Radial distribution model

Predictions of the USNRDL radial distribution model (5) have been added to Figure 10 by Freiling (6) to compare this model with the... 

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

Figure A3.6.13. Density dependence of die photolytic cage effect of iodine in compressed liquid n-pentane (circles), n-hexane (triangles), and n-heptane (squares) [38], The solid curves represent calculations using the diffusion model [37], the dotted and dashed curves are from static caging models using Camahan-Starling packing fractions and calculated radial distribution fiinctions, respectively [38],...

Models with 32 Radial Distribution Function Values and Eight Additional Descriptors... 

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... 

FIGURE 3.1. The oxygen-oxygen radial distribution function of water. The dotted curve represents the experimental result and the other curves correspond to the results calculated with the different models considered in Ref. 10. 

It is seen that the calculation of a radial distribution curve is closely similar to the calculation of a simplified theoretical intensity curve for the usual visual method, the summation being over the rings seen on the photograph instead of over the interatomic distances in the assumed model but whereas in the usual treatment the calculation may need to be repeated for many models, a single curve only is required for the new method. 

It is pointed out that the radial distribution method is particularly satisfying in that it leads directly to the values of the important interatomic distances in the molecule, thus eliminating many possible models and usually limiting the molecule to the structures represented by small ranges of values of the structural parameters. 

The six-term radial distribution function (Fig. 1) has a broad peak at about 1.24 A., a sharper peak at 2.21 A., and another at 3.32 A. These are compatible with the models discussed below, but provide no basis for choice among them. 

Model A cannot be eliminated definitely by the photographs there are, however, some points which make this model improbable. From the curve for this model the first minimum would be expected to be at least as well pronounced as the second minimum, whereas on the photographs the first minimum is not very well defined. That the qualitative appearance of the photographs supports model C rather than model A is further shown by the fact that the photographs resemble those of methyl nitrate more closely than those of carbon tetrafluoride. Some evidence is also provided by the radial distribution curve (Fig. 1), the first peak being displaced by 0.15 A. from the position expected for it for model A. For these reasons and the additional reason that it is difficult to correlate the tetrahedral configuration with an electronic structure involving only completed octets, we consider model A not to be satisfactory.7... 

The radial distribution curves, shown in Fig. 5 (curves E and F), support model A rather than... 

The photographs obtained for pyridine and py-razine are so closely similar to those of benzene as to leave no doubt that the three molecules have nearly identical structures. The radial distribution curve for pyridine (Fig. 1) calculated from the data given in Table II has well-defined peaks at 1.38, 2.39 (= V3 X 1.38), and 2.76 (= 2 X 1.38) A. The sharpness of the 2.39 peak indicates that the six meta distances in the ring are nearly equal. The calculated intensity curve for the model with C-C = 1.39 A., C-N = 1.33 A., C-H = 1.08 A., the angle C-N-C = 119° and the angles C-C-C = 121°, and having nearly equal meta and para distances, is shown in Fig. 3. The comparison of s0 and s values for this model (Table II) leads to the values C-H =... 

The intensity curves I, II, III, and IV of Fig. 4 are calculated for coplanar trans models with C-H = 1.06 A., the angle H - C=C = 115°, and the angle H—C—H = 109.5°. Although these hydrogen parameters are so chosen as to agree as well as possible with minor peaks of the radial distribution function, no great reliance can be placed on them, and indeed it is likely that for this molecule the C-H bond distance is 1.09 A. The models have the following additional parameters... 

The average of the carbon-carbon bond distances has for each model approximately the radial distribution value 1.38 A. and the angle C=C—C is... 

Figure 9. Data reduction and data analysis in EXAFS spectroscopy. (A) EXAFS spectrum x(k) versus k after background removal. (B) The solid curve is the weighted EXAFS spectrum k3x(k) versus k (after multiplying (k) by k3). The dashed curve represents an attempt to fit the data with a two-distance model by the curve-fitting (CF) technique. (C) Fourier transformation (FT) of the weighted EXAFS spectrum in momentum (k) space into the radial distribution function p3(r ) versus r in distance space. The dashed curve is the window function used to filter the major peak in Fourier filtering (FF). (D) Fourier-filtered EXAFS spectrum k3x (k) versus k (solid curve) of the major peak in (C) after back-transforming into k space. The dashed curve attempts to fit the filtered data with a single-distance model. (From Ref. 25, with permission.)...

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