Static Core Polarization

In alkaline earth atoms, on the other hand, the static core polarization model clearly does not reproduce the energies of the atomic levels, because of the magnitude of the nonadiabatic effects/ Several theories have been formulated for the nonadiabatic effects. For example, the validity of the approach of Eissa and Opik has recently been verified by Vaidyanathan and Shorer by comparing calculated and measured quantum defects ofCa. 

Although the frozen-core approximation underlies all ECP schemes discussed so far, both static (polarization of the core at the Hartree-Fock level) and dynamic (core-valence correlation) polarization of the core may accurately and efficiently be accounted for by a core polarization potential (CPP). The CPP approach was originally used by Meyer and co-workers (MUller etal. 1984) for all-electron calculations and adapted by the Stuttgart group (Fuentealba et al. 1982) for PP calculations. The... 

Bond lengths R (A), binding energies D. (eV) and vibrational constants a>e (cm ) of the homonuclear halogen dimers from dl-electron (AE) Douglas-Kroll-HeB (DKH) and valence-only energy-consistent pseudopotential (EC-PP) Hartree-Fock self-consistent field (SCF) calculations. The effects of static and dynamic core-polarization at the valence-only level are modelled by a core-polarization potential (CPP). 

The static dipole polarizabilities of alkali dimers have been calculated as a function of the internuclear distance and of the vibrational index for both their electronic ground state and lowest triplet state. The method is based on /-dependent pseudopotentials for atomic core representation, Gaussian basis sets, effective core potentials to account for core polarization, the evaluation of molecular orbitals by the restricted HF method, and then a full valence Cl treatment. For all alkali pairs, the parallel and perpendicular components of the ground state a at equilibrium distance Rg scale as the cube of Re, which can be related to a simple electrostatic model of an ellipsoidal charge distribution. So, for the ground state, the longitudinal polarizability exhibits a maximum at a distance corresponding to 1.3-1.5 times the equilibrium distance. 

Lawrence (1937) first noted that the site of incorporation of solubilized molecules depended on their relative hydrophobic and hydrophilic tendencies. The solubilizate may be entrapped in the hydrocarbon core of the micelle, be oriented radially in the micelle with its polar group buried (deep penetration) or near the surface (short penetration), or be adsorbed on the surface of the micelle. Additionally, for non-ionic surfactants, incorporation of the solubilizate can occur in the polyoxyethylene shell of the surfactant. Fig. 3 illustrates several modes of solubilizate incorporation. It is important to realize, however, that solubilization, like micelle formation, is not a static but a dynamic equilibrium process. 

The frozen-core approximation is one basic assumption underlying all ECP schemes described so far. Especially for main group elements, where a large-core ECP approximation works fairly well if not too high accuracy is desired, the polarizability of the cores (Fig. 14) has nonnegligible effects for elements from the lower part of the periodic table. Within the ECP approach it is indeed possible to account for both static (polarization of the core at the HF level) and dynamic (core-valence correlation) polarization of the cores in an efficient way. Meyer and coworkers [202] proposed in the framework of AE calculations the... 

Early we noted that if we could measure several A/ intervals we could parametrize all the higher A/ intervals in terms of the polarizability of the ionic core. As an example of the application of the static polarization model let us consider the Cs A/ intervals. It is clear that Eq. (4) may be recast in terms of the intervals between adjacent / states. Let us label the difference in the polarization enei ies of the / and / states, Wp i/ - Wpou- as A,/-. It is convenient to adopt the convention of Edlen and write An- in cm" and define for / the terms Pi = where R is the Rydberg constant and... 

For the amphiphilic block copolymer in the non-polar selective solvent, the unpolar blocks form the corona, which provides solubilization and stabilization, while the polar or hydrophilic and functionalized blocks form the core, which is able to dissolve metal compounds due to coordination, followed by the nucleation and growth of metal particles upon reduction. Also the internal structure of block-copolymer micelles, as given by the size of core and corona and the density profile in each domain, has been carefully characterized by static and dynamic light scattering [146] and by small angle neutron scattering using contrast variation techniques [147], The micellar corona has many of the characteristics of a spherical polymer brush. 

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... 

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