Experimental and calculational consequences of phases in molecules with multiple conical intersections

We set up a theoretical model for treating several degeneracies between two adjacent potential energy surfaces in molecular systems, e.g., cases of four (or more) conically intersecting degeneracies, located in a place formed by two-fold molecular displacement coordinates, in trigonal (or cubic) symmetry and twin conical intersections (CIs) for molecules with two-fold symmetries. When the system circles (in a time-variant manner) entirely inside or entirely outside these CIs, it picks up phases (the geometric phases) that are zero or 2Nπ. Here, N is (to a good approximation) an integer whose value depends on the Hamiltonian for the system, on the distance of circling from the CI, and on the speed of circling. For the adiabatic (slow circling) limit, we develop a simple algebraic-graphical method that yields the value of N in each case. Even in this limit, the value of N is model dependent and is not uniquely given by the number of CIs that are circled. Further, we suggest an experiment (based on a pump-probe method) for tracing the time development of a system that is subject to a periodically varying field carrying it around the CI. The experiment leads to relations between the spectroscopic transition strengths and state amplitudes, such that the (open-path) phases are fully given (including the additive 2Nπ term and excluding the dynamic phase). Finally, we compare geometric phases that differ by integral multiples of 2π and, therefore, cannot be distinguished by phase interference. We show that the different cases have, nevertheless, observational correlates in the number of times that zero or unity occurs in a wave function component amplitude during circling. The rules of correspondence are given.