Catastrophe, polarization

Eo and E (Afi(i)) are respectively the electric fields generated by the permanent and induced multipoles moments. a(i) represents the polarisability tensor and Afi(i) is the induced dipole at a center i. This computation is performed iteratively, as Epoi generally converges in 5-6 iterations. It is important to note that in order to avoid problems at the short-range, the so-called polarization catastrophe, it is necessary to reduce the polarization energy when two centers are at close contact distance. In SIBFA, the electric fields equations are dressed by a Gaussian function reducing their value to avoid such problems. 

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,... 

FERROELECTRIC EFFECT. The phenomenon whereby certain crystals may exhibit a spontaneous dipole moment twhich is called ferroelectric by analogy with ferromagnetic—exhibiting a permanent magnetic moment). The effect in the most typical case, barium manate. seems to he due to a polarization catastrophe, in which the local electric fields due lo the polarizuiion itself increase faster than die elastic restoring forces on the ions in Ihe crystal, thereby leading to an asymmetrical shift in ionic positions, and hence lo a permanent dipole moment. Ferroelectric crystals... 

Note that the energy is the dot product of the induced dipole and the static field, not the total field. Without a static field, there are no induced dipoles. Induced dipoles alone do not interact strongly enough to overcome the polarization energy it takes to create them (except when they are close enough to polarize catastrophically). 

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. 

W. Preyland, K. Garbade, and E. Pfeiffer (1983) Optical study of electron localization approaching a polarization catastrophe in hquid Ka -KCli j . Phys. Rev. Lett. 51, p. 1304... 

We should point out that if we use the potential drop A( ) (or E) to express the properties of the adsorbed layer, the above peculiarity disappears and we observe the expected behavior, which is the decrease in the adsorption of a neutral solute in the presence of specifically adsorbed anions. Thus, the experimental and theoretical study of the effect of the specific adsorption on the adsorption features of an organic compound gives the first indication that the two electrical variables, and A(j), are not equivalent in expressing the properties of the adsorbed layer and that the use of as the independent electrical variable should be handled with care. This issue of the equivalence of the two electrical variables is also discussed from another point of view in the section Polarization Catastrophe and Other Artifacts below. 

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. 

It is seen that the polarization catastrophe is in fact an innocent artifact, because it is now easily detected and amended. In contrast, there are other artifacts appearing in the adsorption isotherms when is used as the independent electrical variable which are difficult to be identified. Note especially the artifact curve (2) in Figure 18, which does not exhibit any sign of phase transition. However, even these artifacts can be amended either by the use of the generalized ensemble A or the procedure suggested above. At any rate we observe once more that the electrode charge density when it is used as an independent electrical variable, is a difficult variable, which should be handled with much care. 

Pauli blockade (p. 842) penetration energy (p. 814) permanent multipoles (p. 826) polarization catastrophe (p. 858) polarization collapse (p. 836) polarization perturbation theory (p. 840) rotaxans (p. 801)... 

This equation has a singularity at y 3/4jt/tv for large enough molecular polarizability the macroscopic susceptibility and, consequently, polarization become infinite. This phenomenon is called polarization catastrophe. In a more subtle approach, the polarization remains finite and exists even in the absence of the external field spontaneous polarization PJ. The spontaneous polarization is responsible for pyro- and ferroelectricity in solid and liquid crystals, however it is not observed in the isotropic liquid (see Chapters 4 and 13). 

This formula is very important for the further discussion because it predicts the polarization catastrophe . For small molecular polarizability y, susceptibility x depends linearly on y. However, when y 3/Annv, the denominator of (13.2) tends to zero and diverges. 

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