Accidental degeneration

Figures 8 and 9 shows a part of the bending region at low temperature containing the components of Vg (150-160 cm ) and Vs (190-200 cm ). The Vg vibration, IR active in the free molecule, has weak components in the Raman spectrum. According to theoretically calculated Raman intensities, which almost perfectly fit the experimental spectrum, the big component has a very low scattering cross-section [87] and is accidentally degenerate with the b2g component at ca. 188 cm. The IR active components of Vg cause strong absorptions in the IR spectrum even if the crystalline sample used for transmission studies is as thin as 400 pm [107, 109].

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... 

From the AOM point of view, there are two rather puzzling features in these observations. The first is the fact that the 2B2g and 2Eg states are not accidentally degenerate, which would be predicted if the equatorial ligands are assumed to have no 7r-overlap capability. In CuN4X2 chromophores, the distant axial ligands might have... 

Thus, in this approximation, the 4" orbital is accidentally degenerate with the E" orbitals and the A orbital is accidentally degenerate with the E orbitals. 

F ERMI RESONANCE. In polyatomic molecules. Hvo vibrational levels belonging to different vibrations lor combinations of vibrations) may happen lo have nearly die same energy, and therefore be accidentally degenerate. As was recognized hy Fermi in the case of CO such a "resonance" leads to a perturbation of the energy levels that is very similar to the vibrational perturbations of diatomic molecules. 

The (n - 71 )1 3 excited states of the pyrazine molecule are a well-known case of wave function symmetry breaking [27,55]. An accidental degeneracy arises when one considers the valence electronic excitations within the equivalent nitrogen lone pairs. The pairs are in opposite and equivalent positions and, when there are two singly occupied orbitals in different symmetries, we will have a pair of accidentally degenerate states as shown below ... 

The total symmetry of the molecule is D2h, so we cannot have point group degenerate states. The accidentally degenerate states represented above have C2v symmetry. The direct product C2v Cs recovers properly the full symmetry of the system. Depending on the particular excitation and final spin state four states can be generated. In Table 4 we compare the results of Hartree-Fock (HF), configuration interaction with singles and doubles excitations (CI-SD) and the two structure GMS calculations, with the available experimental data. 

In Equation 12, Eo(F5) - 0, and E3(ry) E2(F0) since the doubly-degenerate Fy state is accidentally degenerate with the second F0 state (through first order in spin-orbit coupling). The gj are the degeneracy factors (2 or 4), and p E (kpT)"l. 

SO2F2 predict a value near 300 cm (6, 7). It appears that the twist(FSCl) and S(FSCl) are nearly accidentally degenerate, so we assign them arbitrarily to the Raman band at 308 cm" and the infrared band at 300 cm. The adopted assignments are those of Toyuki and Shimizu (6) and Pfeiffer (7). All adopted values lie between the analogous modes for SO Cl and S0 F. 6(FSC1) = 195... 

Figure 2.2 Morse potential energy curves for the neutral and negative-ion states of anthracene. The vertical electron affinity VEa, adiabatic electron affinity AEa, and activation energy for thermal electron attachment E are shown. The two Ea are 0.68 eV and 0.53 eV observed in ECD data. There will be nine other negative ion curves, yielding a total of thirteen anion curves, four each for the different C—H bonds and a polarization curve. Some of these will be accidentally degenerate.

Of the monocyclic alkanes, cyclopropane has been studied by INS [16]. The work was able to resolve an uncertainty in the assignment by showing that two modes were accidentally degenerate. 

The two methyl torsions of the shorter alkane chains couple to produce an in-phase and out-of-phase pair of bands. The longer the chain the weaker this coupling and so the less the frequency splitting of the pair. In the limit, the splitting should completely collapse. The methyl torsion of one end of a chain would then be unaware of the nature, or even the presence, of the other end of the chain and both methyl torsions oscillate independently. Only one accidentally degenerate band is observed. In the alkanes this collapse has already occurred in undecane. Naively then, all methyl torsions, n > 11, will occur at the same frequency. Unfortunately this is not observed. 

Wc shall deal also with the ease in which such relations exist only for certain values of the, ik the mechanical system is then non-dcgcncratc the particular motions in question, for which (ti>)=0, we shall call accidentally degenerate, whilst the motions of a degenerato system [( tv)=0 identically] are spokon of as intrinsically degenerate. 

In the rase of accidental degeneration, tlie manlier of independent frequencies is less for certain motions than for the number of the whole system. We call the former number the degree of periodicity of the motion under consideration. 

The proof of the adiabatic invariance of the J s given here in not altogether free from objection on account of difficulties due to the appearance of accidentally degenerate motions in the course of the adiabatic change. A strict proof has been given by M. v. Laue, Ann. d. Phynik, vol. lxxi, p. 619,1925. 

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