  We benchmark a set of quantum-chemistry methods, including multitrajectory
Ehrenfest, fewest-switches surface-hopping, and multiconfigurational-Ehrenfest
dynamics, against exact quantum-many-body techniques by studying real-time
dynamics in the Holstein model. This is a paradigmatic model in condensed
matter theory incorporating a local coupling of electrons to Einstein phonons.
For the two-site and three-site Holstein model, we discuss the exact and
quantum-chemistry methods in terms of the Born-Huang formalism, covering
different initial states, which either start on a single Born-Oppenheimer
surface, or with the electron localized to a single site. For extended systems
with up to 51 sites, we address both the physics of single Holstein polarons
and the dynamics of charge-density waves at finite electron densities. For
these extended systems, we compare the quantum-chemistry methods to exact
dynamics obtained from time-dependent density matrix renormalization group
calculations with local basis optimization (DMRG-LBO). We observe that the
multitrajectory Ehrenfest method, in general, only captures the ultrashort time
dynamics accurately. In contrast, the surface-hopping method with suitable
corrections provides a much better description of the long-time behavior but
struggles with the short-time description of coherences between different
Born-Oppenheimer states. We show that the multiconfigurational Ehrenfest method
yields a significant improvement over the multitrajectory Ehrenfest method and
can be converged to the exact results in small systems with moderate
computational efforts. We further observe that for extended systems, this
convergence is slower with respect to the number of configurations. Our
benchmark study demonstrates that DMRG-LBO is a useful tool for assessing the
quality of the quantum-chemistry methods.
