Electrons radial electron density function

FIGURE 2.4 Graphs of radial electron density function for Is, 2s, 2p, 3s, 3p, and 3d orbitals. 

Let pj be a vector from the origin of the m,n,p unit cell to the nucleus of the/th atom in the unit cell and let p be a vector from the nucleus to a point in its electron cloud. Let (/r(p) be the radial electron density function of the atom. The vector r extends from the crystal origin to p so that r = Tm p + py + p as shown in Figure 6.8. 

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). 

In addition to this, average properties like (r > or (/> ) play a special role in the formulation of bounds or approximations to different properties like the kinetic energy [4,5], the average of the radial and momentum densities [6,7] and p(0) itself [8,9,10] they also are the basic information required for the application of bounds to the radial electron density p(r), the momentum one density y(p), the form factor and related functions [11,12,13], Moreover they are required as input in some applications of the Maximum-entropy principle to modelize the electron radial and momentum densities [14,15],... 

The probability of finding the electron in the ground state of the hydrogen atom between radii r and r + dr is given by D(r)Ar, where D(r) is the radial probability density function shown in Figure 4.5. The most probable distance of the electron from the nucleus is found by locating the maximum in D(r) (see Problem 4.12 below). It should come as no surprise to discover that this maximum occurs at the value r = ao, the Bohr radius. 

The radial probability density function for the electron in the ground state of the hydrogen atom takes the form D(r) = Nr2e 2r a°, where N is a constant. 

Consider the radial probability density function, D(r), for the ground state of the hydrogen atom. This function describes the probability per unit length of finding an electron at a radial distance between r and r + dr (see Figure 6.5). 

Figure 7.3 Oxygen-oxygen radial density distribution for liquid water. The smooth curve is the experimental result of Head-Gordon and Hura (2002) the dots are the data, collected in bins of width 0.05 A, from ab initio molecular dynamics utilizing the rPBE electron density functional. The estimated temperature is 314 K for the latter case, slightly higher than that of the experiments at 300 K. See Asthagiri et al. (2003c).

For any pD-model p 3), we can develop descriptors of variable dimensionality d. Examples of zero-dimensional descriptors are single numbers such as the radius of gyrationio (used for OD, ID, and 2D models) or the molecular volume 1 (used for 2D models). One-dimensional descriptors such as radial distribution functions or knot polynomials are used in OD and ID models, respectively. Two-dimensional descriptors include distance maps and Rama-chandran torsional-angle maps for some OD and ID models. Similarly, molecular graphs (2D descriptors) can be associated with ID models (contour lines), 2D models (molecular surfaces), or 3D models (e.g., the entire electron density function). Shape descriptors of higher dimensionality can also be constructed. 

Figure 2 displays the SAXS-intensities Io(q) calculated for the radial electron density shown in Fig. 1. Parameter of the curves is the contrast p - Pm expressed as the number of excess electrons per nm. The isoscattering points are clearly visible. Furthermore, the calculation shows that forward scattering for a mono-disperse particle will vanish at zero contrast in accordance with the above deductions. As a consequence of this, the radius of gyration will increase or decrease rapidly as function of contrast in the vicinity of the match point (see below).

Equation(19) demonstrates that Rg of a composite particle diverges at the match point and that may become negative as well. This is shown in Fig. 5 for concentric monodisperse core-shell particles the radial electron density of which is given by Fig. 1 whereas the scattering function has already been discussed in conjunction with Fig. 2. Polydispersity of contrast tends to smear out this feature but it should be kept in mind that may change markedly when conducting measurements in the immediate vicinity of the match point. 

Figure 10.3 The simulated radial electron-density profiles Ap(r) (a), pair-distance distribution function p(r) (b), and the scattering curves l q) (c) for a monodisperse spherical particle having (i) a homogeneous electron-density distribution (circle), (ii) the...

The capability of combined nanoscale spatial and millisecond time resolution provided by SAXS is clearly revealed by a study involving the absorption of bovine serum albumin (BSA) onto spherical polyelectrolyte brushes (SPB). The experiment also highlighted the requirement of an advanced modeling capability for the complete exploration of the time-resolved SAXS data. The quantity of absorbed protein per brush as a function of time was provided from the radial electron density profile of SPB, which has been previously derived from the time-resolved SAXS intensities. Furthermore, an unexpected subdiffusive motion of proteins in the tethered polyelectrolyte brushes has been revealed. A quantitative explanation of this sub-diffusive mode can be approached in terms of a simple model involving direct motions of proteins enclosed in the effective interaction potential of the polyelectrolyte chains. 

The discussion of this technique has been kept short because the diffraction spectra are very similar to Debye - Scherrer patterns the method is becoming very important for molecular structure analysis. An electron beam in a high vacuum (0.1-10 Pa) collides with a molecular beam, and the electrons are diffracted by the molecules. The film reveals washed-out Debye-Scherrer rings, from which a radial electron density distribution function for the molecule can be derived. Together with spectroscopic data, the distribution makes it possible to infer the molecular structure 129]. 

Leonard (1977) traced the transformation sequence kaolinite-metakaolinite-spinel-muUite by studying the radial electron density (RED) distribution of kaolinite and its transition products as a function of temperature, and correlated the results with chemical information obtained with X-ray fluorescence spectroscopy. While the presence of some Si-Al spinel could not completely be ruled out, most of the A1 atoms were found in the configuration of y-Al203 beyond a firing temperature of 900°C. This assumption was corroborated by Sonuparlak et al. (1987), who showed by transmission electron microscopy (TEM) in conjunction with energy-dispersive X-ray (EDX) spectroscopy that the spinel phase formed at 980 °C, indicated by a strong exothermic DTA peak, contains <10 mass% Si02, if any. 

Appendix C Radial Electron Density Distribution (RED) Function 517... 

As an example. Figure C.2 displays the radial electron density (RED) curve p(r) of silica glass as a function of the distance r. 

Electron density function. In order to carry out a self-consistent eneigy band calculation, it is necessary to calculate the electron density function associated with a symmetrized relativistic APW function. For the Bloch state k. A), the spherically averaged radial density function within the j-type APW sphere is given by... 

The magnitude and shape of such a mean-field potential is shown below [21] in figure B3.1.4 for the two 1 s electrons of a beryllium atom. The Be nucleus is at the origin, and one electron is held fixed 0.13 A from the nucleus, the maximum of the Is orbital s radial probability density. The Coulomb potential experienced by the second electron is then a function of the second electron s position along the v-axis (coimecting the Be nucleus and the first electron) and its distance perpendicular to the v-axis. For simplicity, this second electron... 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

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