Conical intersections degeneracy

Nonlinearities That Lead to Multiple Degeneracies A. Conical Intersection Pairs... 

Figure 5, Sketch of a conical intersection. The vectors x and X2 are the GD and DC respectively, that lift the degeneracy of the two adiabatic surfaces, The plane containing these vectors is known as the branching space.

Figure 6. Two-dimensional (top) and 3D (bottom) representations of a peaked (a) and sloped (b) conical intersection topology. There are two directions that lift the degeneracy the GD and the DC. The top figures have energy plotted against the DC while the bottom figures represent the energy plotted in the space of hoth the GD and DC vectors. At a peaked intersection, as shown at the bottom of (a), the probability of recrossing the conical intersection should be small whereas in the case of a sloped intersection [bottom of ( )l, this possibility should be high. [Reproduced from [84] courtesy of Elsevier Publishers.]...

Accepting the Longuet-Higgins rule as the basis for the search of conical intersection, it is necessary to look for the appropriate loop. The -type degeneracy of a Jahn-Teller system is removed by a nonsymmetric motion. 

As shown in Figure 27, an in-phase combination of type-V structures leads to another A] symmetry structures (type-VI), which is expected to be stabilized by allyl cation-type resonance. However, calculation shows that the two shuctures are isoenergetic. The electronic wave function preserves its phase when tr ansported through a complete loop around the degeneracy shown in Figure 25, so that no conical intersection (or an even number of conical intersections) should be enclosed in it. This is obviously in contrast with the Jahn-Teller theorem, that predicts splitting into A and states. 

A final comment concerns the presence of other conical intersections near the central one. They are enclosed by loops consisting of two Ai (type-VI) and one A[ (type-V) species, as depicted in Figure 32. This is a phase-inverting ip loop. Thus, the main Jahn-Teller degeneracy is sunounded by five further degeneracies, arranged in a symmetrical fashion. 

The phase-change nale, also known as the Ben phase [101], the geometric phase effect [102,103] or the molecular Aharonov-Bohm effect [104-106], was used by several authors to verify that two near-by surfaces actually cross, and are not repelled apart. This point is of particular relevance for states of the same symmetry. The total electronic wave function and the total nuclear wave function of both the upper and the lower states change their phases upon being bansported in a closed loop around a point of conical intersection. Any one of them may be used in the search for degeneracies. 

We are now in a position to explain the results of Table I. As a consequence of the degeneracy of , at a conical intersection there are four degenerate functions tl/f, tb and Ttbf = Ttb = tb j. By using Eq. (Ic), an otherwise arbitrary Flermitian matrix in this four function time-reversal adapted basis has the form... 

R % (F1 —H )), also 70 cm , provide prima facie evidence for a conical intersection of Flowever, since numerical degeneracies are never exact, an... 

Let us summarize briefly at this stage. We have seen that the point of degeneracy forms an extended hyperline which we have illnstrated in detail for a four electrons in four Is orbitals model. The geometries that lie on the hyperline are predictable for the 4 orbital 4 electron case using the VB bond energy (Eq. 9.1) and the London formula (Eq. 9.2). This concept can be nsed to provide nseful qualitative information in other problems. Thns we were able to rationalize the conical intersection geometry for a [2+2] photochemical cycloaddition and the di-Jt-methane rearrangement. 

Figure 9.10. The conical intersection hyperline traced out by a coordinate X3 plotted in a space containing the coordinate X3 and one coordinate from the degeneracy-lifting space X1X2. See color insert.

Figure 9.11. Two vectors Xi and X2 that lift the degeneracy of the conical intersection seam at point (a) in Figure 9.6.

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