Polarization shell model

Dynamic models for ionic lattices recognize explicitly the force constants between ions and their polarization. In shell models, the ions are represented as a shell and a core, coupled by a spring (see Refs. 57-59), and parameters are evaluated by matching bulk elastic and dielectric properties. Application of these models to the surface region has allowed calculation of surface vibrational modes [60] and LEED patterns [61-63] (see Section VIII-2). 

Figure 2.8 Shell model of ionic polarizability (a) unpolarized ion (no displacement of shell) (b) polarized (displaced shell) (c) interactions 1, core-core 2, shell-shell 3, core-shell.

The hyperactivity of, for example, lipases at low w -values (shown in Fig. 5) is explained by the water-shell-model [2]. The activity of the enzyme at w -values higher than 5 corresponds to its activity in bulk aqueous solutions. There exist two aqueous regions within a reverse micelle, schematically shown in Fig. 6. One is located in the inner part of the reverse micelle and has the same physical properties as bulk water the other is attached to the polar head groups of the surfactant and differs in its physical properties strongly from bulk water. 

Fig. 5. Water-shell-model schematically drafted location of two water parts in a reverse micelle. One is located in the inner part of the reverse micelle and has the same physical properties as bulk water, the other is attached to the polar headgroups of the surfactant...

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)). 

Using perturbation theory and the lowest seniority shell-model description of the 0+ pair distribution,one calculates a quadrupole polarization energy gain AEn = kL N, ( v-Nv ).F (7)... 

Shell model calculations predict a quasi-shell closure at 96Zr. Therefore, it is of interest to measure g-factors of states in 97Zr and test whether they can be described by simple shell model configurations. The 1264.4 keV level has a half-life of 102 nsec, and its g-factor was measured by the time-differential PAC method at TRISTAN [BER85a]. The result, g-0.39(4), is consistent with the Schmidt value of 0.43, which assumes no core polarization and the free value for the neutron g factor, g g free. This indicates that the 1264.4 keV level is a very pure single-particle state, thus confirming the shell model prediction of a quasi-shell closure at 96Zr. 

While these studies give good estimates for the IPs, it has been shown that the properties of DNA components are affected by the first few waters of hydration which mimic the first hydration shell around the molecule. For example, each water that acts as a net hydrogen bond donor to a base results in an elevation of the IP while each water that acts as a net hydrogen bond acceptor will tend to lower the IP [54], The solvation model, e.g., PCM (polarized continuum model), which takes into account the effect of the bulk solvent on the solute lacks these specific interactions and has the effect of substantially lowering the IP. Nevertheless, these first waters need to be included for a good accounting of IPs and EAs. 

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. 

The shell model has its origin in the Born theory of lattice dynamics, used in studies of the phonon dispersion curves in crystals/ Although the Born theory includes the effects of polarization at each lattice site, it does not account for the short-range interactions between sites and, most importantly, neglects the effects of this interaction potential on the polarization behavior. The shell model, however, incorporates these short-range interactions. 

To the extent that the polarization of physical atoms results in dipole moments of finite length, it can be argued that the shell model is more physically realistic (the section on Applications will examine this argument in more detail). Of course, both models include additional approximations that may be even more severe than ignoring the finite electronic displacement upon polarization. Among these approximations are (1) the representation of the electronic charge density with point charges and/or dipoles, (2) the assumption of an isotropic electrostatic polarizability, and (3) the assumption that the electrostatic interactions can be terminated after the dipole-dipole term. 

In the shell model, as mentioned above, the short-range repulsion and van der Waals interactions are taken to act between the shell particles. This finding has the effect of coupling the electrostatic and steric interactions in the system in a solid-state system where the nuclei are fixed at the lattice positions, polarization can occur not only from the electric field generated by neighboring atoms, but also from the short-range interactions with close neighbors (as, e.g., in the case of defects, substitutions, or surfaces). This ability to model both electrical and mechanical polarizability is one reason for the success of shell models in solid-state ionic materials. 

The polarization energy in the EE models can be compared directly to that in the polarizable point dipole and shell models. Consider the first term in Eq. [43],... 

One important difference between the shell model and polarizable point dipole models is in the former s ability to treat so-called mechanical polarization effects. In this context, mechanical polarization refers to any polarization of the electrostatic charges or dipoles that result from causes other than the electric field of neighboring atoms. In particular, mechanical interactions such as steric overlap with nearby molecules can induce polarization in the shell model, as further described below. These mechanical polarization effects are physically realistic and are quite important in some condensed-phase systems. 

To further illustrate the importance of coupling the electrostatic and short-ranged repulsion interactions, we consider the example of a dimer of polarizable rare gas atoms, as presented by Jordan et al. In the absence of an external electric field, a PPD model predicts that no induced dipoles exist (see Eq. [12]). But the shell model correctly predicts that the rare gas atoms polarize each other when displaced away from the minimum-energy (force-free) configuration. The dimer will have a positive quadrupole moment at large separations, due to the attraction of each electron cloud for the opposite nucleus, and a negative quadrupole at small separations, due to the exchange-correlation repulsion of the electron clouds. This result is in accord with ab initio quantum calculations on the system, and these calculations can even be used to help parameterize the model. ... 

In essence, this difference between shell models and PPD models arises from the former s treatment of the induced dipole as a dipole of finite length. Polarization in physical atoms results in a dipole moment of a small, but finite, extent. Approximating this dipole moment as an idealized point dipole, as in the PPD models, is an attractive mathematical approximation and produces... 

All polarizable models share the ability to polarize, by varying their charge distribution in response to their environment. In addition, shell models and EE models with charge-dependent radii have the ability to modify their polarizability—the magnitude of this polarization response—in response to their local environment. Consequently, it is reasonable to expect that shell models and mechanically coupled EE models may be slightly more transferable to different environments than more standard PPD and EE models. To date, it is not clear whether this expectation has been fully achieved. Although some shell-based models for both ionic and molecular compounds have been demonstrated to be transferable across several phases and wide ranges of phase points, " it is not clear that the transferability displayed by these models is better than that demonstrated in PPD- or EE-based models. And even with an environment-dependent polarizability, it has been demonstrated that the basic shell model cannot fully capture all of the variations in ionic polarizabilities in different crystal environments. ... 

Future directions in the development of polarizable models and simulation algorithms are sure to include the combination of classical or semiempir-ical polarizable models with fully quantum mechanical simulations, and with empirical reactive potentials. The increasingly frequent application of Car-Parrinello ab initio simulations methods " may also influence the development of potential models by providing additional data for the validation of models, perhaps most importantly in terms of the importance of various interactions (e.g., polarizability, charge transfer, partially covalent hydrogen bonds, lone-pair-type interactions). It is also likely that we will see continued work toward better coupling of charge-transfer models (i.e., EE and semiem-pirical models) with purely local models of polarization (polarizable dipole and shell models). 

Jackson, M. D., and R. G. Gordon (1988b). MEG investigation of low pressure silica-shell model for polarization. Phys. Chem. Mineral. 16, 212-20. 

The irreversible dehydration process indicates that the underlying dielectric continuum approach used in the anisotropic primitive model does not hold. Further numerical simulations are presently imdertaken at the atomic scale in the frame of the polarized ion model (so-called shell model ) in order to give a better description of Tobennorite dehydration/rehydration and cohesive property of cement [24]. 

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