Symmetry, axes groups

In symmetry point groups in which x, y and 2 belong to separate nondegenerate representations, a naive extension of the analogy with orbital motion might suggest that the three spin functions associated with the triplet yl i, Aq ) should combine similarly to functions that transform like x, y and 2 , but this is not the case. Due to the unique nature of spin, the three spin functions Ax Ay and A have to be assigned to the irreps of Rx Ry and R respectively [2, p. 206]. 

For an Abelian group, each element is in a class by itself, since X 1AX = AX IX = A. Since rotations about the same axis commute with each other, the group e is Abelian and has n classes, each class consisting of one symmetry operation. 

In order to apply group theory to the physical properties of crystals, we need to study the transformation of tensor components under the symmetry operations of the crystal point group. These tensor components form bases for the irreducible repsensentations (IRs) of the point group, for example x, x2 x3 for 7(1) and the set of infinitesimal rotations Rx Ry Rz for 7(1 )ax. (It should be remarked that although there is no unique way of decomposing a finite rotation R( o n) into the product of three rotations about the coordinate axes, infinitesimal rotations do commute and the vector o n can be resolved uniquely... 

For other point groups this analysis of the symmetry properties of a 7 (3)ax can be repeated, or alternatively tables given by Bhagavantam (1966) or Nowick (1995) may be consulted. The Hall tensor pM (and likewise the Leduc-Righi tensor kud) is also a T(3)ax tensor but differs from the Nemst tensor in that pik is symmetric and pi1d Bt pk,i(—Bi) so that the blocks P uc,i are antisymmetric with respect to i < > k. This follows from the ORRs and is... 

No fluorescence emission was originally detected from C o [9]. Very weak fluorescence 720 nm, quantum yield (Op) 10 — 10 ) has since been reported by several groups there is disagreement about the quantum yield [29,31-34]. Weak, structured fluorescence from C70 (d>p 10 ) occurs at 77 K in a glass with A ax 682 nm and is much broader at room temperature [10,28, 35, 36]. The very low fluorescence yields of Cgo and C70 probably result both from the very short lifetime of the singlet state and from the symmetry-forbidden nature of the lowest-energy transition of Cgo-... 

Many prochiral groups, however, cannot be interchanged by any type of symmetry operation, because of the presence of one or more chiral centers in the molecule. They are diastereotopic and anisochronous, and they constitute AB, AX, etc., systems. For this reason, when encountering prochiral groups, chemists should expect them to be diastereotopic and should be, perhaps, pleasantly surprised when they are not. 

The second example we consider is pyrrole (Fig. 3-3b), a molecule which, in group-theoretical terminology, has C2v symmetry. Of the fifteen a-orbitals, nine are of type ax and six are of type b2. Of the 7t-orbitals, there are two of type bt, and there is one of type a2, giving a total of 3 for the occupied /r-orbitals. The sum-total is therefore 18. It is thus quite evident that by going from a consideration of jr-electrons only (as per Hiickel) to this more-complete treatment, we have increased the amount of work involved enormously, for the number of molecular orbitals which have to be handled has increased from three to 18. Let us specialise to consider the nitrogen atom in pyrrole and work out how many s- and p-electrons are associated with it. This turns out to beR42 R43... 

---