  Choosing an appropriate representation of the molecular Hamiltonian is one of
the challenges faced by simulations of the nonadiabatic quantum dynamics around
a conical intersection. The adiabatic, exact quasidiabatic, and strictly
diabatic representations are exact and unitary transforms of each other,
whereas the approximate quasidiabatic Hamiltonian ignores the residual
nonadiabatic couplings in the exact quasidiabatic Hamiltonian. A rigorous
numerical comparison of the four different representations is difficult because
of the exceptional nature of systems where the four representations can be
defined exactly and the necessity of an exceedingly accurate numerical
algorithm that avoids mixing numerical errors with errors due to the different
forms of the Hamiltonian. Using the quadratic Jahn-Teller model and high-order
geometric integrators, we are able to perform this comparison and find that
only the rarely employed exact quasidiabatic Hamiltonian yields nearly
identical results to the benchmark results of the strictly diabatic
Hamiltonian, which is not available in general. In this Jahn-Teller model and
with the same Fourier grid, the commonly employed approximate quasidiabatic
Hamiltonian led to inaccurate wavepacket dynamics, while the Hamiltonian in the
adiabatic basis was the least accurate, due to the singular nonadiabatic
couplings at the conical intersection.
