Noncyclic geometric phase, coherent states, and the time-dependent variational principle: Application to coupled electron-nuclear dynamics

Kinematical and dynamical aspects of the noncyclic geometric phase for coherent states are analyzed. It is shown for Glauber and SU(2) coherent states that the geometric phase obeys a geodesic closure rule associated with the coset space of the respective Lie group. In the former case the geodesic is a straight line in phase space, while in the latter the geodesic is a great circle on the Bloch sphere $\left({\mathcal{S}}^2\right).\mathrm{}$ An alternative expression for the geometric phase is derived from the time-dependent variational principle by making a specific choice of gauge. The noncyclic geometric phase is numerically calculated in a time-dependent variational treatment of the $E\otimes \varepsilon$ Jahn-Teller model first introduced by Longuet-Higgins and co-workers. The electronic and nuclear degrees of freedom are parametrized by SU(2) and Glauber coherent states respectively. In particular, the adiabatic limit is studied and shown to yield the anticipated Berry geometric phase.