Geometrical phase effect in Jahn–Teller systems: Twofold electronic degeneracies and beyond

The question of how many states are subject to geometrical phase in a N -fold electronic degeneracy is examined for Jahn–Teller systems. It is shown that it can be diagnosed from the sign change of the adiabatic electronic vector states when a rotation operator is applied to them in functional space defined in N -dimensional spherical polar coordinates, rather than by adiabatic transportation along a closed path in the higher-dimensional configuration space. It is also demonstrated that the Longuet-Higgins theorem for twofold conical intersections (namely that the electronic wave function changes sign on completing a closed path around a doubly-degenerate intersection) can be used to rationalize the results, irrespectively of N .