Radial charge density

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...

Plot 4nr Rl against p (or r), as shown in Figure 1.7(c). The quantity 4nr Rl is called the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr, radius r, and volume 4 ir dr. 

FIG. 6.2 Radial charge density 4rcr2p(r) as a function of the radius of the sphere, based on powder data on NH4C1 (a) NH4 (b) Cl . The shaded regions represent the boundaries which lead to populations of 10.0 + 5 and 18.0 + 5 electrons for NH4 and Cl , respectively. Source Vahvaselka and Kurki-Suonio (1975). 

Fig. (5a) Radial charge density chstribution Fig. (5b) Oscillating part of the radial density in Argon. Total density D(r) (solid) and its 5D(r) D(r) - D(r) = 4w 5p(r,r )j. for...

Figure 5.4 Radial charge densities P2p(r) of up electrons as a function of the radial distance r (all quantities are in atomic units). The solid line gives P p(r) with the 6p orbital computed in the field of the 4d95s25p6 electron configuration, the broken lines give P2p(r) with the 6p, 7p, 8p, and 9p orbitals calculated in the field of the 4d105s25p4 electron configuration. The corresponding shake probabilities 2 = 0.801, <7p 6p> 2 = 0.196, <8p 6p> 2 = 0.001, <9p 6p> 2 = 0.0005. From [AAk95] and...

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". 

The Ln(III) cations of the series Ce-Lu possess the extended Xe-core electronic configuration [Xe] 4/1 (n = 1-14), a symbol which perfectly pictures the limited radial extension of the f-orbitals The 4f shell is embedded in the interior of the ion, well shielded by the 5s2 and 5p6 orbitals [63], A plot of the radial charge densities for the 4f, 5s, 5p and 6s electrons for Gd+ visually explains why Ln(III) cations are commonly thought as a tripositively-charged closed shell inert-gas electron cloutf (Fig. 1) [63]. 

In this section we investigate the factors affecting the formation and decay of shape resonances by examining the radial charge density /120/ plots from the Feynman Dyson amplitudes corresponding to the resonant poles identified earlier /22,25,26,40,41/ in sections 3.1 and 3.2 for different atomic and molecular resonances. 

Figure 14 Radial charge density plot from the orhital/Feynman-Dyson (FD) amplitude for the Ss orbital in Be from zeroth order (SCF ), the second order (E2 A) and diagonal 2ph-TDA decouplings /EJph-TDA ). On the scale employed in the main plot, distinguishing the orbital/FD amplitudes from different decouplings is not possible but in the inset the difference between radial charge densities from the second and zeroth order (bi-variaiional SCF) decouplings clearly reveals the role of correlation and relaxation effects in changing the ionization potential from 8.44 cV at the SCF level to 8.79 eV at second order. The maximum in the electron density is at imu = 2.1 a.u.

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 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.

Figure 17. Radial charge density plot for the resonant FD amplitude in e-Mg scattering. Fora = 0.75 considered here, only the root labelled I is resonant. The role of optimal theta (6=0.12 radians) in accumulation of electron density near the nucleus is evident.

Figure 18. Radial charge density plot for the Ss orbital in Mg. The rm is at S.SS a.u.

The lanthanide series of elements differ from the transition metal series in that 4f shell inner electrons are shielded by the 5s2,5p6 closed shells. Consequently f shell electrons interact much less strongly with their environment than the d electrons in the transition series. The radial charge density for Pr3+ is shown in Fig. 8.1. The electronic structure of the f lanthanide ion is dominated by many different interactions than for more familiar d transition metal ions. 

FIGURE 5.13 The radial charge density in the argon atom as calculated by Hartree s method. The charge is arrayed into three shells corresponding to the values 1, 2, and 3 for the principal quantum number n. 

Fig. 2. Plot of the radial charge densities for the 4/-, 5s-, 5p-, and 6s-electrons of Gd+ from [14]...

Under a closed-shell assumption, only the radial charge density needs to be determined (cf chapter 1). To fix the ionic background, we suppose that the density of positive charges follows a Heaviside step function 0 thus ... 

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... 

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

Ionic radii can be defined either by the maximum of the radial charge density, r, or the expectation value, (r , of an outer valence orbital. The DF... 

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. 

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