Free ion polarizability

Free-ion polarizability calculations have been proposed by various authors, based on the values of molecular refraction of incident light according to... 

Free-ion polarizabilities, arranged in isoelectronic series, are listed in table 1.7. A semilogarithmic plot of the listed values (figure 1.6) reveals a functional dependence on atomic number. This dependence, quite marked, allows us to estimate the free-ion polarizability for ions for which there are no precise experimental data (values in parentheses in table 1.7 estimates according to Viellard, 1982). 

Figure 1,6 Free-ion polarizability as a function of atomic number. Curves are drawn for isoelectronic series. Reprinted from Viellard (1982), Sciences Geologiques, Memoir n°69, Universite Louis Pasteur, with kind permission of the Director of Publication.

The dipole-dipole ( / /+ ) and dipole-quadrupole dq+ ) coefficients can be derived from free-ion polarizability a and the mean excitation E by applying... 

Because the effective charge may also be related to free ion polarizability a ... 

Table 1.7 Free-Ion polarizability (o ) arranged in isoelectronic series. Data in A. N = number of electrons (adapted from Viellard, 1982).

We recall from Chapter 1 that for ionic materials, ionic polarizability can be taken into account using the shell model of Dick and Overhauser (1958), which treats each ion as a core and shell, coupled by a harmonic spring. The ion charge is divided between the core and shell such that the sum of their charges is the total ion charge. The free ion polarizability, a, is related to the shell charge, Y, and spring constant, k, by ... 

Expressions for the force constant, i.r. absorption frequency, Debye temperature, cohesive energy, and atomization energy of alkali-metal halide crystals have been obtained. Gaussian and modified Gaussian interatomic functions were used as a basis the potential parameters were evaluated, using molecular force constants and interatomic distances. A linear dependence between spectroscopically determined values of crystal ionicity and crystal parameters (e.g. interatomic distances, atomic vibrations) has been observed. Such a correlation permits quantitative prediction of coefficients of thermal expansion and amplitude of thermal vibrations of the atoms. The temperature dependence (295—773 K) of the atomic vibrations for NaF, NaCl, KCl, and KBr has been determined, and molecular dynamics calculations have been performed on Lil and NaCl. Empirical values for free ion polarizabilities of alkali-metal, alkaline-earth-metal, and halide ions have been obtained from static crystal polarizabilities the results for the cations are in agreement with recent experimental and theoretical work. 

The most popular model which takes into account both the ionic and electronic polarizabilities is the shell model of DICK and OVERHAUSER [4.12]. It is assumed that each ion consists of a spherical electronic shell which is isotropically coupled to its rigid ion-core by a spring. To begin with we consider a free ion which is polarized by a static field E. The spring constant is k, the displacement of the shell relative to its core is v and the charge of the shell is ye (Fig.4.7). In equilibrium, the electrostatric force yeE is equal to the elastic force kv yeE = kv. The induced dipole moment is d = yev = aE from which we obtain the free ion polarizability... 

The chemical behavior of ions, ion pairs, and polarizable molecules partakes of the same indistinctness as the definitions of these species. Any attempt to make a complete catalog of the reactions of ions will almost certainly include borderline reactions whose intermediates are in fact ion-pairs or even covalent molecules. For many purposes the identification of a reaction as carbonium ion-like, or what the Germans would call Krypto-ionenreaktion, is as useful as the certain knowledge that the intermediate is actually a carbonium ion. Many of the ionic reaction mechanisms in the literature do not represent actual free ions and were not so intended by their authors. The ionic representation is often merely a convenient simplification if it is an oversimplification it is one that is easily rectified when the pertinent data become available. The value of such approximate mechanisms is that... 

If the electron-cloud radius yrms were exactly equal to the structural radius r, Wasastjerna s criterion would be obviously true. But in fact, for ions r ce. 2 yrms (Table 3). Hence the criterion needs justification. It is obviously most probable for isoelectronic ions (cp. Eauling (/)), but the electron-cloud radii should refer to the ions in the crystals, not to the free ions. For, with a gross difference between crystal and free-ion electron-cloud radii for the hydride ion, there may be significant differences for others 40). For the crystals the electron-cloud radii could be obtained either from polarizeability or from magnetic susceptibility. The theory of polarizeability is less certain and there is a considerable correction to infinite wavelength. We therefore adopt the magnetic evidence. But this must be corrected for the inner shell contribution (Table 3). 

Accordingly, the polarizability of the ions will, in general, change upon molecule formation. Taking into account the polarizabilities and polarizing abilities of the ions in Li+I- and Cs+F , it follows that the actual a + ac valid for the application of Eq. (4) is, relative to the free ions, diminished for Lil and increased for CsF. In fact, it was shown on the occasion of an earlier application of Eq. (4) to 9 alkali hahdes, including the only fluoride CsF, that Eq. (5) represents the experimental data better than Eq. (4). 

In the case of a polarizable monomer like propylene sulfide, cryptated ion pairs are not only much more reactive than the corresponding non-complexed ion pairs but they are even more reactive than free ions In THF. For ethylene oxide, the results are different since free alkoxide ions are significantly more reactive than cryptated Ion pairs which are themselves slightly less reactive than the corresponding non complexed ion pairs. 

The earliest calculations of the dipole and quadrupole contributions to the crystal field used free-ion values for the dipole polarizability, a,-, and for the quadrupole polarizability (Hutchings and Ray, 1963). It is generally recognized today, however, that the polarizability of an ion in a solid is generally much less than the corresponding value for the free ion (Chakrabarti et al., 1976 Bogomolova et al., 1977). How much less, however, is not clear. For this reason, therefore, computations of dipole and quadrupole contributions to the crystal field should be regarded as imprecise, even if they are performed with reduced polarizability values. 

A wide variety of experimental and theoretical results have shown that the lAST is a very powerful tool for multicomponent adsorption equihbria of polar or nonpolar adsorptives adsorbed on activated carbon however, comprehensive plysical models (for instance the MSPDM) are necessary in the case of polar (dipole and quadrupole moment, polarizability) adsorptives adsorbed on zeolites with free ions (Mailonaim 1999). 

Serr A, Netz RR (2006) Polarizabilities of hydrated and free ions derived from DPT calculations. Int J Quantum Chem 106 2960-2974... 

A full quantum-mechanical description of the Menshutkin reaction has been obtained for gas phase and solution by using density functional theory (DFT) and the self-consistent isodensity polarizable continuum model (SCI-PCM). Ammonia and pyridine were the nucleophiles and methyl chloride and methyl bromide, the electrophiles. In the gas phase an initial dipole complex intermediate is followed by a transition state leading to an ion pair. In the solvent-effect calculations, the dipole complex disappears with both cyclohexane and DMSO. The transition state is stabilized compared with the gas phase. The ion-pair product is strongly stabilized and in DMSO it is dissociated into free ions. 

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