The Geometric Phase Effect

Kuppermann A and Wu Y S M 1993 The geometric phase effect shows up in chemical reactions Chem. 

Figure 1. Quasiclassical cross-sections for the reaction D -I- H2 (w — 1,2 — 1) DH (v — 1, /) -f H at 1.8-eV total energy as a function of/. The solid line indicates results obtained without including the geometric phase effect. Boxes show the results with the geometric phase included using either 9o = 0 (dashed) or 9o = 11.5 " (dotted).

As demonstrated in [53] it is convenient to incorporate the geometrical phase effect by adding the vector potential in hyperspherical coordinates. Thus we found that the vector potential gave three terms, the first of which was zero, the second is just a potential term... 

In order to incorporate the geometric phase effect in a formulation based on an expansion in G-H basis functions we need to consider the operation of the momentum operator on a basis function, that is, to evaluate terms as... 

Since the geometric phase effect is related to the angle 4> we express as... 

Note that in this TDGH-DVR formulation of quantum dynamics, the inclusion of the geometric phase effects through the addition of a vector potential is veiy simple and the calculations can be carried out with about the same effort as what is involved in the ordinary scattering case. 

Figure 3 shows the results with and without including the geometric phase effect for the D + H2 reaction. The basis set is taken as 1,1,1,15,15,15, that is. 

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. 

Han S, Yarkony DR (2003) Conical intersections of three states. Energies, derivative couplings, and the geometric phase effect in the neighborhood of degeneracy subspaces. Application to the allyl radical. J Chem Phys 119 11562... 

Thus, the matrix elements are those that should be added in order to incorporate the geometric phase effect. 

Another subtle consequence of conical intersections is the geometric phase effect [42,51], which occurs even when the dynamics is confined to low energies avoiding the neighbourhood of the CL It is the result of transporting the electronic wave function on a closed loop around the CL This leads to a sign change in the electronic wave function when it returns to its initial position... 

Mead and Truhlar [52] introduced an elegant way of incorporating the geometric phase effect, namely the vector potential approach. In this method, the real electronic wave function 4>(a), where a is any internal angular coordinate describing the motion around the Cl, is multiplied by a complex phase factor c(a) to ensure the single-valuedness of the new complex electronic wave function ... 

Topological Interpretation of the Geometric Phase Effect in the Dynamics of H3... 

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