Radial charge density distribution

The radial charge density from the resonant orbital is displayed in fig. 15. and the number of radial nodes identify this as a 4p orbital and not the lowest 2p that one would have expected from the successful qualitative correlation of LUMOs with resonances in e-molecule scattering /18/. However, the accumulation of the radial charge density distribution at small r values is a strong reminder of the 2p type orbital density distribution /120/. This feature,... 

Figure 9.12 The radial charge density distribution of Rb+ (tf0 is a con-stant=0.532 A). Based on Hartree (1933).

Figure 1.1 Radial charge density distribution ofPr(lll). Reproduced from [13] with permission from Elsevier...

Figure 1.7 Plots of (a) the radial wave function (b) the radial probability distribution function and (c) the radial charge density function 4nr Rl( against p...

Additionally, from Fig. 29 one sees that, if, as proposed by Frost 42), a spherical gaussian function is a fair representation of the distribution of charge within an electride ion, there should he, as found by Slater 97>, a very good correlation, and in many cases practically an equality, between the atomic radii. . . and the calculated radius of maximum radial charge density in the outermost shell of the atom". 

Fig. 3. Comparison of the radial electron density distribution of Na+ and Cl- obtained from experimental results of Schoknecht (solid corves) with the corresponding results obtained from the theories of Hartree and Debye (broken curves). The distance be-ween the centres of the oppositely charged ions corresponds to ro in NaCl (c)...

Figure 15. Radial charge density plot for the resonant p-type virtual orbital for dilation angles 9 = 0.0 and 6 = 90pt (0.42 radians) in e-Be scattering. The role of optimal theta in the accumulation of electron density near the nucleus is clearly seen. In the inset, the maximum is seen to occur at rmaz — 2.5 a.u., very close to that for the rmax of the outer valence 2s orbital, seen in fig. 14- Though a cursory look at the nodal pattern identifies this as a 4P orbital, the dominant contribution to the charge density distribution is mainly of 2p-iype.

Considering the outermost atomic orbitals, the effects of relativistic corrections on one-electron binding energies and the spatial distribution of the radial charge densities are illustrated by the results displayed in Fig. 4. From the strong... 

The normalization condition for the charge density distribution now reduces to a radial integral,... 

A radial expectation value of the normalized charge density distribution function for an arbitrary function of the radius, f r), is obtainable from the general formula... 

Net atomic charges of about -0.2 at each H were calculated with an ab initio MO-SCF method [2], with the semiempirical CNDO/2 method [11], and with another semiempirical method using localized bond orbitals for Cl [12]. A lower value came from an EH calculation [3]. A radial electron density distribution was calculated within the united-atom approximation [10]. Two different dipole moments were obtained with an MO-SCF calculation (yielding also quadrupole and octupole moments) [2] and with the electron propagator theory (EPT) [13]. 

It is worthwhile to demonstrate the competition between interactions by means of a qualitative evaluation of the strengths of the various interactions. This ev iluation is based on the properties of the radieil wavefunctions Rni(r) of the 4f, 5d, 6s and 6p electrons. In fig. 1.20 the radial charge densities Rh(r) are plotted as functions of r for the 4f, 5s, 5p, 5d, 6s and 6p electrons of Ce I 4f5d6s6p. These charge distributions, which are characteristic of all lanthanides were obtained by Z.B. Goldschmidt (1972) by performing Hartree-Fock calculations. 

Fig. 7 Radial monomer density distribution for Ka = 1 and the surface charge densities cr/(12n (Tp / .kBTl)) = 3,5,10,25, and 100 (from right to left) [60]...

Fig. 8 Radial monomer density distribution Iot the charge density al( 2neP ap l(ek Tr)) = 3 and the Debye-Huckel screening parameters Kfl = 0.1,0.5,0.8,1.0, and 1.1 (from left to right) [60]...

The differences between the actinide and lanthanide metals can be rationalized by a consideration of the differences between the 4f- and 5f-electron shells [25]. In the 4f series, all the 4f electrons (added after cerium) are buried in the interior of the electron cloud. The 4f electrons are thus confined to the core of the atom, and experience relatively little interaction with electrons in the 5d shell. The maxima in the radial charge density occur well inside the usual interatomic distances in solids, and consequently the 4f electron properties of the free atoms are retained in the metallic as well as ionic lanthanide solids. Cerium is the only 4f metal that does not conform to this generalization, presumably because its 4f-electron shell is not yet fully stabilized. The actinide 5f electrons behave quite differently. For the early members of the actinide series, the Sf electrons have a greater radial distribution than do their 4f homologs. The first few 5f electrons are not confined to the core of the atom, and they can therefore interact or mix with the other valence electrons to affect interatomic interactions in the solid state. Beyond plutonium, all the 5f electrons are localized within the atomic core, and the resemblance between the f-block elements becomes closer. Americium is the first actinide metal whose crystal structure resembles that of the lanthanide metals. In the transcurium metals, the resemblance to the lanthanide metals becomes increasingly stronger. The room-temperature crystal structure for the elements for Am to Cf is dhep, just as it is in the light lanthanides. 

Fig. 10 shows the radial particle densities, electrolyte solutions in nonpolar pores. Fig. 11 the corresponding data for electrolyte solutions in functionalized pores with immobile point charges on the cylinder surface. All ion density profiles in the nonpolar pores show a clear preference for the interior of the pore. The ions avoid the pore surface, a consequence of the tendency to form complete hydration shells. The ionic distribution is analogous to the one of electrolytes near planar nonpolar surfaces or near the liquid/gas interface (vide supra). 

FIG. 11 Cation (full), anion (dashed), and oxygen (dotted) radial density distributions in polar pores with embedded surface charges. Top NaCl solution bottom KCl solution. 

A simple modification of the IAM model, referred to as the K-formalism, makes it possible to allow for charge transfer between atoms. By separating the scattering of the valence electrons from that of the inner shells, it becomes possible to adjust the population and radial dependence of the valence shell. In practice, two charge-density variables, P , the valence shell population parameter, and k, a parameter which allows expansion and contraction of the valence shell, are added to the conventional parameters of structure analysis (Coppens et al. 1979). For consistency, Pv and k must be introduced simultaneously, as a change in the number of electrons affects the electron-electron repulsions, and therefore the radial dependence of the electron distribution (Coulson 1961). 

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