Phase-change rule and

C, Anchors, Molecules and Independent Quantum Species II. The Phase-Change Rule and the Construction of Loops... 

We can now proceed to discuss the phase-change rule and its use to locate conical intersections. 

In this section, we apply the phase-change rule and the loop method to some representative photochemical systems. The discussion is illustiative, no comprehensive coverage is intended. It is hoped that the examples are sufficient to help others in applying the method to other systems. This section is divided into two parts in the first, loops are constructed and a qualitative discussion of the photochemical consequences is presented. In the second, the method is used for an in-depth, quantitative analysis of one system—photolysis of 1,4-cyclohexadiene. 

The phase-change rule, also known as the Berry 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 transported in a closed loop around a point of conical intersection. Any one of them may be used in the search for degeneracies. 

To apply the phase-change rule discussed in the previous section, we choose the reactant A with wave function A> and the two chemically equivalent products B and C with wave functions B> and C>, which differ by a rotation of the methylene group through 180, as anchors as indicated in Figure 6.7. The transition from B to C involves a HOMO-LUMO crossing and hence is phase-inverting, whereas the transitions from A to B or C are equivalent the total number of phase changes on the closed-loop A-B-C-A is thus odd, and a conical intersection must be contained within the loop. The... 

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