Points of degeneracy

In Section III.D, we shall investigate when this happens. For the moment, imagine that we are at a point of degeneracy. To find out the topology of the adiabatic PES around this point, the diabatic potential matrix elements can be expressed by a hrst order Taylor expansion. 

We consider a case where in the vicinity of a point of degeneracy between two electronic states the diabatic potentials behave linearly as a function of the coordinates in the following way [16-21]... 

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. 

Consequently, the dispersion relation is usually displayed by plotting co/q) along high-symmetry directions in the Brilliouin zone. If q) is degenerate, then the RS of eq. (27) should read q), where / labels one of the modes degenerate with loj q). However, the choice f =j is a convenient one that ensures that points of degeneracy can be treated in the same way as points where degeneracy is absent. However, eq. (28) would then be true only to within a phase factor, so that in this form eq. (28) implies that this phase factor has been chosen to be unity (Maradudin et al. (1971)). 

Along this analysis we hardly referred to points of degeneracy - also known as the points of conical intersections or simply conical intersections (Cl) [15,27]. It can be shown that line integrals as presented in equation (28) yields a unit matrix if the closed... 

The A A q term is an electric gauge potential giving rise to an inverse cubed force directed away from the point of degeneracy [29]. 

Without any symmetry, or if A = A is nondegenerate, the coupling mode g is totally symmetric (possibly trivially so). No restriction on the point of degeneracy is given by symmetry alone. This is also called accidental intersection and plays a central role, for example, in organic photochemistry (see the contribution by Robb... 

The word adiabatic is not completely justified here since the transformation proceeds through a point of degeneracy. 

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