Finite point charge

A variation of this method is the finite point charge method, used by Maroulis and Thakkar (1988), in which the external electric field or field gradient is simulated by an appropriate arrangement of point charges. This method is even simpler to implement, since it only requires the option to include centres with a charge but no basis functions, rather than a modified one-electron Hamiltonian. 

The electric field can be incorporated in the Flamiltonian via a finite field term or approximated by a set of point charges. This allows the computation of corrections to the dipole only, which is generally the most significant contribution. 

Terms up to order 1/c are normally sufficient for explaining experimental data. There is one exception, however, namely the interaction of the nuclear quadrupole moment with the electric field gradient, which is of order 1/c. Although nuclei often are modelled as point charges in quantum chemistry, they do in fact have a finite size. The internal structure of the nucleus leads to a quadrupole moment for nuclei with spin larger than 1/2 (the dipole and octopole moments vanish by symmetry). As discussed in section 10.1.1, this leads to an interaction term which is the product of the quadrupole moment with the field gradient (F = VF) created by the electron distribution. 

If the solute were simply a collection of point charges surrounded by a continuous dielectric medium with the bulk dielectric constant of the solvent, the self-energy and the strength of charge-charge interactions in the solute would be reduced by a factor of . This is called dielectric screening. However, the solute itself occupies a finite volume, and solvent is excluded from this volume. This reduces the dielectric screening and is called... 

The two electron coulomb integral aa/bb) is smaller than the point charge integral (aa/Ri1) at all finite distances and therefore the use of Eq. (17) introduces an underestimation of an attractive contribution to the intermolecular energy of interaction or, if we put it the other way round, a net repulsion between the molecules results. 

The Extended Debye-Hiickel Equation. This exercise reminds us that the Debye-Hiickel limiting law is not sufficiently accurate for most physicochemical studies. To estimate the calculated activity coefficient more accurately, one must consider the fact that ions are not point charges. To the contrary, ions are of finite size relative to the distance over which the ions interact electrostatically. This brings us to the extended Debye-Hiickel equation ... 

The above equation assumes that the ions are point charges and interact in a continuous dielectric. It is essentially correct in the limit, but problems arise when considering finite concentrations of solute where an extended Debye-Htickel expression may be more appropriate... 

In this connection there is an important question concerning the infinite selfenergy of a point charge in classical as well as in quantum field theory. The latter uses a renormalization process to solve the problem, namely, by subtracting two infinities to end up with a finite result. Despite the success of such a procedure, a more physically satisfactory way is needed [80]. Possibly the present theory may provide such an alternative, by tackling the divergence problem in a more surveyable manner. The finite result of a difference between two infinities due to renormalization theory would then be replaced by a finite result obtained from the product of an infinity and a zero, as demonstrated by the present analysis. 

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. 

In the first two chapters we have seen that the Na atom, for example, differs from the H atom because the valence electron orbits about a finite sized Na+ core, not the point charge of the proton. As a result of the finite size of the Na+ core the Rydberg electron can both penetrate and polarize it. The most obvious manifestation of these two phenomena occurs in the lowest states, which are substantially depressed in energy below the hydrogenic levels by core penetration. Core penetration is a short range phenomenon which is well described by quantum defect theory, as outlined in Chapter 2. 

The Debye-Huckel Theory The Finite-Ion-Size Model. If the approximation of the point charge is removed, the extended form of the Debye-Huckel law is obtained ... 

The treatment of the diffuse double layer outlined in the last section is based on an assumption of point charges in the electrolyte medium. The finite size of the ions will, however, limit the inner boundary of the diffuse part of the double layer, since the centre of an ion can only... 

The HF equations are approximate mainly because they treat electron-electron repulsion approximately (other approximations are mentioned in the answer suggested for Chapter 5, Harder Question 1). This repulsion is approximated as resulting from interaction between two charge clouds rather than correctly, as the force between each pair of point-charge electrons. The equations become more exact as one increases the number of determinants representing the wavefunctions (as well as the size of the basis set), but this takes us into post-Hartree-Fock equations. Solutions to the HF equations are exact because the mathematics of the solution method is rigorous successive iterations (the SCF method) approach an exact solution (within the limits of the finite basis set) to the equations, i.e. an exact value of the (approximate ) wavefunction l m.. 

A defining feature of the models discussed in the previous section, regardless of whether they are implemented via matrix inversion, iterative techniques, or predictive methods, is that they all treat the polarization response in each polarizable center using point dipoles. An alternative approach is to model the polarizable centers using dipoles of finite length, represented by a pair of point charges. A variety of different models of polarizability have used this approach, but especially noteworthy are the shell models frequently used in simulations of solid-state ionic materials. 

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