Dick-Overhauser shell model

We have estimated each of the parameters in Eq. (2) in a unified manner by combining the strengths of several previous molecular modeling studies. We used the force field of Karasawa and (ioddard" to model the atomic potential energy surface and to describe the charge distribution at the atomic level. This force field includes the effects of electronic polarization via the shell model of electronic polarization, originally developed by Dick and Overhauser." By direct minimization of total crystal free energy with respect to both the atomic and shell... 

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)). 

We recall from Chapter 1 that for ionic materials, ionic polarizability can be taken into account using the shell model of Dick and Overhauser (1958), which treats each ion as a core and shell, coupled by a harmonic spring. The ion charge is divided between the core and shell such that the sum of their charges is the total ion charge. The free ion polarizability, a, is related to the shell charge, Y, and spring constant, k, by ... 

This includes the Pauli repulsion and (attractive) dispersion terms. The polarizability of the ions is included using the shell model (Dick and Overhauser, 1964) which, as discussed in Chapter 3, models the polarizability using a massive core linked to a mass-less shell by a spring. The theoretical basis of this model is uncertain, but its practical success has been attested over 20 years. Probably the best way to consider it is as a sensible model for linking the electronic polarizability of the ions to the forces exerted by the surrounding lattice. It is therefore a many-body term, a fact that should be remembered if one wishes to consider three-body potentials in the description of the crystal. A recent development in the field has been the use of quantum calculations. These are discussed in detail elsewhere (Chapter 8) but some results will be compared with the classical simulations in this chapter. 

The final term in this potential model is that due to polarisation effects. In the solid environment there is likely to be some distortion of electron clouds due to the surrounding electric field, and this must be taken into account when modelling the interactions of an essentially ionic system. The polarisability in this case is modelled using the shell model of Dick and Overhauser. Here the atom is considered to consist of a massless charged shell, for the valence electrons, and a charged core. The two components are linked via a harmonic spring, and displacement of the valence electrons takes place with respect to the following equation... 

Polarization of ions can be included in one of two ways. The natural approach is to use point ion polarizabilities, which has been successfully explored by Wilson and Madden (1996). An alternative, which has been used for many decades, is the so-called shell model (Dick and Overhauser 1958) as illustrated schematically in Figure 1. This is a simple mechanical model, in which an ion is represented by two particles-a core and a shell-where the core can be regarded as the representing the nucleus and inner electrons, while the shell represents the valence electrons. As such, all the mass is assigned to the core, while the total ion charge (qt = qc + qs) is split between both of the species. The core and shell interact by a harmonic spring constant, Kcs, but are Coulombically screened from each other. The polarizability is then given by ... 

The first step in modeling the mineral-water interface is to develop a reliable and consistent model for the interaction of water with solid surfaces. There is a wealth of different water potentials available (e.g., Duan et al. 1995 Jorgensen et al. 1983 Brodholt et al. 1995a,b). However, we require a potential that simulates polarizability and is compatible with our potential models for solid phases. Thus, we included polarizability by using the shell model (Dick and Overhauser 1958) for the oxygen atom of the water molecule. 

The shell model the ions are described by a core, including the nucleus and the inner electrons, and a zero-mass shell representing the valence electrons (Dick and Overhauser, 1958). The core and the shell bear opposite charges. They are harmonically coupled by a spring of stiffness k. The electric field / exerted by neighbouring ions shifts the shell with respect to the core position. If Y is the shell charge, induces a dipole moment equal to SY Ik. The ion polarizability ot in the model, is thus equal to o = Y jk. The value of k may be deduced from the value of the optical dielectric constant eClausius-Mosotti relationship ... 

The most popular model which takes into account both the ionic and electronic polarizabilities is the shell model of DICK and OVERHAUSER [4.12]. It is assumed that each ion consists of a spherical electronic shell which is isotropically coupled to its rigid ion-core by a spring. To begin with we consider a free ion which is polarized by a static field E. The spring constant is k, the displacement of the shell relative to its core is v and the charge of the shell is ye (Fig.4.7). In equilibrium, the electrostatric force yeE is equal to the elastic force kv yeE = kv. The induced dipole moment is d = yev = aE from which we obtain the free ion polarizability... 

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