  The choice of the electronic representation in on-the-fly quantum dynamics is
crucial. The adiabatic representation is appealing since adiabatic states are
readily available from quantum chemistry packages. The nuclear wavepackets are
then expanded in a basis of Gaussian functions, which follow trajectories to
explore the potential energy surfaces and approximate the potential using a
local expansion of the adiabatic quantities. Nevertheless, the adiabatic
representation is plagued with severe limitations when conical intersections
are involved: the diagonal Born-Oppenheimer corrections (DBOCs) are
non-integrable, and the geometric phase effect on the nuclear wavepackets
cannot be accounted for unless a model is available. To circumvent these
difficulties, the moving crude adiabatic (MCA) representation was proposed and
successfully tested in low energy dynamics where the wavepacket skirts the
conical intersection. We assess the MCA representation in the case of
non-adiabatic transitions through conical intersections. First, we show that
using a Gaussian basis in the adiabatic representation indeed exhibits the
aforementioned difficulties with a special emphasis on the possibility to
regularize the DBOC terms. Then, we show that MCA is indeed able to properly
model non-adiabatic transitions. Tests are done on linear vibronic coupling
models for the bis(methylene) adamantyl cation and the butatriene cation.
