Point groups symmetry elements

For sites of cubic symmetry the point-group symmetry elements mix the spherical harmonic basis functions. As a result, linear combinations of spherical harmonic functions, referred to as Kubic harmonics (Von der Lage and Bethe 1947), must be used. 

Upon appropriate reduction in the number and nature of the independent tensorial components of i/i(s) (= j/, ), resulting from the common point-group symmetry elements of the sphere and cube (applied to fourth-rank tensors), the material tensor can be shown quite generally to be of the form (Zuzovsky et al., 1983)... 

A few pairwise interfaces in the range 1300-1900 are nevertheless observed in crystals. Almost all result from the presence of twofold and other point-group symmetry elements, which are relatively uncommon in crystals of monomeric proteins. Their occurrence suggests that the formation of dimers or other small oligomers in solution precedes crystallization under the conditions where these particular crystals are obtained at protein concentrations typically in the range 10 —10 M. The large majority of the crystal contacts are associated with lattice translations and screw rotations not found in oligomeric proteins. Their size distribution resembles that of the transient interfaces created by the random collision of two small proteins simulated in the computer... 

Point group Symmetry element (h order) Example... 

Fig. 9. Schematic representation of non-coordinated and coordinated CCh ion and the corresponding point group symmetry elements. The changes in the Vj and V3 IR vibrations of the COs " ion upon coordination are also shown. For simplicity, only monodentate coordination is presented. Notations I - identity, Cn - n-fold axis of rotation, Oh, a, - mirror planes perpendicular and parallel to the principal axis, respectively, Sn - n-fold rotation-reflection operation. The number preceding the symmetry operation symbol refers to number of such symmetry elements that the molecule possesses. For further details consult Nakamoto, 1997.

In the phase, where some point group symmetry element is lost, there are at least two orientational states [3]. The regions, where these states occur, are called domains, and the regions separating the domains are called domain walls. 

Definition (Orthogonal group, point group, symmetry elements)... 

The ideal crystal is a rigid, three-dimensional array of molecules extending infinitely in all directions. This is the model used to evaluate the symmetry of a group of real atoms. The infinite extent of this array allows us to add new symmetry operations to our list of point group symmetry elements (Section 6.1). Previously, we counted only operations that leave the center of mass unchanged. However, the center of mass is not defined for an infinite number of atoms, so we can ignore that constraint now by adding translational symmetry elements to the list. 

Molecular point-group symmetry can often be used to determine whether a particular transition s dipole matrix element will vanish and, as a result, the electronic transition will be "forbidden" and thus predicted to have zero intensity. If the direct product of the symmetries of the initial and final electronic states /ei and /ef do not match the symmetry of the electric dipole operator (which has the symmetry of its x, y, and z components these symmetries can be read off the right most column of the character tables given in Appendix E), the matrix element will vanish. 

It is assumed that the reader has previously learned, in undergraduate inorganie or physieal ehemistry elasses, how symmetry arises in moleeular shapes and struetures and what symmetry elements are (e.g., planes, axes of rotation, eenters of inversion, ete.). For the reader who feels, after reading this appendix, that additional baekground is needed, the texts by Cotton and EWK, as well as most physieal ehemistry texts ean be eonsulted. We review and teaeh here only that material that is of direet applieation to symmetry analysis of moleeular orbitals and vibrations and rotations of moleeules. We use a speeifie example, the ammonia moleeule, to introduee and illustrate the important aspeets of point group symmetry. 

The structure of cyclo-Sio is shown in Fig. 15.5(b).The molecule belongs to the very rare point group symmetry Do (three orthogonal twofold axes of rotation as the only symmetry elements). ITie mean interatomic distance and bond angle are close to those in cyclo-Su (Table 15.5) and the molecule can be regarded as composed of two identical S5 units obtained from the S 2 molecule (Fig. 15.6). 

It is instructive to add to these examples from the numerous instances of point group symmetry mentioned throughout the text. In this way a facility will gradually be acquired in discerning the various elements of symmetry present in a molecule. 

