Effective interaction between ions

To describe the simple phenomena mentioned above, we would hke to have only transparent approximations as in the Poisson-Boltzmann theory for ionic systems or in the van der Waals theory for non-coulombic systems [14]. Certainly there are many ways to reach this goal. Here we show that a field-theoretic approach is well suited for that. Its advantage is to focus on some aspects of charged interfaces traditionally paid little attention for instance, the role of symmetry in the effective interaction between ions and the analysis of the profiles in terms of a transformation group, as is done in quantum field theory. 

For most purposes the band-structure energy is conveniently calculated by performing the sum over wave number space directly as indicated in Eq. (17-4). We shall use that procedure when we discuss phonons. There is, however, conceptual value in making a transformation back into position space where we may see that the band-structure energy can be written as a contribution to an effective interaction between ions. We rewrite the band-structure energy of Eq. (17-4), inserting the explicit form of the structure factors from Eq. (16-5) ... 

B The Effective Interaction Between Ions, and Higher-Order Terms 387... 

The effective interaction between ions in aluminum. Also shown is the distribution of neighbors as a function of distance in the face-centered cubic structure. [After Harrison, 1964.]... 

The perturbation of electron states by the pseudopotential modifies the total energy by a term called the band-structure energy, which is obtained to second order in the pseudopotential it is essential to an understanding of the bonding properties, and it can be directly evaluated for any specified positions of the metallic ions. It may also be thought of as contributing to an effective interaction between ions the interaction shows Friedel oscillations at large separations. 

In this section, we discuss the correlations and effective interactions between ions in aqueous solution. For this purpose we typically look at simulation boxes with about 2000 water molecules and add between 1 and 200 salt pairs in the solution. The main output from the simulation is the radial distribution function jij (r) between the ions. In order to obtain good statistics for further analysis, long simulation runs of about 200 ns are needed. The potential of mean force (PMF) follows by Boltzmann inversion from the radial distribution function according to... 

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