochemically-informative and can be extremely useful for filtering out low-quality structural regions from inclusion into systematic analyses that span large numbers of PDB entries. Furthermore, these new metrics will improve the ability of non-crystallographers to evaluate regions of interest within PDB entries, since only the PDB structure and the associated electron density maps are needed. These new methods are available as a well-documented Python package on GitHub and the Python Package Index under a modified Clear BSD open source license.Author summaryElectron density maps are very useful for validating the x-ray structure models in the Protein Data Bank (PDB). However, it is often daunting for non-crystallographers to use electron density maps, as it requires a lot of prior knowledge. This study provides methods that can infer chemical information solely from the electron density maps available from the PDB to interpret the electron density and electron density discrepancy values in terms of units of electrons. It also provides methods to evaluate regions of interest in terms of the number of missing or excessing electrons, so that a broader audience, such as biologists or bioinformaticians, can also make better use of the electron density information available in the PDB, especially for quality control purposes.Software and full results available athttps://github.com/MoseleyBioinformaticsLab/pdb_eda (software on GitHub)https://pypi.org/project/pdb-eda/ (software on PyPI)https://pdb-eda.readthedocs.io/en/latest/ (documentation on ReadTheDocs)https://doi.org/10.6084/m9.figshare.7994294 (code and results on FigShare)

Maximum Entropy Method Applied in the Experimental Visualization of Electron Density Distributions in BiFeO3

2015 ◽  
Vol 166 (1) ◽  
pp. 168-174 ◽  
Author(s):  
I. B. Catelani ◽  
G. S. Dias ◽  
I. A. Santos ◽  
R. Guo ◽  
A. S. Bhalla ◽  
...  
Keyword(s):  
Maximum Entropy ◽  
Electron Density ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Density Distributions

Electron-density distribution from X-ray powder data by use of profile fits and the maximum-entropy method

1990 ◽  
Vol 23 (6) ◽  
pp. 526-534 ◽  
Author(s):  
M. Sakata ◽  
R. Mori ◽  
S. Kumazawza ◽  
M. Takata ◽  
H. Toraya
Keyword(s):  
Density Distribution ◽  
Maximum Entropy ◽  
Electron Density ◽  
Electron Density Distribution ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Distribution Map ◽  
X Ray ◽  
Profile Decomposition ◽  
Structure Factors

Following the profile decomposition of CeO2 X-ray powder data into individual structure factors, the maximum-entropy method (MEM) has been used to obtain an electron-density-distribution map. In the profile decomposition process, it is impossible to avoid the problems of overlapping peaks which have the same magnitude of reciprocal vectors, such as d*(511) and d*(333), for a cubic crystal, or very severely overlapping reflections. The formalism to treat such overlapping reflections in the MEM analysis is to introduce combined structure factors. The maximum value of the scattering vector, 4π(sinθ)/λ, which was used in the present analysis is small (about 7.8 Å−1) but the resulting electron-density-distribution map is of a high quality and much superior to the conventional map. As a consequence, the ionic charge of Ce and O ions can be obtained with reasonable accuracy from the MEM density map. Furthermore, the map reveals the existence of electrons around the supposedly vacant site surrounded by eight O atoms, which is probably related to the high ionic conductivity of this substance.

Imaging of the electron density distributions of hydrogen in LiH and LiOH by maximum entropy method

1999 ◽  
Vol 60 (10) ◽  
pp. 1721-1724 ◽  
Author(s):  
Shigefumi Yamamura ◽  
Satoshi Kasahara ◽  
Masaki Takata ◽  
Yoko Sugawara ◽  
Matoto Sakata
Keyword(s):  
Maximum Entropy ◽  
Electron Density ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Density Distributions

Pressure Dependence of Electron Density Distribution of Ferroielectric KNbO3 Polymorphs by Maximum Entropy Method (MEM) Using Single Crystal Diffraction Study

2006 ◽  
Vol 987 ◽  
Author(s):  
Takamitsu Yamanaka ◽  
Taku Okada ◽  
Yuki Nakamoto ◽  
Kenji Ohi
Keyword(s):  
High Pressure ◽  
Density Distribution ◽  
Single Crystal ◽  
Maximum Entropy ◽  
Electron Density ◽  
Electron Density Distribution ◽  
Maximum Entropy Method ◽  
Crystal Structure Analysis ◽  
Entropy Method ◽  
Cubic Phase

