The ionic potential

In order to develop the pseudopotenfial for a specific atom we consider it as isolated, and denote by the single-particle states which are the solutions of the single-particle equations discussed earlier, as they apply to the case of an isolated atom. In principle, we need to calculate these states for all the electrons of the atom, using as an external potential that of its nucleus. Let us separate explicitly the single-particle states into valence and core sets, identified as and respectively. These satisfy the SchrMinger type equations 

Applying the single-particle hamiltonian Wp to this equation, we obtain 

Therefore, the new states obey a single-particle equation with a modihed potential, but have the same eigenvalues as the original valence states The modified potential for these states is called the pseudopotential , given by 

Why is this a useful approach First, consider the definition of the pseudo-wavefunctions through Eq. (2.114) what this definition amounts to is projecting out of the valence wavefunctions any overlap they have with the core wavefunctions. In fact, the quantity 

There are some aspects of the pseudopotential, at least in the way that was formulated above, that make it somewhat suspicious. First, it is a non-local potential  

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The data given should serve only as reference values following the rule, the higher the ionic potential, the thicker the hydration layer of the water molecules around the ion, and the slower the ionic diffusion. Cations generally diffuse more rapidly than anions. 

Cartledge, G. H. (1928a). Studies on the periodic system. I. The ionic potential as a periodic function. Journal of the American Chemical Society, 50, 2855-63. 

The ionic potentials can be experimentally determined either with the use of galvanic cells containing interfaces of the type in Scheme 7 or electroanalytically, using for instance, polarography, voltammetry, or chronopotentiometry. The values of and Aj f, obtained with the use of electrochemical methods for the water-1,2-dichloroethane, water-dichloromethane, water-acetophenone, water-methyl-isobutyl ketone, o-nitrotol-uene, and chloroform systems, and recently for 2-heptanone and 2-octanone [43] systems, have been published. These data are listed in many papers [1-10,14,37]. The most probable values for a few ions in water-nitrobenzene and water-1,2-dichloroethane systems are presented in Table 1. 

The recombination of He is a special case. We include it here because of the similarities with H3 and because it is the only known example where three-body recombination of a diatomic molecular ion dominates over the binary process. The literature on the helium afterglow is quite large and we will not be able to do justice to all aspects of this problem. Mulliken71 had predicted that fast dissociative recombination of Hej should not occur due to a lack of a suitable curve crossing between the ionic potential curve and repulsive curves of He. Afterglow experiments in pure helium, at sufficient pressure to enable formation of Hej ions, have confirmed this expectation. It does not appear that the true binary recombination... 

Figure 2.2. F.lemental enrichment factors in soils, plotted on a log scale against the ionic potential of the elements...

Figure 2.3. Elemental enrichment factors in baterial and fungi, plotted on a log scale against the ionic potential of the elements (after Banin and Navrot, 1975. Reprinted from Science, 189, Banin A. and Navrot J., Origin of Life Clues from relations between chemical compositions of living organisms and natural environments, pp 550-551, Copyright (1975), with permission from AAAS)...

A), an intrusion of the ionic potential happens that causes a drastic increase in the weight of lu ) and forms a deep minimum at fairly large R ( 2.3 A). 

Table 2.3 gives the self-diffusion coefficients of some important ions in submerged soils and Figure 2.2 shows the values for the elemental ions plotted against ionic potential ( z /r where z is the absolute ionic charge and r the crystal ionic radius). As the ionic potential increases the hydration layer of water molecules around the ion increases, and therefore the mobility tends to decrease. Also, at the same ionic potential, cations diffuse faster than anions. The mobilities... 

The Thomas-Fermi approximation is, unfortunately, a poor approximation for the sp-valent metals. It is based on the assumption that the potential varies much more slowly than the screening length of the electrons themselves, so that the local approximation for the kinetic energy, eqn (6.6), is valid. In practice, however, the variation in the ionic potential is measured by the core radius, Rc (cf Fig. 5.11), which is not large but of the same size as the screening length, XTF. Thus, we do not satisfy the criterion for the validity... 

In coal, copper shows an organic affinity different from the Mellor and Maley (14) series, and zinc appears to be completely associated with the inorganic matter. Copper in the physicochemical environment of coal deposition can be reduced to the univalent state (5). Univalent copper cannot be expected to behave like the bivalent metals. The ionic potential of univalent copper fits reasonably well into the organic affinity series (Figure la). 

The similarities in electronegativities are not so close as that of the ionic potentials for Be2+ and Al3+. TTie heavier element in the diagonal pair always has a lower electronegativity. but the effect is still noticeable. Thus when considering elements that resemble carbon, phosphorus is often as good a choice as silicon, and the resemblance is sufficient to establish a base from which notable differences can be formulated.7... 

Figure 15.10 Enthalpies of formation of orthosilicates versus the ionic potential (z/r) of , alkaline earth, and O, transition metal cations.

Various semi-empirical relationships have been developed to describe the ionic potential actually observed by the valence electrons in an A-B bond. An example of such a relationship is shown in Eq. (2.3). 

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