Determination of the Bonding and Valence Distribution in Inorganic Solids by the Maximum Entropy Method

1998 ◽  
Vol 54 (3) ◽  
pp. 221-230 ◽  
Author(s):  
G. H. Rao ◽  
I. D. Brown
Keyword(s):  
Maximum Entropy ◽  
Bond Graph ◽  
Entropy Method ◽  
Missing Information ◽  
Bond Graphs ◽  
Acta Cryst ◽  
Coordination Numbers ◽  
Inorganic Solids ◽  
Cation Coordination

The distribution of valence among the bonds in the bond graph of an inorganic compound is used to calculate an `entropy'. We show that the distribution of valence that maximizes this entropy (ME) is similar, but not identical, to that obtained using the equal-valence rule (EVR) proposed by Brown [Acta Cryst. (1977), B33, 1305–1310]. Since the ME solutions are maximally non-committal with regard to missing information, they give better predictions of the observed valence distributions than the EVR solutions when lattice constraints or electronic anisotropies are present, but worse predictions when these effects are absent. Since valences calculated using ME are necessarily positive, they give significantly better predictions in cases where EVR predicts a negative bond valence. In the absence of electronic distortions the observed bond graph is either the graph with the highest maximum entropy or it has an entropy within 1% of this value. Since the entropy depends on the oxidation states of the atoms, compounds with the same stoichiometry and cation coordination numbers but different atomic valences may adopt different bond graphs and hence different structures.

Modulation functions of incommensurately modulated Cr2P2O7 studied by the maximum entropy method (MEM)

2010 ◽  
Vol 66 (2) ◽  
pp. 130-140 ◽  
Author(s):  
Liang Li ◽  
Andreas Schönleber ◽  
Sander van Smaalen
Keyword(s):  
Maximum Entropy ◽  
Diffraction Data ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Chromium Atom ◽  
Structure Model ◽  
X Ray Diffraction ◽  
Acta Cryst ◽  
X Ray

The maximum entropy method (MEM) has been used to determine electron density in superspace of incommensurately modulated chromium pyrophosphate from X-ray diffraction data measured by Palatinus et al. [(2006), Acta Cryst. B62, 556–566]. Chromium pyrophosphate, Cr2P2O7, contains ordered regions (83% of the volume) and regions with disorder. Analysis of the MEM density has allowed the determination of the displacive modulation functions within ordered regions. The disordered regions can be described as the alternate occupation of two conformations of the pyrophosphate group and two positions of the chromium atom, with occupational probabilities that depend continuously on the phase of modulation t. A structure model based on the interpretation of the MEM density provides a fit to the diffraction data of the same quality as the model given by Palatinus et al. (2006). The failure to find a model that better fits the data is attributed to the intrinsic inaccuracy of ∼ 0.01 Å for positions derived from the MEM and to the difficulties in constructing an appropriate model for the anharmonic ADPs and their modulation functions from electron densities.

scholarly journals Determination of the time origin by 
the maximum entropy method in 
time-domain terahertz emission spectroscopy

2010 ◽  
Vol 18 (15) ◽  
pp. 15853 ◽  
Author(s):  
Takeya Unuma ◽  
Yusuke Ino ◽  
Makoto Kuwata-Gonokami ◽  
Erik M. Vartiainen ◽  
Kai-Erik Peiponen ◽  
...  
Keyword(s):  
Maximum Entropy ◽  
Time Domain ◽  
Emission Spectroscopy ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Terahertz Emission ◽  
Terahertz Emission Spectroscopy ◽  
Time Origin

Improved maximum entropy method for the analysis of fluorescence spectroscopy data: evaluating zero-time shift and assessing its effect on the determination of fluorescence lifetimes

The Analyst ◽  
2015 ◽  
Vol 140 (24) ◽  
pp. 8138-8147 ◽  
Author(s):  
Rosario Esposito ◽  
Giuseppe Mensitieri ◽  
Sergio de Nicola
Keyword(s):  
Maximum Entropy ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Lifetime Distribution ◽  
Time Shift ◽  
Fluorescence Lifetimes ◽  
Time Resolved Fluorescence ◽  
Time Resolved ◽  
Zero Time

A new algorithm based on the Maximum Entropy Method (MEM) is proposed for recovering the lifetime distribution and the zero-time shift from experimental time-resolved fluorescence decays.

Application of the maximum entropy method to electron density determination

1985 ◽  
Vol 18 (6) ◽  
pp. 442-445 ◽  
Author(s):  
W. Wei
Keyword(s):  
Maximum Entropy ◽  
Electron Density ◽  
Entropy Method ◽  
X Ray Diffraction ◽  
Resolution Electron ◽  
Density Map ◽  
Electron Density Map ◽  
Principle Of Maximum ◽  
Better Than

The principle of maximum entropy is adopted to derive a procedure for obtaining the electron density distribution in crystals from incomplete X-ray diffraction data. This method was applied to cementite and the result proved to be better than the conventional Fourier inversion in resolution as well as in the absence of ripples. The potential advantages of this method are: (1) the amount of subjective judgment imposed on unavailable data is significantly limited, and (2) the result of this method is consistent with the known information and maximally noncommittal with regard to the unknowns. It is shown that the method is especially well suited to the problem of the determination of a high-resolution electron density map from insufficient experimental data.

Electron densities by the maximum entropy method (MEM) for various types of prior densities: a case study on three amino acids and a tripeptide

2013 ◽  
Vol 69 (2) ◽  
pp. 203-213 ◽  
Author(s):  
Siriyara Jagannatha Prathapa ◽  
Swastik Mondal ◽  
Sander van Smaalen
Keyword(s):  
Maximum Entropy ◽  
Dynamic Model ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Topological Descriptors ◽  
Acta Cryst ◽  
Density Maps ◽  
Large Spread ◽  
Better Than

Dynamic model densities according to Mondalet al.[(2012),Acta Cryst.A68, 568–581] are presented for independent atom models (IAM), IAMs after high-order refinements (IAM-HO), invariom (INV) models and multipole (MP) models of α-glycine, DL-serine, L-alanine and Ala–Tyr–Ala atT≃ 20 K. Each dynamic model density is used as prior in the calculation of electron density according to the maximum entropy method (MEM). We show that at the bond-critical points (BCPs) of covalent C—C and C—N bonds the IAM-HO and INV priors produce reliable MEM density maps, including reliable values for the density and its Laplacian. The agreement between these MEM density maps and dynamic MP density maps is less good for polar C—O bonds, which is explained by the large spread of values of topological descriptors of C—O bonds in static MP densities. The density and Laplacian at BCPs of hydrogen bonds have similar values in MEM density maps obtained with all four kinds of prior densities. This feature is related to the smaller spatial variation of the densities in these regions, as expressed by small magnitudes of the Laplacians and the densities. It is concluded that the use of the IAM-HO prior instead of the IAM prior leads to improved MEM density maps. This observation shows interesting parallels to MP refinements, where the use of the IAM-HO as an initial model is the accepted procedure for solving MP parameters. A deconvolution of thermal motion and static density that is better than the deconvolution of the IAM appears to be necessary in order to arrive at the best MP models as well as at the best MEM densities.

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