The Electrostatics Approximation

Consider a homogeneous, isotropic sphere that is placed in an arbitrary medium in which there exists a uniform static electric field E0 = E0ez (Fig. 5.3). If the permittivities of the sphere and medium are different, a charge will be induced on the surface of the sphere. Therefore, the initially uniform field will be distorted by the introduction of the sphere. The electric fields inside and outside the sphere, Ej and E2, respectively, are derivable from scalar potentials 0,(r, 6) and 02(r, 8) 

Because of the symmetry of the problem, the potentials are independent of the azimuthal angle j . At the boundary between sphere and medium the potentials must satisfy 

Consider now two point charges q and -q which are separated by a distance d (Fig. 5.4). This configuration of charges is called a dipole with dipole moment p = pez, where p = qd. If the charges are embedded in a uniform unbounded medium with permittivity em, the potential of the dipole at any point P is 

If we let d approach zero in such a way that the product qd remains constant, we obtain the potential of an ideal dipole 

Let us return now to the problem of a sphere in a uniform field. We note from (5.13) and (5.14) that the field outside the sphere is the superposition of the applied field and the field of an ideal dipole at the origin with dipole moment 

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The condition (12.6) has on occasion been attributed to Mie, presumably because it can be obtained from the Mie theory. But it is sobering to realize that it follows from simple electrostatics. For we showed in Section 5.2 that the absorption efficiency in the electrostatics approximation is... 

Notable progress in analyzing nonspherical particles has been made by Fuchs (1975), who calculated absorption by cubes in the electrostatics approximation and applied the results to experimental data for MgO and NaCl. We shall discuss Fuchs s results at the end of Section 12.3. Langbein (1976) also did calculations for rectangular parallelepipeds, including cubes, which give valuable insights into nonspherical shape effects. Because the cube is a common shape of microcrystals, such as MgO and the alkali halides, these theoretical predictions have been used several times to interpret experimental data. We shall do the same for MgO. Our theoretical treatment of nonsphericity, however, is based on ellipsoids. Despite its simplicity, this method predicts correctly many of the nonspherical effects. 

Fuchs s (1975) result for volume-normalized absorption by randomly oriented cubes in the electrostatics approximation is... 

In the hypersurface portions where polarization effects may be considered inessential to the understanding of the physical phenomenon under investigation, the model may be limited to electrostatic interactions. This further reduction is less well justified than the preceding ones, although in some cases (see, e.g. Section IX) a mutual cancellation of other effects enhances the reliability of the electrostatic approximation. 

A correct forecast of differences in electrophilic reactivity among the carbon atoms of the ring constitutes a classical testing bench for all theories on chemical reactivity. To exploit the electrostatic approximation, we need some information on the trend of W (r) in the regions... 

In pyrrole (XV), a wide approach channel for electrophilic reagents leads to positions 3 and 4 (see Figs. 33 and 34). This finding is in accordance with the experimental evidence24) that protonation in 3,4 is faster than in 2,5, though the 2-protonated salts are more stable. As has been repeated many times, the electrostatic approximation can give at most a picture of the first part of the reaction and it is not able to predict the energetically most stable final product. 

Fig. 44. Proton interaction energy trend along an approach path to the aziridine N. as obtained a) by the electrostatic approximation, b) by the Hartree approximation, c) by SCF computations (GTO wave function)...

An analytical expansion of the electrostatic molecular potential could be very useful for actual utilizations of the electrostatic approximation. The choice of the best analytical form is in general dictated by the specific problem. For the applications considered in the present paper we have selected multipole expansions into spherical harmonics ... 

The monohydration associates A-H O, where A is a neutral molecule containing polar groups, represent a typical association to which the electrostatic approximations of Section II. E may be applied. The object is to obtain by relatively inexpensive methods a first-order de-... 

Note added in proofs. It should be clear that this paper was not designed to offer a review of the applications of the electrostatic approximations to the chemical reactivity or molecular interaction problems. However, it may be of some interest to add a few quotations of further developments and direct applications of the electrostatic molecular potential method performed or noticed by the authors after the completion of the present paper. 

Here it should be noted that the function E( ) in the electrostatic approximation (1) is not a function in the usual sense and its continuity is broken up at any rational point =Q/p (where p and q are mutually simple numbers). This discontinuity appears by the following reason. At zero temperature the system of electrons and holes is arranged in the Wigner crystal with a cell consisting of TCNQ (or TTP) molecules. Let us study Ejj( ) with a little deviation of from Q p i.e. = q/P + where 1. Assume that /> 0. ... 

The solid particle can have an arbitrary shape, but attention for now will focus on the case of a sphere. The radius of the sphere is denoted by a and its complex dielectric function by s((o). The dielectric function will be assumed to be local in this discussion so there is no dependence on the wave-vector of the photon. The dielectric function for the solvent is denoted by the local function Ss( ) particle size is assumed to be sufficiently small compared to the wave length of the relevant photons that the electrostatic approximation to electrodynamics is warranted. Thus retardation effects will be neglected here. 

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