Symmetry rising

We have described here one particular type of molecular synnnetry, rotational symmetry. On one hand, this example is complicated because the appropriate symmetry group, K (spatial), has infinitely many elements. On the other hand, it is simple because each irreducible representation of K (spatial) corresponds to a particular value of the quantum number F which is associated with a physically observable quantity, the angular momentum. Below we describe other types of molecular synnnetry, some of which give rise to finite synnnetry groups. 

Similar to the fullerene ground state the singlet and triplet excited state properties of the carbon network are best discussed with respect to the tliree-dimensional symmetry. SurjDrisingly, the singlet excited state gives rise to a low emission fluorescence quantum yield of 1.0 x 10 [143]. Despite the highly constrained carbon network,... 

If, instead of a configuration like that treated above, one had a 52 configuration, the above analysis would yield F, and symmetries (because the two 5 orbitals m values could be combined as 2 + 2, 2 - 2, -2 + 2, and -2 -2) the wavefunctions would be identical to those given above with the 7ii orbitals replaced by 82 orbitals and 71.1 replaced by 5.2. Likewise, dp- gives rise to H, and symmetries. 

This redueible representation (the oeeupaney of two e orbitals in the anion gives rise to more than one state, so the direet produet e x e eontains more than one symmetry eomponent) ean be deeomposed into pure symmetry eomponents (labels F are used to denote the irredueible symmetries) by using the deeomposition formula given in Appendix... 

In the alkaline earth atom ease, the polarized orbital pairs are formed by mixing the ns and np orbitals (aetually, one must mix in equal amounts of pi, p.i, and po orbitals to preserve overall S symmetry in this ease), and give rise to angular eorrelation of the eleetron pair. Use of an (n+l)s2 CSF for the alkaline earth ealeulation would eontribute in-out or radial eorrelation beeause, in this ease, the polarized orbital pair formed from the ns and (n+l)s orbitals would be radially polarized. 

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... 

Sensitivity to process gas inlet temperature. Figure 7-13b shows TTE variation with changes in process gas inlet temperature. The sensitivity of variable speed machines to temperature variation is less than constant speed machines. The pattern and symmetry around the design point, however, are the same for constant speed machines. For example, note that a 3% decrease in the inlet temperature causes TTE to drop to 98% and that the same percentage drop in TTE occurs when gas inlet temperature rises by 3%. 

Since the optical transitions near the HOMO-LUMO gap are symmetry-forbidden for electric dipole transitions, and their absorption strengths are consequently very low, study of the absorption edge in Ceo is difficult from both an experimental and theoretical standpoint. To add to this difficulty, Ceo is strongly photosensitive, so that unless measurements arc made under low light intensities, photo-induced chemical reactions take place, in some cases giving rise to irreversible structural changes and polymerization of the... 

Compounds in which one or more carbon atoms have four nonidentical substituents are the largest class of chiral molecules. Carbon atoms with four nonidentical ligands are referred to as asymmetric carbon atoms because the molecular environment at such a carbon atom possesses no element of symmetry. Asymmetric carbons are a specific example of a stereogenic center. A stereogenic center is any structural feature that gives rise to chirality in a molecule. 2-Butanol is an example of a chiral molecule and exists as two nonsuperimposable mirror images. Carbon-2 is a stereogenic center. 

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