Polar catastrophe model

The development of the methods described in Section 9.2 was an important step in modeling polarization because it led to accurate calculations of molecular polarizability tensors. The most serious issue with those methods is known as the polarization catastrophe since they are unable to reproduce the substantial decrease of the total dipole moment at distances close to contact as obtained from ab initio calculations. As noted by Applequist et al. [49], and Thole [50], a property of the unmodified point dipole is that it may originate infinite polarization by the cooperative interaction of the two induced dipoles in the direction of the line connecting the two. The mathematical origins of such singularities are made more evident by considering a simple system consisting of two atoms (A and B) with isotropic polarizabilities, aA and c b. The molecular polarizability, has two components, one parallel and one perpendicular to the bond axis between A and B,... 

A large number of polarizable models have been developed for water, many of them with one polarizable site (with a = 1.44 A ) on or near the oxygen position. For these models, the polarizable sites do not typically get close enough for polarization catastrophes (4aa) = 1.4 A, see comments after Eq. [16], so screening is not as necessary as it would be if polarization sites were on all atoms. However, some water models with a single polarizable site do screen the dipole field tensor. Another model for water places polarizable sites on bonds. Other polarizable models have been used for monatomic ions and used no screening of T or gO 15,16,27,34 Polarizable models have been developed for proteins as well, by Warshel and co-workers (with screening of T but not and by Wodak and... 

One feature of the semiempirical models is that because the polarization is described by a set of coefficients that have a normalization condition, for example, Eq. [69], there will be no polarization catastrophe like there can be with dipole polarizable or fluctuating charge models. With a finite basis set, the polarization response is limited and can become only as large as the state with the largest dipole moment. 

Although the first theoretical treatment of the surface phase transitions appeared within the frames of the TPC modef " a few years after the first experimental evidences of this phenomenon," " problems closely related to phase transitions tantalized the relevant research area for more than two decades. These are the interrelated issues of the polarization catastrophe, the equivalence or not of the electrical variables as well as the equivalence or not of the various statistical mechanical treatments. Due to their significance, these issues are discussed separately in the section below. Here, we focus our attention on the types and properties of the phase transitions predicted by the models for electrosorption. 

Under normal circumstances water oxygens will never approach each other this closely, and the polarization catastrophe should not be a problem. If, on the other hand, the polarization density is modeled by point polarizations located at both oxygen and hydrogen positions, the occurrence of unphysical polarization energies should be monitored. 

The Thole model is based on a modified dipole-dipole interaction, which can be reformulated in terms of the interaction of smeared charge densities. This eliminates the divergence of the head-to-taU dipole-dipole interaction at small interseparations (Angstrom scale) [103-105]. Smearing out the charge distribution mimics the nature of the quantum mechanical wavefunction, which effectively guards against this unphysical polarization catastrophe. 

In the (110) direction, SrTiOs is composed of alternating layers of SrTiO" and 02 (Fig. 6). This leads to a polar surface, similar to the (111) surface of SrTiOs a nominal charge of 2 per surface unit cell will maintain valence neutrality. Perhaps the simplest way of solving the polar catastrophe is to terminate the (110) surface in a half-occupied O2" layer (half02. Fig. 6) [32 0], or variations thereupon [32-36]. Other commonly considered models for the (110) surface are the TiO faceted model and the Sr faceted model. The TiO faceted model has TiO rows in bulk-like positions atop an 02" termination and is equivalent to the (00 X 1) member of the homologous series (see below). The Sr facet model consists of an Sr -adatom in a bulk-like position atop at O2" termination [32 0]. 

These observations were the basis for the proposal that polymers, like ionic crystals, exhibit shock-induced polarization due to mechanically induced defects which are forced into polar configurations with the large acceleration forces within the loading portion of the shock pulse. Such a process was termed a mechanically induced, bond-scission model [79G01] and is somewhat supported by independent observations of the propensity of polymers to be damaged by more conventional mechanical deformation processes. As in the ionic crystals, the mechanically induced, bond-scission model is an example of a catastrophic shock compression model. 

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