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Ehrenfest dynamics is a useful approximation for ab initio mixedquantum-classical molecular dynamics that can treat electronically nonadiabaticeffects. Although a severe approximation to the exact solution of the moleculartime-dependent Schr\"odinger equation, Ehrenfest dynamics is symplectic,time-reversible, and conserves exactly the total molecular energy as well asthe norm of the electronic wavefunction. Here, we surpass apparentcomplications due to the coupling of classical nuclear and quantum electronicmotions and present efficient geometric integrators for "representation-free"Ehrenfest dynamics, which do not rely on a diabatic or adiabatic representationof electronic states and are of arbitrary even orders of accuracy in the timestep. These numerical integrators, obtained by symmetrically composing thesecond-order splitting method and exactly solving the kinetic and potentialpropagation steps, are norm-conserving, symplectic, and time-reversibleregardless of the time step used. Using a nonadiabatic simulation in the regionof a conical intersection as an example, we demonstrate that these integratorspreserve the geometric properties exactly and, if highly accurate solutions aredesired, can be even more efficient than the most popular non-geometricintegrators.

High-order geometric integrators for representation-free Ehrenfest dynamics