Symmetry polar

There are a number of other technical details associated with HF and other ah initio methods that are discussed in other chapters. Basis sets and basis set superposition error are discussed in more detail in Chapters 10 and 28. For open-shell systems, additional issues exist spin polarization, symmetry breaking, and spin contamination. These are discussed in Chapter 27. Size-consistency and size-extensivity are discussed in Chapter 26. 

We note that the bilayer smectic phase which may be formed in main-chain polymers with two odd numbered spacers of different length (Fig. 7), should also be polar even in an achiral system [68]. This bilayer structure belongs to the same polar symmetry group mm2 as the chevron structure depicted in Fig. 17b, and macroscopic polarization might exist in the tilt direction of molecules in the layer. From this point of view, the formation of two-dimensional structure of the type shown in Fig. 7, where the polarization directions in neighbouring areas have opposite signs, is a unique example of a two dimensional antiferroelectric structure. 

Macroscopic polarity (polar symmetry) in materials is an extremely important structural feature providing a dizzying array of useful properties including... 

Polar symmetry (point group Coov or lower) is quite familiar at the molecular level as the symmetry required for the existence of a molecular dipole moment. Molecules possessing higher, nonpolar symmetry, cannot possess a permanent molecular dipole moment even when there are bond dipoles. Until the 1970s, no LC phases were known to possess polar symmetry, in spite of the fact that most mesogens are polar. 

Polar structures may have rotation symmetry and reflection symmetry. However, there can be no rotation or reflection normal to the principal rotation axis. Thus, the presence of the mirror plane normal to the C2 axis precludes any properties in the SmC requiring polar symmetry the SmC phase is nonpolar. 

This idea is elegant for its simplicity and also for its usefulness. While often in phenomenological theories of materials, control of parameters with molecular structure would provide useful properties, but the parameters are not related in any obvious way to controllable molecular structural features. Meyer s idea, however, is just the opposite. Chemists have the ability to control enantiomeric purity and thus can easily create an LC phase lacking reflection symmetry. In the case of the SmC, the macroscopic polar symmetry of this fluid phase can lead to a macroscopic electric dipole, and such a dipole was indeed detected by Meyer and his collaborators in a SmC material, as reported in 1975.2... 

In fact, as also indicated in Figure 8.12, an achiral SmC phase possesses antiparallel polarized sheets, in this case with a pitch of half the layer spacing. Photinos and Samulski have made much of this polar symmetry of the SmC phase,28 but neither the SmCA nor the SmC phases have net macroscopic polar symmetry (the SmCA is Di/, while the SmC is C21, as mentioned above), and thus neither shows properties associated with polar materials (e.g., a pyroelectric effect). 

Figure 8.12 Longitudinal sheets with antiparallel polar symmetry are illustrated for achiral SmCA and SmC phases. Since it is not possible to switch to ferroelectric state in such system upon application of electric field, these structure should not be considered antiferroelectric.

This volume of Topics in Stereochemistry could not be complete without hearing about ferroelectric liquid crystals, where chirality is the essential element behind the wide interest in this mesogenic state. In Chapter 8, Walba, a pioneering contributor to this area, provides a historical overview of the earlier key developments in this field and leads us to the discovery of the unique banana phases. This discussion is followed by a view of the most recent results, which involve, among others, the directed design of chiral ferroelectric banana phases, which display spontaneous polar symmetry breaking in a smectic liquid crystal. 

In principle, an anisotropic reaction performed on a crystal of polar symmetry may fix the absolute direction of the polar axis. In the case of an asymmetric reaction carried out in a centrosymmetric (enantiopolar) crystal, one may establish the absolute configuration of the chiral product. The degree of reliability of the assignment will depend on knowledge of the various states of the reaction pathway. Here we briefly describe some heterogeneous reactions in polar and enantiopolar crystals that illustrate this approach. 

The existence of the layers and director tilt in the achiral smectic C liquid crystal phase are experimental facts. Given these, the maximum possible symmetry of the phase would be Ci, with a C2 axis normal to the tilt plane, and a a plane congruent with the tilt plane. In fact, there is no fundamental reason why a given C phase must possess either of these symmetry elements. But, breaking of either of the symmetry elements would afford polar symmetry, and no C phase has ever been shown to possess any property associated with polar symmetry (e.g. pyroelectricity). Therefore, we can say that all known C phases indeed possess the maximum possible symmetry consistent with the layers and tilt. 

