Point groups properties

A typical characteristic of hypersymmetry operations is that they exercise their influence in well-defined discrete domains. These domains do not overlap—they do not even touch each other. The usual hypersymmetry elements lead to point-group properties. This means that no infinite molecular chains could be selected, for example, to which these hypersymmetry operations would apply. They affect, instead, pairs of molecules or very small groups of molecules. Thus, they can really be considered as local point-group operations. These hypersymmetry elements, accordingly, divide the whole crystalline system into numerous small groups of molecules, or transform the crystal space into a layered structure. 

Note that Equation [51] represents formally the tensorial relationship, while Equation [52] expresses this relation by the (Cartesian) components of P and E and some coefficients whereas Equation [53] states this relation by using Einstein s summation convention. Here, the coefficients are the components of the electric susceptibility tensor which is a tensor of rank 2. The tensor % is an example of what is usually called a property tensor or matter tensor. Strictly speaking, property tensors describe physical properties of the static crystal which belong to the totally symmetric irreducible representation of the relevant point group. Properties, however, that depend on vibrations of the crystal lattice are described by tensors which belong to the different irreducible representations. The corresponding tensors are then often designated as tensorial covariants. 

Each such nonual mode can be assigned a synuuetry in the point group of the molecule. The wavefrmctions for non-degenerate modes have the following simple synuuetry properties the wavefrmctions with an odd vibrational quantum number v. have the same synuuetry as their nonual mode 2the ones with an even v. are totally symmetric. The synuuetry of the total vibrational wavefrmction (Q) is tlien the direct product of the synuuetries of its constituent nonual coordinate frmctions (p, (2,). In particular, the lowest vibrational state. 

The Couplitig-Coefficierits lJ ABC abc) for the Complex Form of a Doubly Degenerate Representation in the Octahedral Group, Following G. F. Koster et al.. Properties of tke Thirt i-Two Point Groups, MIT Press, MA, 1963, pp, 8, 52. 

In fact, there is a hierarchy in calculating molecular properties by additivity of atomic, bond, or group properties, as was pointed out some time ago by Benson [1, 2]. The larger the substructures that have to be considered, the larger the number of inaements that can be derived and the higher the accuracy in the values obtained for a molecular property. 

Most ah initio calculations use symmetry-adapted molecular orbitals. Under this scheme, the Hamiltonian matrix is block diagonal. This means that every molecular orbital will have the symmetry properties of one of the irreducible representations of the point group. No orbitals will be described by mixing dilferent irreducible representations. 

The total electron density contributed by all the electrons in any molecule is a property that can be visualized and it is possible to imagine an experiment in which it could be observed. It is when we try to break down this electron density into a contribution from each electron that problems arise. The methods employing hybrid orbitals or equivalent orbitals are useful in certain circumsfances such as in rationalizing properties of a localized part of fhe molecule. Flowever, fhe promotion of an electron from one orbifal fo anofher, in an electronic transition, or the complete removal of it, in an ionization process, both obey symmetry selection mles. For this reason the orbitals used to describe the difference befween eifher fwo electronic states of the molecule or an electronic state of the molecule and an electronic state of the positive ion must be MOs which belong to symmetry species of the point group to which the molecule belongs. Such orbitals are called symmetry orbitals and are the only type we shall consider here. 

For the orbital parts of the electronic wave functions of two electronic states the selection rules depend entirely on symmetry properties. [In fact, the electronic selection rules can also be obtained, from symmetry arguments only, for diatomic molecules and atoms, using the (or and Kf point groups, respectively but it is more... 

Crystals with one of the ten polar point-group symmetries (Ci, C2, Cs, C2V, C4, C4V, C3, C3v, C(, Cgv) are called polar crystals. They display spontaneous polarization and form a family of ferroelectric materials. The main properties of ferroelectric materials include relatively high dielectric permittivity, ferroelectric-paraelectric phase transition that occurs at a certain temperature called the Curie temperature, piezoelectric effect, pyroelectric effect, nonlinear optic property - the ability to multiply frequencies, ferroelectric hysteresis loop, and electrostrictive, electro-optic and other properties [16, 388],... 

Spin Warns.—In Application to Point Groups, above, we considered the irreducible representations of magnetic point groups. These would be useful in obtaining the symmetry properties of localized states in magnetic crystals, as impurity or single ion states in the tight... 

Invariance principle, 664 Invariance properties of quantum electrodynamics, 664 Inventory problem, 252,281,286 Inverse collisions, 11 direct and, 12 Inverse operator, 688 Investment problem, 286 Irreducible representations of crystallographic point groups, 726 Isoperimetric problems, 305 Iteration for the inverse, 60... 

Magnetic ordering, 746 Magnetic point groups, 738, 739 international notation, 739 properties of, 740 Schonflies notation, 739 Shubnikov notation, 739 Magnetic point symmetry, determination of, 744... 

Papalexi, N., 356,371,377,381 Paramagnetic crystals point groups for, 737 symmetry properties of eigenstates, 745... 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

Coordinates such as these, which have the symmetry properties of the point group are known as symmetry coordinates. As they transform in the same manner as the IRs when used as basis coordinates, they factor the secular determinant into block-diagonal form. Thus, while normal coordinates most be found to diagonalize the secular determinant, the factorization resulting horn the use of symmetry coordinates often provides considerable simplification of the vibrational problem. Furthermore, symmetry coordinates can be chosen a priori by a simple analysis of the molecular structure. 

Podolsky method, Renner-Teller effect, triatomic molecules, Hamiltonian equations, 612—615 Poincare sphere, phase properties, 206 Point group symmetry ... 

This exercise may appear pointless until one asks about the effect of the symmetry operations on other properties associated with the point group, such as cartesian coordinates and rotations. 

Vibrations may be decomposed into three orthogonal components Ta (a = x, y, z) in three directions. These displacements have the same symmetry properties as cartesian coordinates. Likewise, any rotation may be decomposed into components Ra. The i.r. spanned by translations and rotations must clearly follow the appropriate symmetry type of the point-group character table. In quantum formalism, a transition will be allowed only if the symmetry product of the initial and final-state wave functions contains the symmetry species of the operator appropriate to the transition process. Definition of the symmetry product will be explained in terms of a simple example. 

Crystal lattices can be depicted not only by the lattice translation defined in Eq. (7.2), but also by the performance of various point symmetry operations. A symmetry operation is defined as an operation that moves the system into a new configuration that is equivalent to and indistinguishable from the original one. A symmetry element is a point, line, or plane with respect to which a symmetry operation is performed. The complete ensemble of symmetry operations that define the spatial properties of a molecule or its crystal are referred to as its group. In addition to the fundamental symmetry operations associated with molecular species that define the point group of the molecule, there are additional symmetry operations necessary to define the space group of its crystal. These will only be briefly outlined here, but additional information on molecular symmetry [10] and solid-state symmetry [11] is available. 

It is evident that methods analogous to the ones developed here could be applied to molecular properties which, instead of being pseudoscalar, belong to some other representation of the skeleton point group (vector, tensor, etc. properties). To treat such properties, one needs only to induce from a different representation of than the chiral one. 

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