AbstractSingle-crystal structure analysis of KNbO3 has been executed under high pressure through diamond anvil cell installed in the four-circle diffractometer using synchrotron radiation at Photon Factory, KEK in order to clarify the dielectric property. KNbO3 has three structural transitions with increasing pressure at ambient temperature: from orthorhombic structure with the space group Cm2m (Amm2) to tetragonal structure (P4mm) at about 7.0 GPa, to cubic structure (Pm3m) at about 10.0 GPa. The highest-pressure cubic phase is paraelectric, and the other two phases are ferroelectric. The dielectric changes in KNbO3 are clarified by the successive pressure-change of the electron density distribution observed by maximum entropy method (MEM) using high-pressure diffraction data. The MEM electron density maps suggest that the tetragonal phase designates the largest polarization among three polymorphs. The maps also indicate that the localization of the valence electron around the cation position is more enhanced under higher pressure.

The Electron Density Distribution in Be Metal Obtained from Synchrotron-Radiation Powder Data by the Maximum-Entropy Method

1993 ◽  
Vol 48 (1-2) ◽  
pp. 75-80 ◽  
Author(s):  
Masaki Takata ◽  
Yoshiki Kubota ◽  
Makoto Sakata
Keyword(s):  
Synchrotron Radiation ◽  
Density Distribution ◽  
Maximum Entropy ◽  
Electron Density ◽  
Electron Density Distribution ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Imaging Plate ◽  
Acta Cryst ◽  
Structure Factors

Abstract The nature of the bonding in Be metal was studied by investigating the MEM map, which is the electron density distribution obtained by the Maximum-Entropy Method. In order to avoid extinc-tion effects, 19 Bragg reflections were measured by a new powder-diffraction experiment that utilizes Synchrotron Radiation as an incident X-ray and an Imaging Plate as detector. The experiment was carried out at the Photon Factory BL6A2. In spite of the limited number of reflections used in the MEM analysis, the electron density distribution of Be was obtained accurately and reliably. The structure factors for unmeasured reflections were calculated and compared with the values observed by Larsen and Hansen [Acta Cryst. B40, 169 (1984)]. The agreement is very good. Furthermore, the MEM map revealed that Be metal forms an electronic layer in the shape of a honeycomb that is parallel to the basal plane.

A synchrotron X-ray study of the electron density in C-type rare earth oxides

1996 ◽  
Vol 52 (3) ◽  
pp. 414-422 ◽  
Author(s):  
E. N. Maslen ◽  
V. A. Streltsov ◽  
N. Ishizawa
Keyword(s):  
Heavy Metal ◽  
Rare Earth ◽  
Electron Density ◽  
Approximate Symmetry ◽  
Data Set ◽  
X Ray ◽  
Metal Atoms ◽  
Structure Factors ◽  
Deformation Electron Density ◽  
Synthetic Crystals

Structure factors for small synthetic crystals of the C-type rare earth (RE) sesquioxides Y2O3, Dy2O3 and Ho2O3 were measured with focused λ = 0.7000 (2) Å, synchrotron X-radiation, and for Ho2O3 were re-measured with an MoKα (λ = 0.71073 Å) source. Approximate symmetry in the deformation electron density (Δρ) around a RE atom with pseudo-octahedral O coordination matches the cation geometry. Interactions between heavy metal atoms have a pronounced effect on the Δρ map. The electron-density symmetry around a second RE atom is also perturbed significantly by cation–anion interactions. The compounds magnetic properties reflect this complexity. Space group Ia{\bar 3}, cubic, Z = 16, T = 293 K: Y2O3, Mr = 225.82, a = 10.5981 (7) Å, V = 1190.4 (2) Å3, Dx = 5.040 Mg m−3, μ 0.7 = 37.01 mm−1, F(000) = 1632, R = 0.067, wR = 0.067, S = 9.0 (2) for 1098 unique reflections; Dy2O3, Mr = 373.00, a = 10.6706 (7) Å, V = 1215.0 (2) Å3, Dx = 8.156 Mg m−3, μ 0.7 = 44.84 mm−1, F(000) = 2496, R = 0.056, wR = 0.051, S = 7.5 (2) for 1113 unique reflections; Ho2O3, Mr = 377.86, a = 10.606 (2) Å, V = 1193.0 (7) Å3, Dx = 8.415 Mg m−3, μ 0.7 = 48.51 mm−1 F(000) = 2528, R = 0.072, wR = 0.045, S = 9.2 (2) for 1098 unique reflections of the synchrotron data set.

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