Groups, symmetry

We collect syimnetry operations into various syimnetry groups , and this chapter is about the definition and use of such syimnetry operations and symmetry groups. Symmetry groups are used to label molecular states and this labelling makes the states, and their possible interactions, much easier to understand. One important syimnetry group that we describe is called the molecular symmetry group and the syimnetry operations it contains are pemuitations of identical nuclei with and without the inversion of the molecule at its centre of mass. One fascinating outcome is that indeed for... 

Energy minimisation and normal mode analysis have an important role to play in the study of the solid state. Algorithms similar to those discussed above are employed but an extra feature of such systems, at least when they form a perfect lattice, is that it is can be possible to exploit the space group symmetry of the lattice to speed up the calculations. It is also important to properly take the interactions with atoms in neighbouring cells into account. 

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. 

For all point, axial rotation, and full rotation group symmetries, this observation holds if the orbitals are equivalent, certain space-spin symmetry combinations will vanish due to antisymmetry if the orbitals are not equivalent, all space-spin symmetry combinations consistent with the content of the direct product analysis are possible. In either case, one must proceed through the construction of determinental wavefunctions as outlined above. 

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... 

Any perturbation from ideal space-group symmetry in a crystal will give rise to diffuse scattering. The X-ray diffuse scattering intensity at some point (hkl) in reciprocal space can be written as... 

Inspired by experimental observations on bundles of carbon nanotubes, calculations of the electronic structure have also been carried out on arrays of (6,6) armchair nanotubes to determine the crystalline structure of the arrays, the relative orientation of adjacent nanotubes, and the optimal spacing between them. Figure 5 shows one tetragonal and two hexagonal arrays that were considered, with space group symmetries P42/mmc P6/mmni Dh,), and P6/mcc... 

These properties are illustrative of the unique behavior of ID systems on a rolled surface and result from the group symmetry outlined in this paper. Observation of ID quantum effects in carbon nanotubes requires study of tubules of sufficiently small diameter to exhibit measurable quantum effects and, ideally, the measurements should be made on single nanotubes, characterized for their diameter and chirality. Interesting effects can be observed in carbon nanotubes for diameters in the range 1-20 nm, depending... 

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. 

---