Shell model covalent crystals

I have not described the calculation of the eigenvalues, which requires the solution of the equations of motion and therefore a knowledge of the force constants. The shell model for ionic crystals, introduced by Dick and Overhauser (1958), has proved to be extremely useful in the development of empirical crystal potentials for the calculation of phonon dispersion and other physical properties of perfect and imperfect ionic crystals. There is now a considerable literature in this field, and the following references will provide an introduction Catlow etal. (1977), Gale (1997), Grimes etal. (1996), Jackson et al. (1995), Sangster and Attwood (1978). The shell model can also be used for polar and covalent crystals and has been applied to silicon and germanium (Cochran (1965)). 

In the first sum, indices i and denote shells and cations, the second term runs over all point charges in the system, and the third term accounts for interactions of the shells with their cores. The shell model takes into account the polarization of the anions by the crystal field of the solid, which is an important feature. To better reproduce the characteristics of systems with partly covalent bonds, such as zeolites, Eqs. [15] and [16] are supplemented with a term... 

In this chapter we have shown how force fields can be utilized in materials science applications. There are similarities between force fields used in life science and in materials science. Owing to the variety of molecules studied in materials science, however, there are several complementary approaches to modeling such systems. Molecular mechanics force fields as used in life science (i.e., in biomolecules) can also be applied to organic materials such as polymers or liquid crystals. Ionic materials such as oxides are better described by means of ion pair or shell model potentials. For some systems with ionic as well as covalent character in their bonds (e.g, zeolites), both approaches are feasible. 

In the case of organic molecules chemists consider a logical choice to express the potential in terms of valence bonds and valence electrons [20] while physicists often proceed with an unbiassed cartesian space which is sometimes transformed in that of a suitable model aimed at a specific physical property. Often the potentials used in treating the vibrations of ionic crystals or metals are transferred and constrained to the case of organic molecules. However the chemical reality of molecular covalent bonds and of their interactions is different from any type of "ionic" system. For example the popular "shell model" has not been as successful as the classical "valence force field" in the study of the lattice dinamics of "covalent crys-... 

The general influence of covalency can be qualitatively explained in a very basic MO scheme. For example, we may consider the p-oxo Fe(III) dimers that are encountered in inorganic complexes and nonheme iron proteins, such as ribonucleotide reductase. In spite of a half-filled crystal-field model), the ferric high-spin ions show quadrupole splittings as large as 2.45 mm s < 0, 5 = 0.53 mm s 4.2-77 K) [61, 62]. This is explained... 

The reduction of the free-ion parameters has been ascribed to different mechanisms, where in general two types of models can be distinguished. On the one hand, one has the most often used wavefunction renormalisation or covalency models, which consider an expansion of the open-shell orbitals in the crystal (Jprgcnscn and Reisfeld, 1977). This expansion follows either from a covalent admixture with ligand orbitals (symmetry-restricted covalency mechanism) or from a modification of the effective nuclear charge Z, due to the penetration of the ligand electron clouds into the metal ion (central-field covalency mechanism). 

Because charge defects will polarize other ions in the lattice, ionic polarizability must be incorporated into the potential model. The shell modeP provides a simple description of such effects and has proven to be effective in simulating the dielectric and lattice dynamical properties of ceramic oxides. It should be stressed, as argued previously, that employing such a potential model does not necessarily mean that the electron distribution corresponds to a fully ionic system, and that the general validity of the model is assessed primarily by its ability to reproduce observed crystal properties. In practice, it is found that potential models based on formal charges work well even for some scmi-covalent compounds such as silicates and zeolites. 

Probably, 99.99 % of all X-ray structure determinations to-date have been done on this approximation. Of course, the real distribution of the electron density in an actual crystal/molecule is different from such model (often called pro-crystal/pro-molecule) its topology can be best understood within the framework of the AIM theory, developed by Bader [8]. Electron density is concentrated between atoms which are linked by a covalent bond, and is depleted between atoms which participate in closed-shell interactions (ionic or van der Waals). A good quantitative measure of such effects is the Laplacian of the electron density (V p), equal to the sum of its principal curvatures (second derivatives) at a given point ... 

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