Geometric phase effect electronic wave function

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. 

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 ... 

In the paraxial approximation, the aperture function A is simply a mathematical device defining the area of integration in the aperture plane A = 1 inside the pupil and A = 0 outside the pupil. If we wish to include the effect of geometric aberrations, however, we can represent them as a phase shift of the electron wave function at the exit pupil. Thus, if the lens suffers from spherical aberration, we write... 

The geometric phase effect is concerned with the X dependence of the adiabatic electronic wave function. Consider the result of transporting a j(x X) aroimd a small circle. The expected result is that should return to itself, that is, shoiffd be single-valued. That is indeed the result when the circle does not contain a conical intersection. However, consider the same process applied to the eigenfunctions of W, Eqs. (5b) and (5c), with 2A = A. Then since tan A = h/g)tdXi9 with h,g > 0 as 9 increases by 27t so does A. Therefore cos((A + 2n)/2) = cos(A/2 + tt) = — cos(A/2) and sin((A + 2n)f2) = sin(A/2 + tt) = — sin(A/2), so that... 

II electronic states, 638-640 vibronic coupling, 628-631 triatomic molecules, 594-598 Hamiltonian equations, 612-615 pragmatic models, 620-621 Kramers doublets, geometric phase theory linear Jahn-Teller effect, 20-22 spin-orbit coupling, 20-22 Kramers-Kronig reciprocity, wave function analycity, 201 -205 Kramers theorem ... 

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