Madelung problem

Computer modeling of GBs can give considerable insights, but be careful when using older results in the literature. The complexity of ceramic structures and the Madelung problem cause difficulties and some older simulations may not be reliable, in part, because they used too little material in the calculation because of the capabilities of the computers. 

Wolf D (1992) Reconstruction of NaCl surfaces from a dipolar solution to the Madelung problem. Phys Rev Lett 68(22) 3315-3318... 

Many-body problems wnth RT potentials are notoriously difficult. It is well known that the Coulomb potential falls off so slowly with distance that mathematical difficulties can arise. The 4-k dependence of the integration volume element, combined with the RT dependence of the potential, produce ill-defined interaction integrals unless attractive and repulsive mteractions are properly combined. The classical or quantum treatment of ionic melts [17], many-body gravitational dynamics [18] and Madelung sums [19] for ionic crystals are all plagued by such difficulties. 

Although the quantum problem seems to be solved by the hydrodynamics of a continuous distribution of electricity with charge density proportional to mass density, this approach has never been accepted as a serious alternative, largely because of doubts raised by Madelung himself. The most important of these, concerns the self-interaction between the charge elements of an extended electron. 

Embedding techniques at various levels have been suggested to close the gap between the cluster and the periodic treatment. A physical approach to the electronic structure problems of solids and surfaces contrasts sharply with the intuition of chemists that local interactions dominate the properties of surfaces and adsorption complexes. Hence, it is very desirable to replace the infinite solid, which is very difficult to treat quantum-chemically, by finite models of the sites of interest. In this way it is easy to describe a local site as cluster having a relatively small number of atoms which interact with a potentially infinite number of surrounding atoms through, for example, the Madelung potential which is treated as perturbation.26,44 These lead to terms such as... 

As a solution to this problem we have tried the application of Madelung potential [2] to the semiconductor crystals. Till now, the trial has not been carried out in the semiconductor crystals to our knowledge except our presentation at the fourth international conference on DV-Xa method (2006) [3]. The difficulty in the application to the semiconductor crystals might be related with a kind of a sea with + electronic charges used in the Madelung potential technique. This is partly so because the semiconductor crystals such as Si and diamond are of covalent bond structure and these crystals are composed of a single element. These two difficulties make the application of a Madelung potential to the semiconductor crystals difficult. 

It should be realized that taking the sum over all ions in a crystal is a nontrivial task, because the sum converges very slowly. This is a classic problem addressed early in this century by Madelung (Madelung, 1909) for a recent discussion, see Brown (1967, p. 93 ff). 

For further discussion of the problem of defining Madelung constants, see Quane. D. J. Chem. Educ. 1970. 47, 396. 

---