Conical intersections Longuet-Higgins loops

Figure 31 shows the proposed Longuet-Higgins loop for the cyclopentadienyl cation. It uses the type-VI Ai anchors, with the type-VII B structures as transition states between them. This situation is completely analogous to that of the radical (Fig. 23). Since the loop is phase inverting, a conical intersection should be located at its center—as required by the Jahn-Teller theorem.

In this section, the systematic search for conical intersections based on the Longuet-Higgins phase-change rule is described. For conciseness sake, we limit the present discussion to Hiickel-type systems only, unless specifically noted otherwise. The first step in the antilysis is the determination of the LH loops containing a conical intersection for the reaction of interest. 

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. 

Reciprocally, the Longuet-Higgins theorem states that if an adiabatic electronic wave function changes sign upon circulation around a closed loop in nuclear configuration space, somewhere on or inside that loop there must exist a conical intersection. Such a theorem has been used to localize at the ab initio level a conical intersection in LiNaK, and more recently in other systems such as... 

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