The mathematical expression of N(6, q>, i//) is complex but, fortunately, it can be simplified for systems displaying some symmetry. Two levels of symmetry have to be considered. The first is relative to the statistical distribution of structural units orientation. For example, if the distribution is centrosymmetric, all the D(mn coefficients are equal to 0 for odd ( values. Since this is almost always the case, only u(mn coefficients with even t will be considered herein. In addition, if the (X, Y), (Y, Z), and (X, Z) planes are all statistical symmetry elements, m should also be even otherwise = 0 [1]. In this chapter, biaxial and uniaxial statistical symmetries are more specifically considered. The second type of symmetry is inherent to the structural unit itself. For example, the structural units may have an orthorhombic symmetry (point group symmetry D2) which requires that n is even otherwise <>tmn = 0 [1], In this theoretical section, we will detail the equations of orientation for structural units that exhibit a cylindrical symmetry (cigar-like or rod-like), i.e., with no preferred orientation around the Oz-axis. In this case, the ODF is independent of t/z, leading to n — 0. More complex cases have been treated elsewhere [1,4]... 

All the possible combinations of these symmetry elements result in 32 crystallographic point-group symmetries or crystal classes their symbols are listed in Table 3.3. Notice that in putting together the symbols to denote the symmetries of any crystal classes the convention is to give the symmetry of the principal axis first for instance 4 or 4, for tetragonal classes. If there is a plane of symmetry perpendicular to the principal axis, the two symbols are associated as in 4 m or Aim (4 over m), then the symbols for the secondary axes, if any, follow, and then any other symmetry planes. In a symbol such as Almmm, the second and third m refer to planes parallel to the four-fold axis. 

Since Eqs. (B.7) and (B.8) are equal, this implies that a rotational symmetry element of direct space is also a rotational symmetry element of reciprocal space. This result must be correct since X-ray scattering is a physical property of the crystal, it must at least have the point-group symmetry of the crystal. 

In addition, group theory can be used to assess when transition dipole moments must be zero. The product of the irreducible representations of the two wave functions and the dipole moment operator within the molecular point group symmetry must contain the totally symmetric representation for the matrix element to be non-zero (note that, if the molecule does not contain an inversion center, the operator r does not belong to any single irrep, except for the trivial case of Ci symmetry see Appendix B for more details). A consequence of this consideration is that, for instance, electronic transitions between states of the same symmetry are forbidden in molecules possessing inversion centers. 

The thirty-two point-group symmetries or crystal classes. All the possible point-group symmetries—the combinations of symmetry elements exhibited by idealized crystal shapes—are different combinations of the symmetry elements already described, that is, the centre of symmetry (T), the plane of symmetry (m), the axes of symmetry (2, 3, 4, and 0), and the inversion axes (3, 4, and 6). 

Figure 16.9. Location of some of the equivalent points and symmetry elements in the unit cell of space group Pnali. An open circle marked + denotes the position of a general point xyz, the + sign meaning that the point lies at a height z above the xy plane. Circles containing a comma denote equivalent points that result from mirror reflections. The origin is in the top left comer, and the filled digon with tails denotes the presence of a two-fold screw axis at the origin. Small arrows in this figure show the directions of a1 a2, which in an orthorhombic cell coincide with x, y. The dashed line...

The 32 crystallographic point groups result from combinations of symmetry based on a fixed point. These symmetry elements can be combined with the two translational symmetry elements the screw... 

Most electronic transitions between different states of the f-electrons are dominated by electric dipole transitions. Only in exceptional cases like Eu(III), magnetic dipole transitions are found to be as strong as electric dipole transitions. However, in the case of an f element, electric-dipole transitions between the 4fw states are forbidden because the parity of initial and final state is conserved. Only when the f element is embedded in a crystal providing a point group symmetry that does not contain the inversion operation, these transitions can be observed readily. 

This group has symmetry element E, a principal Cn axis, n secondary C2 axes perpendicular to Cn, and a ah also perpendicular to C . The necessary consequences of such combination of elements are a S axis coincident with the Cn axis and a set of n ctv s containing the C2 axes. Also, when n is even, symmetry center i is necessarily present. The BrF molecule has point group symmetry D4h, as shown in Fig. 6.1.8. Examples of other molecules belonging to point groups >2h, D3h, Z>5h and D6h are given in Fig. 6.2.6. 

---