Degeneracy present

Calculations of such spectra have been reported assuming the molecule to be planar with C2v symmetry [36]. The lower symmetry of corrole removes the degeneracy present in porphyrins so that degenerate transitions are no longer expected and more transitions are allowed in the case of corrole than in that of porphyrin. The la2 and 8fc, levels in corrole correspond to the two components of the lowest vacant MO in porphyrins, the 4eg pair (LUMO). Similarly the 6a2 and 7 >1 orbitals correspond to the highest filled porphyrin orbital, alu and a2u (HOMO). In the case of porphyrins the Q and Soret (thus called because it was discovered in 1883 by Soret in the optical spectrum of hemoglobin) bands arise mainly from the alu, a2u and eg pair and the spectra of corroles closely resemble those of porphyrins (Table 6). 

Figure 2 shows the thermal equilibrium spin population for a three-spin system. Generally there is no degeneracy present. When an appropriate polarizing agent is used, the energy levels IIV > and IV > or IVI > and IIII > become degenerated (Fig. 2a). Irradiation of EPR transition and CE transitions leads to positive (Fig. 2b) or negative enhancement (Fig. 2c) of the nuclear polarization.

In addition to the change of coordinates, lowering the symmetry to C3V lifts the threefold degeneracy present in Oh symmetry (Fig. 8c). Under the C3V symmetry, the absence of a horizontal mirror plane (orthogonal to z-axis in C3v) allows us to split the six hybrid ( sp ) o-acceptor orbitals into two orbital clusters, each... 

For these reasons, in the MCSCF method the number of CSFs is usually kept to a small to moderate number (e.g. a few to several thousand) chosen to describe essential correlations (i.e. configuration crossings, near degeneracies, proper dissociation, etc, all of which are often tenned non-dynamicaI correlations) and important dynamical correlations (those electron-pair correlations of angular, radial, left-right, etc nature that are important when low-lying virtual orbitals are present). 

Magnetic circular dicliroism (MCD) is independent of, and thus complementary to, the natural CD associated with chirality of nuclear stmcture or solvation. Closely related to the Zeeman effect, MCD is most often associated with orbital and spin degeneracies in cliromophores. Chemical applications are thus typically found in systems where a chromophore of high symmetry is present metal complexes, poriihyrins and other aromatics, and haem proteins are... 

Introduction of the vibronic coupling (s / 0) causes removal of the above degeneracy and leads to the general vibronic-spin-orbit pattern presented in the central part of Figure 3. Each vibronic level is characterized by a particular K... 

Figure 3. Molecular-orbital diagrams as obtained by the ROHF method. Dashed lines indicate MOs dominated by the metal d-orbitals, the solid lines stand for doubly occupied or virtual ligand orbitals. Orbitals which are close in energy are presented as degenerate the average deviation from degeneracy is approximately 0.01 a.u. In the case of a septet state (S=3), the singly occupied open-shell orbitals come from a separate Fock operator and their orbital energies do not relate to ionization potentials as do the doubly occupied MOs (i.e. Koopmann s approximation). For these reasons, the open-shell orbitals appear well below the doubly occupied metal orbitals. Doubly occupying these gives rise to excited states, see text.

In Eigure 9.9, we have presented another cartoon that attempts to show the potential energy surfaces in the space of the degeneracy lifting coordinates as one traverses a third coordinate X3. Of conrse, this is really a fonr-dimensional pictnre, which is not easy to assimilate. Nevertheless, it shonld be clear that the degeneracy... 

As stated in the introduction, we present the derivation of an extended BO approximate equation for a Hilbert space of arbitary dimensions, for a situation where all the surfaces including the ground-state surface, have a degeneracy along a single line (e.g., a conical intersection) with the excited states. In a two-state problem, this kind of derivation can be done with an arbitary t matrix. On the contrary, such derivation for an N > 2 dimensional case has been performed with some limits to the elements of the r matrix. Hence, in this sence the present derivation is not general but hoped that with some additional assumptions it will be applicable for more general cases. 

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