Time development of geometric phases in the Longuet-Higgins model

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Abstract
Using a time-dependent variational method, we study the evolution of nonstationary states in Longuet-Higgin’s model of a Jahn–Teller molecule. Conditions are found for the nuclear motion to be adiabatic. The effects of wave-packet spreading are neglected upon specializing to the case of nearly harmonic motion. It is shown explicitly how the effective vector potential introduced by Mead and Truhlar gives rise to an electronic Berry phase. In a semiclassical approximation sufficient to produce the electronic adiabatic phase anticipated from the result for a given sequence of nuclear configurations, it is demonstrated that the effective vector potential has a negligible effect on the nuclear motion; the effective vector potential, the source of an effective field proportional to ℏ, is seen to affect the nuclear trajectory only in higher order. For the special case of periodic nuclear motion the electronic adiabatic phase is seen as a contribution to the geometric phase attending an arbitrary cyclic evolution. It is demonstrated that a molecular state prepared with identically pseudorotating nuclear wave packets in both electronic levels corresponds, in a weak coupling limit, to a spin 1/2 in a conically varying external field. Geometrical phase differences are shown to make discernible contributions to the frequencies of oscillation of the electronic charge and current densities, which may serve as classical sources for superradiant emission. Our results are shown to be gauge invariant.