Polarizability charge density distribution

As Fig. 12 shows, the inner shell electrons of the alkaline ions behave classically like a polarizable spherical charge-density distribution. Therefore it seemed promising to apply a "frozen-core approximation in this case 194>. In this formalism all those orbitals which are not assumed to undergo larger changes in shape are not involved in the variational procedure. The orthogonality requirement is... 

In this expression, the dipole dipole interactions are included in the electrostatic term rather than in the van der Waals interactions as in Eq. (9.43). Of the four contributions, the electrostatic energy can be derived directly from the charge distribution. As discussed in section 9.2, information on the nonelectrostatic terms can be deduced indirectly from the charge density. The polarizability a, which occurs in the expressions for the Debye and dispersion terms of Eqs. (9.41) and (9.42), can be expressed as a functional of the density (Matsuzawa and Dixon 1994), and also obtained from the quadrupole moments of the experimental charge density distribution (see section 12.3.2). However, most frequently, empirical atom-atom pair potential functions like Eqs. (9.45) and (9.46) are used in the calculation of the nonelectrostatic contributions to the intermolecular interactions. 

The determination of the number of the SHG active complex cations from the corresponding SHG intensity and thus the surface charge density, a°, is not possible because the values of the molecular second-order nonlinear electrical polarizability, a , and molecular orientation, T), of the SHG active complex cation and its distribution at the membrane surface are not known [see Eq. (3)]. Although the formation of an SHG active monolayer seems not to be the only possible explanation, we used the following method to estimate the surface charge density from the SHG results since the square root of the SHG intensity, is proportional to the number of SHG active cation com-... 

Robinson (1967) has used the Unsold approximation for the energy levels to express the polarizabilities in terms of the electrostatic moments of the ground-state electron distribution. The expressions have been applied to X-ray charge densities by Zyss, Baert, and coworkers (Fkyerat et al. 1995 F. Hamzaoui, F. Baert and J. Zyss, private communication). A detailed description of the derivation and the approximations involved is beyond the scope of this treatise. However, it should be mentioned that the severe approximations are made that all excited-state energy levels are equal, and that the exciting light frequency is equal to zero. 

Full knowledge of the charge distribution of a molecule requires specification of the charge density at all points. For some purposes the charge density provides excess information thus, the potential outside a sodium ion is independent of the distribution of the electrons, and the interaction of a molecule with a uniform external field is determined by its dipole moment and dipole polarizabilities. The electric multipole moments characterize the charge distribution the first three are defined as follows ... 

Here the position r. of the point charges located on the solvent molecules is determined by the stmcture of the solvent shell and the electron density distribution within the solvent molecule. In this t5q)e of model, the latter is assumed to be fixed, i.e. the solvent molecules are considered non-polarizable while solving the Schrodinger equation for the coupled system. 

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