Radial Distribution Function Method

Korsunsky, V.I. (2000) The investigation of structure of heavy metal clusters and polynuclear complexes in powder samples with the radial distribution function method. Coord. Chem. Rev., 199, 55. 

The Investigation of Heavy Metal Cluster Structures in Powder Samples by the Radial Distribution Function Method... 

The coarse-graining approach is commonly used for thermodynamic properties whereas the systematic or random sampling methods are appropriate for static structural properties such as the radial distribution function. 

A very important aspect of both these methods is the means to obtain radial distribution functions. Radial distribution functions are the best description of liquid structure at the molecular level. This is because they reflect the statistical nature of liquids. Radial distribution functions also provide the interface between these simulations and statistical mechanics. 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

Carbon Tetrachloride.—By the usual visual method and by other methods involving microphotometer records, we have assigned8 to the carbon tetrachloride molecule the value 1.760 0.005 A. for the C-Cl distance, a value supported by other recent work.9 The radial distribution function for this molecule calculated by Equation 6, using the ten terms for which data are given in Table I, is shown in Fig. 1. 

The power of X-ray methods can be extended to investigate the local structure on a scale of a few angstroms by means of the analysis of the fine structure and the radial distribution function. 

In this paper we have endeavored to present a review of some characterization methods of metal nanoclusters, focusing, among the extremely vast array of methods and techniques, on two of them, XRD and TEM, on which we have direct experience, and emphasizing also some recent developments, like the radial distribution function in XRD and EH in TEM. 

Fig. 5.12 Normalized radial distribution function for the neutral Mn atom (3d 4s as calculated by the UHF method...

In structure matching methods, potentials between the CG sites are determined by fitting structural properties, typically radial distribution functions (RDF), obtained from MD employing the CG potential (CG-MD), to those of the original atomistic system. This is often achieved by either of two closely related methods, Inverse Monte Carlo [12-15] and Boltzmann Inversion [5, 16-22], Both of these methods refine the CG potentials iteratively such that the RDF obtained from the CG-MD approaches the corresponding RDF from an atomistic MD simulation. 

Both the inverse Monte Carlo and iterative Boltzmann inversion methods are semi-automatic since the radial distribution function needs to be re-evaluated at... 

In amorphous solids there is a considerable disorder and it is impossible to give a description of their structure comparable to that applicable to crystals. In a crystal indeed the identification of all the atoms in the unit cell, at least in principle, is possible with a precise determination of their coordinates. For a glass, only a statistical description may be obtained to this end different experimental techniques are useful and often complementary to each other. Especially important are the methods based on diffraction experiments only these will be briefly mentioned here. The diffraction pattern of an amorphous alloy does not show sharp diffraction peaks as for crystalline materials but only a few broadened peaks. Much more limited information can thus be extracted and only a statistical description of the structure may be obtained. The so-called radial distribution function is defined as ... 

The second category of methods uses a more general approach, based on fundamental concepts in statistical mechanics of the liquid state. As mentioned above, the Hwang and Freed theory (138) and the work of Ayant et al. (139) allow for the presence of intermolecular forces by including in the formulation the radial distribution function, g(r), of the nuclear spins with respect to the electron spins. The radial distribution function is related to the effective interaction potential, V(r), or the potential of mean force, W(r), between the spin-carrying particles through the relation (138,139) ... 

The group in Grenoble has used the radial distribution function approach in a series of papers on intermolecular relaxation. We wish to mention in particular some of their papers from the 1990s, where the radial distribution functions were obtained through different approximate methods and a relatively simple description of the electron spin relaxation was applied (150-154). This work has also been reviewed (155,156). In a recent communication from the same group, the improved description of the electron spin relaxation in Gd(III) complexes (120,121) was included in the model and applied for... 

Scanning Tunneling Microscopy. One way to get estimates for lateral interactions is STM. Using STM one can determine the statistical distribution of the adsorbates over the surface. This distribution can be converted into a radial distribution function, which in turn can be converted into effective lateral interactions. The accuracy of the value of a certain lateral interaction depends on the number of times it has been observed on the surface. Strongly repulsive interactions (larger than a few times k T) can therefore not be determined using this method. 

One has to be able to image the exact position of each adsorbate to obtain the radial distribution function. This requires a low adsorbate mobility, since a full scan typically takes one minute to do. This method is therefore mainly applied to atoms on low-temperature surfaces. The consequence of keeping the temperature low to suppress surface mobility is that the absolute value of the repulsive interactions that can be determined is also small, typically less than 10 kJ/mol at room temperature. 

A number of approximate integral equations for the radial distribution function g(r) of fluids have been proposed in recent years. Two particularly useful approximations are the Percus-Yevick (PY)1,2 and the Convolution Hypernetted Chain (CHNC)3-4 equations. In this paper an efficient numerical method of solving these equations is described and the results obtained bv applying the method to the PY equation are discussed. A later paper will describe the behavior of the... 

Studies of solvent structure are usually carried out by analyzing radial distribution functions that are obtained by X-ray or neutron diffraction methods. Monte Carlo (MC) or molecular dynamics (MD) calculations are also used. Studies of the structure of lion-aqueous and mixed solvents are not extensive yet but some of the results have been reviewed. Pure and mixed solvents included in the reviews... 

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