  Ab initio molecular dynamics in a magnetic field requires solving equations
of motion with velocity-dependent forces -- namely, the Lorentz force arising
from the nuclear charges moving in a magnetic field and the Berry force arising
from the shielding of these charges from the magnetic field by the surrounding
electrons. In this work, we revisit two existing propagators for these
equations of motion, the auxiliary-coordinates-and-momenta (ACM) propagator and
the Tajima propagator (TAJ), and compare them with a new exponential (EXP)
propagator based on the Magnus expansion. Additionally, we explore limits (for
example, the zero-shielding limit), the implementation of higher-order
integration schemes, and series truncation to reduce computational cost by
carrying out simulations of a HeH$^+$ model system for a wide range of field
strengths. While being as efficient as the TAJ propagator, the EXP propagator
is the only propagator that converges to both the schemes of Spreiter and
Walter (derived for systems without shielding of the Lorentz force) and to the
exact cyclotronic motion of a charged particle. Since it also performs best in
our model simulations, we conclude that the EXP propagator is the recommended
propagator for molecules in magnetic fields.
