Point charge arrays

Figure 9.1 A Ba8Ti706 cluster shown embedded at the center of a three-dimensional point charge array.

The effects of particle size on the critical thickness for ferroelectricity of nanoscale BaTiCh particles were investigated using a first-principles method and population analysis. Models composed of a small Ba8Ti706 cluster and a point charge array surrounding the cluster were used for the calculations, with the size of each... 

Calculations with point-charge arrays representing zeolite unit cells have, indeed, confirmed that the BE trends described above are closely mirrored by the trends of fhe average Madelung potential felt by the emitting atom type (Si, 0, Al,... 

The situation becomes more complicated when, instead of point-charge arrays, one encounters continuous charge distributions of known functional forms, as occurs when orbitals are introduced to describe localized electron distributions in systems with periodicity in one or more dimensions. It is possible to use EMM methods for such systems, as was shown, for example, by Strain et al. [7]. However, in addition to the possibility of simply representing an orbital by its moments (and thereby foregoing any processing based on its specific form), one may alternatively be able to use properties of the orbitals to make further mathematical analyses that lead to gains in computational efficiency. Such an approach is represented in two efforts of which we are aware ... 

Another way of calculating the electrostatic component of solvation uses the Poisson-Boltzmann equations [22, 23]. This formalism, which is also frequently applied to biological macromolecules, treats the solvent as a high-dielectric continuum, whereas the solute is considered as an array of point charges in a constant, low-dielectric medium. Changes of the potential within a medium with the dielectric constant e can be related to the charge density p according to the Poisson equation (Eq. (41)). 

Molecules do not consist of rigid arrays of point charges, and on application of an external electrostatic field the electrons and protons will rearrange themselves until the interaction energy is a minimum. In classical electrostatics, where we deal with macroscopic samples, the phenomenon is referred to as the induced polarization. I dealt with this in Chapter 15, when we discussed the Onsager model of solvation. The nuclei and the electrons will tend to move in opposite directions when a field is applied, and so the electric dipole moment will change. Again, in classical electrostatics we study the induced dipole moment per unit volume. 

Those electrons must not only avoid each other but also the negatively charged anionic environment. In its simplest form, the crystal field is viewed as composed of an array of negative point charges. This simplification is not essential but perfectly adequate for our introduction. We comment upon it later. 

We are concerned with what happens to the (spectral) d electrons of a transition-metal ion surrounded by a group of ligands which, in the crystal-field model, may be represented by point negative charges. The results depend upon the number and spatial arrangements of these charges. For the moment, and because of the very common occurrence of octahedral coordination, we focus exclusively upon an octahedral array of point charges. 

Their angular parts are shown in Fig. 3-1. Let us consider the six point charges in an octahedral array to be disposed along the positive and negative x, y and axes to which these d orbitals are referred. This is conveniently drawn, as shown in Fig. 

It is generally unwise to think of ionic compounds as holding together with physical bonds it is better to think of an array of point charges, held together by the balance of their mutual electrostatic interactions. (By mutual here, we imply equal numbers of positive and negative ions, which therefore impart an overall charge of zero to the solid.)... 

A structure may be imposed on the ionic cloud by supposing that dq in the volume element dv = r2 sin OdOdipdr has a finite number, n, of maxima similarly situated at k 1 from the surface of the central ion (Figure lb). By analogy, this non-radial atmosphere is reducible to a corresponding array of point charges, and this device later enables us to formulate the necessary boundary conditions. 

Figure 2 Octahedral (a) and tetrahedral (b) arrays of point charges. For equal origin-point distances, a = (8/3) /2a. For a cube arrangement add points at ( a d -a ), ( a a a )...

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