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We benchmark a set of quantum-chemistry methods, including multitrajectoryEhrenfest, fewest-switches surface-hopping, and multiconfigurational-Ehrenfestdynamics, against exact quantum-many-body techniques by studying real-timedynamics in the Holstein model. This is a paradigmatic model in condensedmatter theory incorporating a local coupling of electrons to Einstein phonons.For the two-site and three-site Holstein model, we discuss the exact andquantum-chemistry methods in terms of the Born-Huang formalism, coveringdifferent initial states, which either start on a single Born-Oppenheimersurface, or with the electron localized to a single site. For extended systemswith up to 51 sites, we address both the physics of single Holstein polaronsand the dynamics of charge-density waves at finite electron densities. Forthese extended systems, we compare the quantum-chemistry methods to exactdynamics obtained from time-dependent density matrix renormalization groupcalculations with local basis optimization (DMRG-LBO). We observe that themultitrajectory Ehrenfest method, in general, only captures the ultrashort timedynamics accurately. In contrast, the surface-hopping method with suitablecorrections provides a much better description of the long-time behavior butstruggles with the short-time description of coherences between differentBorn-Oppenheimer states. We show that the multiconfigurational Ehrenfest methodyields a significant improvement over the multitrajectory Ehrenfest method andcan be converged to the exact results in small systems with moderatecomputational efforts. We further observe that for extended systems, thisconvergence is slower with respect to the number of configurations. Ourbenchmark study demonstrates that DMRG-LBO is a useful tool for assessing thequality of the quantum-chemistry methods.

Real-time non-adiabatic dynamics in the one-dimensional Holstein model: Trajectory-based vs exact methods