In the early 1950’s Purcell and Ramsey hypothesized that the electron might have a CP-violating electric dipole moment proportional to its spin.  This hypothesis set in motion a hunt for the electron’s electric dipole moment (e-EDM) that is still ongoing.   In a prototypical experiment, one creates an atom or molecule in a coherent superposition of two states that differ only by the sign of the projection of total angular momentum on the axis of an electric field.   One then determines if this phase evolves due to an energy difference -2p­_EDM E between the two states.  In such analysis, one must consider carefully the possibility that a geometric effect, and not an energy difference, causes this phase to evolve.  Such a geometric phase is the result of the evolution of the quantization axis in a non-uniform electric field and is analogous to the Renner-Teller effect.  Here we adapt the formalism of Longuet-Higgens to derive a general Hamiltonian to that describes the geometric phase for the case of a molecule evolving adiabatically in a non-uniform electric field.  We then apply this formalism first to the determination of the coherence time of molecules in a Stark trap and second to the construction of a phase rotator to be used in a beam-resonance experiment.