Pair distribution function methods

It is also possible to prepare crystalline electrides in which a trapped electron acts in effect as the anion. The bnUc of the excess electron density in electrides resides in the X-ray empty cavities and in the intercoimecting chaimels. Stmctures of electri-dides [Li(2,l,l-crypt)]+ e [K(2,2,2-crypt)]+ e , [Rb(2,2,2-crypt)]+ e, [Cs(18-crown-6)2]+ e, [Cs(15-crown-5)2]" e and mixed-sandwich electride [Cs(18-crown-6)(15-crown-5)+e ]6 18-crown-6 are known. Silica-zeolites with pore diameters of vA have been used to prepare silica-based electrides. The potassium species contains weakly bound electron pairs which appear to be delocalized, whereas the cesium species have optical and magnetic properties indicative of electron locahzation in cavities with little interaction between the electrons or between them and the cation. The structural model of the stable cesium electride synthesized by intercalating cesium in zeohte ITQ-4 has been coirfirmed by the atomic pair distribution function (PDF) analysis. The synthetic methods, structures, spectroscopic properties, and magnetic behavior of some electrides have been reviewed. Theoretical study on structural and electronic properties of inorganic electrides has also been addressed recently. ... 

The driving force behind the rapid development of powder diffraction methods over the past 10 years is the increasing need for structural characterization of materials that are only available as powders. Examples are zeolite catalysts, magnets, metal hydrides, ceramics, battery and fuel cell electrodes, piezo- and ferroelectrics, and more recently pharmaceuticals and organic and molecular materials as well as biominerals. The emergence of nanoscience as an interdisciplinary research area will further increase the need for powder diffraction, pair-distribution function (PDF) analysis of powder diffraction pattern allows the refinement of structural models regardless of the crystalline quality of the sample and is therefore a very powerful structural characterization tool for nanomaterials and disordered complex materials. 

In spite of the great success of the computer simulation methods in the determination of the microscopic properties of the solutions, the capacity of the traditional MD and MC simulations is always limited by the choice of the suitable potential functions to describe the interatomic interactions. The potentials are most often checked by comparison of the structural properties calculated from the simulation with those determined experimentally. The reverse Monte Carlo (RMC) method, developed by McGreevy and Pusztai [41] does not rely upon knowledge of any interaction potential, instead it generates a large set of atomic configurations on the condition that the difference between the experimental and calculated structure functions (or pair-distribution functions) should be minimum. The same structural... 

The modified potential function method is the implicit way to include the polarizability in the calculation by modifying the potential function based on the physical properties in solid state so that it reproduces the pair distribution function obtained by the X-ray and neutron diffraction measurements in the liquid state. 

The use of difference methods offers a means whereby a detailed picture of ionic hydration can be obtained 22). For neutron diffraction, the first-order isotopic difference method (see Section III,A) provides information on ionic hydration in terms of a linear combination of weighted ion-water and ion-ion pair distribution functions. Since the ion-water terms dominate this combination, the first-order difference method offers a direct way of establishing the structure of the aquaion. In cases for which counterion effects are known to occur, as, for example, in aqueous solutions of Cu + or Zn +, it is necessary to proceed to a second difference to obtain, for example, gMX and thereby possess a detailed knowledge of both the aquaion-water and the aquaion-coun-terion structure. 

It can be seen from Eq. (18) that the improvement for the potentials Ujj(r) depends only on the difference between the predicted and experimentally observed pair distribution functions instead of any properties related to the reference state. Therefore, the iterative method does not face the reference state problem encountered by traditional mean-force/knowledge-based scoring functions. 

The accuracy of the analysis presented in this paper is determined by the validity of two key approximations (1) the description of the energy transfer dynamics by the first order cumulant expansion method, (2) the use of a Gaussian chromophore pair distribution function. Although originally developed for, and successfully applied to, the problem of energy transfer in disordered infinite volume systems, the cumulant method can be modified to provide a highly accurate description of energy transfer in finite volume systems such as polymer coils (2 ). ... 

D17.6 The pair distribution function is a statistical method for studying the complex properties of liquids. 

With m atomic species, there are m(m + l)/2 partial pair distribution functions gap(r) that are distinct from each other. When only a single intensity function I(q) is available from experiment, no method of ingenious analysis can lead to determination of all these separate partial pair distribution functions from it. Different and independent intensity functions I(q) may be obtained experimentally when measurements are made, for example, with samples prepared with some of their atoms replaced by isotopes. When a sufficient number of such independent intensity functions is available, it is then possible to have all the partial pair distribution functions gap(r) individually determined, as will be elaborated on shortly. When only a single intensity function is available from x-ray or neutron scattering, however, what can be obtained from a Fourier inversion of the interference function is some type of weighted average of all gap (r) functions. The exact relationship between such an averaged function and gap(r)s is as follows. 

The first method, is based on the use of pair potentieds related to atoms or groups of atoms of the solvent S and the solute M, supplemented by a simplified description of the corresponding pairs distribution functions. The pairs potentials are independent on solute charge distribution they are not involved in the QM description of the system, but they affect only the total free energy value. 

The differences in the CMC and EMC predictions can be traced to the different pair probability densities estimated by these methods from the given time series. In fig. 9.4, we show a contour plot of the pair distribution function p Xi, Xg) as calculated by a semi-nonparametric (SNP) method [10] with the time-series simulations for these two species superimposed. In comparison, we show in fig. 9.5 a Gaussian pair probability distribution, as is consistent with CMC, with the same means and variances as those of the EMC distribution. The deviations of the EMC from the Gaussian distribution show that higher than second moments contribute. Since the information entropy for... 

Another approach that is gaining acceptance in the catalysis world is the use of total scattering formalisms and, particularly, the analysis of the nanoscale structural order from the atomic pair distribution function (PDF). Although this approach has a long history, the advent of synchrotron and novel detectors has provided a powerful tool to analyse nanoscale. The PDF method can yield precise short and long range structural and size information provided that special care is applied to the measurement and handling of data. The atomic PDF, G(g), is defined as ... 

A fundamental approach to liquids is provided by the integral equation methods (sometimes called distribution function methods), initiated by Kirkwood and Yvon in the 1930s. As we shall show below, one starts by writing down an exact equation for the molecular distribution function of interest, usually the pair function, and then introduces one or more approximations to solve the problem. These approximations are often motivated by considerations of mathematical simplicity, so that their validity depends on a posteriori agreement with computer simulation or experiment. The theories in question, called YBG (Yvon-Bom-Green), PY (Percus-Yevick), and the HNC (hypemetted chain) approximation, provide the distribution functions directly, and are thus applicable to a wide variety of properties. 

Dixon, M. and P Hutchinson. 1977. Method for extrapolation of pair distribution functions. Molecular Physics. 33, 1663. 

The second line of inquiry alluded to was the use of modem computational power, both hardware and software, for the evaluation of pair distribution functions. When the Kirkwood-Buff paper was published, the use of computers for this kind of scientific computation was in its infancy. Indeed, one can say that it was in its prenatal stage. It is difficult to put a date on the time when computers became powerful enough to compute pair correlation functions and, consequently, KB integrals with sufficient accuracy for application to real systems. They have certainly reached that stage at the time of the writing of these words. The computational method of choice in carrying out these calculations is the molecular dynamics method. Since this kind of calculation is discussed in detail in several of the later chapters of this work, we eschew discussion here. 

---