Eleven acentric crystal classes are chiral, i.e., they exist in enantiomorphic forms, whereas ten are polar, i.e., they exhibit a dipole moment. Only five (1,2, 3, 4, and 6) have both chiral and polar symmetry. All acentric crystal classes except 432 possess the same symmetry requirements for materials to display piezoelectric and SHG properties. Both ferroelectricity and pyroelectricity are related to polarity a ferroelectric material crystallizes in one of ten polar crystal classes (1, 2, 3,4, 6, m, mm2, 3m, 4mm, and 6mm) and possesses a permanent dipole moment that can be reversed by an applied voltage, but the spontaneous polarization (as a function of temperature) of a pyroelectric material is not. Thus all ferroelectric materials are pyroelectric, but the converse is not true. 

For dipolar chromophores that are the subject of this chapter, only one component of the molecular hyperpolarizability tensor, Pzzz, is important. Thus, the summation in Eq. (8) disappears. Electric field poling induces Cv cylindrical polar symmetry. Assuming Kleinman [12] symmetry, only two independent components of the macroscopic second-order nonlinear optical susceptibility tensor... 

Are there further manifestations of a polar axis in crystals Obviously, the morphology,the crystal growth speed,the form of etch figures,and the chemical reactivity of faces are properties that can express a polar symmetry. 

Polarization that exists within a material in the absence of the application of an external field. It requires nonenantiomorphic polar symmetry. It is usually used in the context of electric fields and electrical polarization. Materials that exhibit spontaneous polarization and are piezoelectric are able to retain an ionic polarizarimi and therefore said to be ferroelectric. 

The reaction is useful in preparations of isoregic ordered chains with translational polar symmetry. It can also be applied in polymerizations of functional or chiral monomers. 

The isotropic phase formed by achiral molecules has continuous point group symmetry Kh (spherical). According to the group representations [5], upon cooling, the symmetry Kh lowers, at first, retaining its overall translation symmetry T(3) but reduces the orientational symmetry down to either conical or cylindrical. The cone has a polar symmetry Coov and the cylinder has a quadrupolar one Dooh- The absence of polarity of the nematic phase has been established experimentally. At least, polar nematic phases have not been found yet. In other words, there is a head-to-tail symmetry taken into account by introduction of the director n(r), a unit axial vector coinciding with the preferred direction of molecular axes dependent on coordinate (r is radius-vector). 

A molecule that, in principle, may form a bowl phase should itself have the bowl form like that seen in Fig. 4.15a [7]. Molecular bowls may have different symmetry as shown on the top of Fig. 4.15b and the corresponding phases could be either uniaxial or biaxial. The packing of bowls into the columns may have specific features. For example, when all molecules in the colunui are oriented bottom down then the head-to-to tail symmetry is broken and the column has conical, i.e. polar symmetry Coov> Fig- 4.15c. Only polar colunuis may form ferroelectric or antiferroelectric phases shown in Fig. 4.15d. 

Note that the polar vector reflects only polar symmetry of the interfacial layer and may be associated with the conical (not rod-like) form of the molecules. However, when the electric charges are involved in the game, the same polar order may results in appearance of the macroscopic surface electric polarization Psurf that is the dipole moment of a unit volume [units CGS(charge) cm/cm = CGSQ/cm = StatV/cm, or C/m in SI system]. When an electric field is applied to a liquid crystal the surface polarization contributes to the free energy of a surface layer... 

The new polar symmetry allows for the existence of macroscopic polarization, large or small, depending on the magnitude of the strain and molecular dipole moments shown by small arrows. Due to the distortions, the densest packing of our pears and bananas results in some preferable ahgimient of molecular skeletons in such a way that molecular dipoles look more up than down. By definition, the dipole moment of the unit volume is electric polarization. These simple arguments brought R. Meyer to the brilliant idea of piezoelectric polarization [25] ... 

Since discovery of chiral ferroelectrics in 1975, a search for the achiral analogues of liquid crystal ferro- and antiferroeiectrics was a challenge to researchers, both theoreticians and experimentalists and recently there was a great progress in this area. The idea was to find a way to break non-polar symmetry Dooh or C2h of... 

Like solid ferroelectrics, the ferroelectric liquid crystals, particularly the FLCPs, show a pyroelectric effect and a piezoelectric effect and are capable of switching polarization direction (dielectric hysteresis). Moreover, they can switch propagating or reflected polarized light. Finally, the polar symmetry of the phase leads to nonlinear optical properties of the FLCPs such as second-harmonic generation, the Pockels effect, and the Kerr effect. These physical properties of the ferroelectric LC polymers are discussed in the following sections. 

As mentioned above, the symmetry of the ferroelectric smectic C phase corresponds to the polar symmetry group C2, Fig. 7.1, so that when going along the -coordinate parallel to a helix axis and perpendicular to the smectic layers the director L and the polarization vector P, directed along the C2 axis, rotate such as... 

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