Point-Group Symmetry

For an undistorted octahedral metal-oxide molecule MOg (Oj point-group symmetry), a treatment similar to that for the MO4 tetrahedron may be carried out using 

A generally applicable relationship has been developed by Brown and Wu for relating Pauling bond strengths to bond lengths for metal-oxygen bonds  

As we will demonstrate, luminescent properties, radiative transition characteristics as well as emission under site selective excitation depend on the local environment s)rmmetry of the luminescent center. Therefore it is necessary to take into account and to describe the different local symmetry. There are two systems commonly used in describing symmetry elements of punctual groups  

The Hermann-Mauguin (1935) or international notation preferred by crys-tallographers. 

Schonflies notation is widely used to describe molecules or assemblages of atoms (polyhedron) such as the local environment of an atom. Thus, it is widely used to describe the symmetry of structural sites. It is a more compact notation but less complete than the Hermann-Mauguin notation. It consists generally of one capital letter, followed by one subscript number and one final letter. 

The concept of symmetry is equally important for understanding properties of individual molecules, crystals and liquid crystals [1]. The symmetry is of special importance in physics of liquid crystal because it allows us to distinguish numerous liquid crystalline phases from each other. In fact, all properties of mesophases are determined by their symmetry [2], In the first section we consider the so-called point group symmetry very often used for discussion of the most important hquid crystalline phases. A brief discussion of the space group symmetry will be presented in Section 2.2. 

It follows from eq. (3) that the gradient of E(k) normal to the zone boundary vanishes at a face center, 

Let k c BZ and let 8 be a small increment in k normal to the face which is parallel to the symmetry plane through T. The perpendicular distance from T to the center of the face is /lb,. Then 

Since the physical system (crystal) is indistinguishable from what it was before the application of a space-group operator, and a translational symmetry operator only changes the phase of the Bloch function without affecting the corresponding energy E(k), 

Equation (2) shows that the band energy has the symmetry of the point group of the wave vector. As R runs over the whole R P(k), it generates a set of degenerate eigenfunctions 

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Operators that eommute with the Hamiltonian and with one another form a partieularly important elass beeause eaeh sueh operator permits eaeh of the energy eigenstates of the system to be labelled with a eorresponding quantum number. These operators are ealled symmetry operators. As will be seen later, they inelude angular momenta (e.g., L2,Lz, S, Sz, for atoms) and point group symmetries (e.g., planes and rotations about axes). Every operator that qualifies as a symmetry operator provides a quantum number with whieh the energy levels of the system ean be labeled. 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

In a second example, the three CH bonds, three CH antibonds, CO bond and antibond, and three 0-atom non-bonding orbitals of the methoxy radical H3C-O also cluster into ai and e orbitals as shown below. In these cases, point group symmetry allows one to identify degeneracies that may not have been apparent from the structure of the orbital interactions alone. 

Using the hybrid atomie orbitals as labeled above (funetions fi-f/) and the D3h point group symmetry it is easiest to eonstruet three sets of redueible representations ... 

Electronic Wavefunctions Must Also Possess Proper Symmetry. These Include Angular Momentum and Point Group Symmetries... 

If the atom or moleeule has additional symmetries (e.g., full rotation symmetry for atoms, axial rotation symmetry for linear moleeules and point group symmetry for nonlinear polyatomies), the trial wavefunetions should also eonform to these spatial symmetries. This Chapter addresses those operators that eommute with H, Pij, S2, and Sz and among one another for atoms, linear, and non-linear moleeules. 

The method of vibrational analysis presented here ean work for any polyatomie moleeule. One knows the mass-weighted Hessian and then eomputes the non-zero eigenvalues whieh then provide the squares of the normal mode vibrational frequeneies. Point group symmetry ean be used to bloek diagonalize this Hessian and to label the vibrational modes aeeording to symmetry. 

A. The Electronic Transition Dipole and Use of Point Group Symmetry... 

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. 

For a function to transform according to a specific irreducible representation means that the function, when operated upon by a point-group symmetry operator, yields a linear combination of the functions that transform according to that irreducible representation. For example, a 2pz orbital (z is the C3 axis of NH3) on the nitrogen atom... 

We now return to the symmetry analysis of orbital produets. Sueh knowledge is important beeause one is routinely faeed with eonstrueting symmetry-adapted N-eleetron eonfigurations that eonsist of produets of N individual orbitals. A point-group symmetry operator S, when aeting on sueh a produet of orbitals, gives the produet of S aeting on eaeh of the individual orbitals... 

For the nanotubes, then, the appropriate symmetries for an allowed band crossing are only present for the serpentine ([ , ]) and the sawtooth ([ ,0]) conformations, which will both have C point group symmetries that will allow band crossings, and with rotation groups generated by the operations equivalent by conformal mapping to the lattice translations Rj -t- R2 and Ri, respectively. However, examination of the graphene model shows that only the serpentine nanotubes will have states of the correct symmetry (i.e., different parities under the reflection operation) at the K point where the bands can cross. Consider the K point at (K — K2)/3. The serpentine case always sat-... 

Several sections of the diffraction space of a chiral SWCNT (40, 5) are reproduced in Fig. 11. In Fig. 11(a) the normal incidence pattern is shown note the 2mm symmetry. The sections = constant exhibit bright circles having radii corresponding to the maxima of the Bessel functions in Eq.(7). The absence of azimuthal dependence of the intensity is consistent with the point group symmetry of diffraction space, which reflects the symmetry of direct space i.e. the infinite chiral tube as well as the corresponding diffraction space exhibit a rotation axis of infinite multiplicity parallel to the tube axis. 

The 180° trans structure is only about 2.5 kcal/mol higher in energy than the 0° conformation, a barrier which is quite a bit less than one would expect for rotation about the double bond. We note that this structure is a member of the point group. Its normal modes of vibration, therefore, will be of two types the symmetrical A and the non-symmetrical A" (point-group symmetry is maintained in the course of symmetrical vibrations). 

Figure 13.17 Molecular structure of some sulfides of arsenic, stressing the relationship to the AS4 tetrahedron (point group symmetry in parentheses).

Cation Formal oxidation state Cluster structure 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). 

Figure 15.6 Various representations of the molecular structure of ryclc-Si2 showing S atoms in three parallel planes. I he idealized point group symmetry is and the mean dihedral angle is 86.1 5.5 . In the crystal the symmetry is slightly distorted to C21, and the central group of 6 S atoms deviate from eoplanarily by 14pm.

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. 

A convenient scheme for identifying the point group symmetry of any given species is set out in the flow chart. Starting at the top of the chart... 

Figure A2.1 Point group symmetry flow chart.

Point group symmetry, notation and representations, and the group theoretical condition for when an integral is zero. 

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