Electron radial distribution functions

The presence of local cation ordering in Mg2Ga and MgsGa - CO3 LDHs noted in Sect. 3.3.1 has been confirmed by means of both EXAFS and by calculation of the electron radial distribution function from the Fourier transform of the diffracted X-ray intensity. In each case the gallium was found to have six magnesium ions and no galhum ions as next-nearest neighbors [39]. 

The Hartree-Fock or self-consistent field (SCF) method is a procedure for optimizing the orbital functions in the Slater determinant (9.1), so as to minimize the energy (9.4). SCF computations have been carried out for all the atoms of the periodic table, with predictions of total energies and ionization energies generally accurate in the 1-2% range. Fig. 9.2 shows the electronic radial distribution function in the argon atom, obtained from a Hartree-Fock computation. The shell structure of the electron cloud is readily apparent. 

The XRD of bulk liquid ethanol is also shown for comparison. The slight difference is observed in the s range from 10 to 30 nm. The peak of ethanol confined in P5 can be seen at s = 35 nm in the inset. This can be attributed to the specific structure of adsorbed ethanol in narrow spaces. However, these diffraction patterns cannot provide the detailed structure of adsorbed molecule. Then we transformed the structure function derived from XRD patterns into electron radial distribution function (ERDF) by Fourier transformation. 

Germanium(iv) bromide has been reinvestigated by electron diffraction. Constraining the molecule symmetry to afforded a value of 2.272(1) The electronic radial distribution functions for germanium(iv) and tin(iv) chlorides in the liquid state at 23 °C have been calculated from X-ray diffraction intensity distributions obtained by use of theta-theta reflection diffractometry. Both liquids show intermolecular effects at distances equivalent to the Cl—Cl intermolecular distance. Values of 0.9 D (Si—Cl), 1.5 D (Ge—Cl), and 2.7 D (Sn—Cl) have been derived for the bond dipole moments of these bonds in the metal(iv)... 

Fig. 111.19. Reduced intensities and differential electronic radial distribution function l> r) for 2.0 M pbcnyl-...

The corresponding so-called atomo-electronic radial distribution function (AERDF) is p(r). 

Additionally, Ohkubo and Kaneko (2001) obtained densities of adsorbed alcohols, water and CCI4 on these two activated carbon fibers, P5 and P20. These authors used the technique of electron radial distribution function (ERDF), so allowing this function to elucidate... 

Brady [5, 6] first undertook a detailed study of the structure of Te02-based glass by electron radial distribution function [RDF] methods using X-ray diffraction [XRD] analysis. Brady found that there were two well-resolved primary peaks, one at about 1.95 A and the other at about 2.75 A, and two other well-resolved secondary peaks, at 3.63 A and 4.38 A. The first two peaks correspond to the two sets of equatorial and axial Te-0 bonds. The latter two peaks overlap, and their extracted distances correspond to the two preferred sets of Te-Te distances. 

This factor is reminiscent of the radial distribution function for electron probability in an atom and the Maxwell distribution of molecular velocities in a gas, both of which pass through a maximum for similar reasons. 

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... 

The radial distribution function tells us, through P(r)f>r, the probability of finding the electron in the range of radii 8r, at a particular value of the radius, summed over all values of 0 and < >. 

FIGURE 1.32 The radial distribution function tells us the probability density for finding an electron at a given radius summed over all directions. The graph shows the radial distribution function for the 1s-, 2s-, and 3s-orbitals in hydrogen. Note how the most probable radius icorresponding to the greatest maximum) increases as n increases. 

FIGURE 1.42 The radial distribution functions for s-, p-, and cf-orbitals in the first three shells of a hydrogen atom. Note that the probability maxima for orbitals of the same shell are close to each other however, note that an electron in an ns-orbital has a higher probability of being found close to the nucleus than does an electron in an np-orbital or an nd-orbital. 

It is shown by empirical tests that the radial distribution function given by a sum of Fourier terms corresponding to the rings observed on an electron diffraction photograph of gas molecules... 

The radial distribution function Dniir) is the probability density for the electron being in a spherical shell with inner radius r and outer radius r -h dr. For the Is, 2s, and 2p states, these functions are... 

The s-states have spherical symmetry. The wave functions (probability amplitudes) associated with them depend only on the distance, r from the origin (center of the nucleus). They have no angular dependence. Functionally, they consist of a normalization coefficient, Nj times a radial distribution function. The normalization coefficient ensures that the integral of the probability amplitude from 0 to °° equals unity so the probability that the electron of interest is somewhere in the vicinity of the nucleus is unity. 

In this equation g(r) is the equilibrium radial distribution function for a pair of reactants (14), g(r)4irr2dr is the probability that the centers of the pair of reactants are separated by a distance between r and r + dr, and (r) is the (first-order) rate constant for electron transfer at the separation distance r. Intramolecular electron transfer reactions involving "floppy" bridging groups can, of course, also occur over a range of separation distances in this case a different normalizing factor is used. 

Fig. 3. Z-scaled electron-nuclear distribution functions for H, He, Li, and Ne (a) radial probability distribution D(r ) Z (b) radial density /o(ri)/Z. The curves can be identified from the fact that higher maxima correspond to higher Z.

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... 

The first satisfactory definition of crystal radius was given by Tosi (1964) In an ideal ionic crystal where every valence electron is supposed to remain localised on its parent ion, to each ion it can be associated a limit at which the wave function vanishes. The radial extension of the ion along the connection with its first neighbour can be considered as a measure of its dimension in the crystal (crystal radius). This concept is clearly displayed in figure 1.7A, in which the radial electron density distribution curves are shown for Na and Cl ions in NaCl. The nucleus of Cl is located at the origin on the abscissa axis and the nucleus of Na is positioned at the interionic distance experimentally observed for neighboring ions in NaCl. The superimposed radial density functions define an electron density minimum that limits the dimensions or crystal radii of the two ions. We also note that the radial distribution functions for the two ions in the crystal (continuous lines) are not identical to the radial distribution functions for the free ions (dashed lines). 

X-ray diffraction studies yield radial distribution function data which are dominated by the much greater scattering power of the more electron-rich oxygen atoms in water. These diffraction results tell us something about... 

The average local electrostatic potential V(r)/p(r), introduced by Pohtzer [57], led Sen and coworkers [58] to conjecture that the global maximum in V(r)/p(r) defines the location of the core-valence separation in ground-state atoms. Using this criterion, one finds N values [Eq. (3.1)] of 2.065 and 2.112 e for carbon and neon, respectively, and 10.073 e for argon, which are reasonable estimates in light of what we know about the electronic shell structure. Politzer [57] also made the significant observation that V(r)/p(r) has a maximum any time the radial distribution function D(r) = Avr pir) is found to have a minimum. 

With reference to the minima of the radial distribution function D r), SCF analyses [61] using the near-Hartree-Fock wavefunctions of dementi [64] indicate that the numbers of electrons found in the inner shell extending up to the minimum of D r) amount to = 2.054 e (Be), 2.131 (C), 2.186 (O), 2.199 (F) and 2.205 electron (Ne). The results of Smith et al. [65] bearing on the boundaries in position space that enclose the exact number given by the Aufbau principle support the idea of physical shells compatible with that principle. The maxima of D r), on the other hand, also appear to be topological features indicative of shells, their positions correlate well with the shell radii from the Bohr-Schrodinger theory of an atom... 

So, on the basis of these results, we shall keep in mind that the radial distribution function offers a vivid pictorial reference suggesting an involvement of the electronic... 

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