Accidental symmetry

Intersections are symmetry-required when the two electronic states form the components of a degenerate irreducible representation. The Jahn-Teller intersection of the two lowest electronic states in Nas which correspond to the components of an E irreducible representation of the point group Csv, provides an example of this class of conical intersection. Conical intersections which are not required by symmetry are accidental intersections. Accidental symmetry-allowed different symmetry) intersections correspond... 

The electronic state symmetry classifications described in the Introduction are contained in the properties of G and W. The intersection is symmetry required if Eqs. (6a) and (6b) are satisfied, provided only that X has the correct symmetry. The intersection is accidental symmetry-allowed, provided Eq. (6b) is satisfied when X has the correct symmetry. The intersection is accidental same-symmetry, provided symmetry does not guarantee that Eq. (6b) is satisfied. 

The Sx and Sy, Agh and Sgh describe the tilt, asymmetry and pitch of the double cone, and Sy describe the tilt of the principal axis of the cone. Agh describes the asymmetry in the pitch of the cone, which is measured by 6gh- The electronic state symmetry classification of the conical intersection is reflected in these topographical parameters. The syirmietry required double cone characteristic of the extensively studied Jahn Teller problem " has Sx = Sy = 0, and g = hhy symmetry so that q = g and Agh = 0. It is therefore a vertical (non-tilted) symmetric, Agh = 0) cone. For the accidental symmetry-allowed conical intersection only Sy... 

This is shown schematically in Figure 6.1. For such molecules, the only degeneracies that occur are accidental ones and all the i/ s have the same spatial symmetry (their irreducible representation is a). 

We believe that our conclusions can be accepted with considerable certainty, for the agreement between the predicted structure and the experimental results in regard to space-group symmetry, size of the unit of structure, and intensities of reflections on rotation photographs is so striking as to remove nearly completely from consideration the possibility of its being accidental. 

Most of the many modes of a-Ss have been assigned to their symmetry class. However, some strong infrared absorptions (V4 240 cm V5 470 cm ) and weak Raman lines (Vn 250 cm ) as well as signals originating from accidental degeneracies (e.g., Vi, V5, and Vy at around 475 cm ) were difficult to assign. 

Matsika S, Yarkony DR (2002) Accidental conical intersections of three states of the same symmetry. I. Location and relevance. J Chem Phys 117 6907... 

Optical Spectra. The optical spectra of the trans and cis porphyrazines are surprisingly different. While the trans pz (76) exhibits D2h symmetry with a split LUMO resulting in a split Q band, Ni = 605 and 722 nm, Mn = 654 and 780 nm, the cis pz (77) exhibits C2v symmetry, which also has a split LUMO, however, the Q band is observed as a single band, Ni = 660 nm, Mn = 654 nm, because the LUMO is accidentally degenerate. Compound 76 shows a split Q band because the molecular symmetry is further lowered the... 

The Hamiltonians of the previous sections describe realistic vibrational spectra of linear triatomic molecules except when accidental degeneracies (resonances, cf. Section 3.3) occur. A particularly important case is that in which the bending overtone (02°0) is nearly degenerate with the stretching fundamental (10°0) of the same symmetry Fermi, 1929, resonance). This situation occurs when the coefficient in Eq. (4.67) is nearly equal to -A (Figure 4.13). The Majorana... 

Otherwise, it would mean that there are at least two subspaces of the space of eigenfunctions fi,..., 4>i, - , 0gn, each of them closed under the symmetry operations of G. This would mean that there are no symmetry operations connecting these two subspaces, in spite of the fact that they have the same energy This, of course, seems to be unreasonable except in the case of accidental coincidence of two energy levels. 

Each ifi nucleus is shielded or screened by the electrons that surround it. Consequently each nucleus feels the influence of the main magnetic field to a different extent, depending on the efficiency with which it is screened. Each nucleus with a different chemical environment has a slightly different shielding and hence a different chemical shift in the H NMR spectrum. Conversely, the number of different signals in the iff NMR spectrum reflects the number of chemically distinct environments for iff in the molecule. Unless two iff environments are precisely identical (by symmetry) their chemical shifts must be different. When two nuclei have identical molecular environments and hence the same chemical shift, they are termed chemically equivalent or isochronous nuclei. Non-equivalent nuclei that fortuitously have chemical shifts that are so close that their signals are indistinguishable are termed accidentally equivalent nuclei. 

Furthermore, room-temperature CW EPR only picks up the isotropic g value that represents the average of the three principal g values. Two complexes with entirely different surroundings, and hence entirely different g tensors, may accidentally have the same isotropic g value. For instance, the example in Fig. 1 may stem from a complex with gx = 1.9825, gy = gz= 1.9792 or from a complex with gr =1.9825, gy= 1.9801, gz= 1.9784. The first concerns a molecule where the unpaired electron senses a local axial symmetry, the latter is due to a Cr(V) site with rhombic surroundings. From the room-temperature CW-EPR spectrum, both cases would be considered identical if no additional proton hyperfine value is resolved. The full g tensor can only be obtained from low-temperature CW-EPR, preferably at higher microwave frequencies.42,48... 

Note here that in some instances, even when the molecule has lower symmetry, the value of E can be so small as to be indistinguishable from zero, especially with a randomly oriented sample. In that case again, only four of the expected six lines may be observed. With this caution in mind, we can see that a non-zero E value may be interpreted confidently as indicative of a carrier with low symmetry, but the converse, an approximately zero value, could be due to true symmetry, or to an accidental equivalence of two axes. We return to this point in the discussion... 

Nevertheless, just that kind of accidental hyperfine equivalence may occur in the case of TME (5) (see below), which shows a nine-line pattern " suggestive of eight equivalent hydrogens. Neither planar nor twisted TME can have eight tmly equivalent hydrogens, so the hyperfine pattern is due to some cause other than molecular symmetry. 

Nuclear magnetic resonance chemical shift differences can serve as an indicator of molecular symmetry. If two groups have the same chemical shift, they are isochronous. Isochrony is a property of homotopic groups and of enantiotopic groups under achiral conditions. Diastereotopic or constitutionally heterotopic groups will have different chemical shifts (be anisochronous), except by accidental equivalence and/or lack of sufficient resolution. 

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